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Detection of quantum entanglement
in physical systems
Carolina Moura Alves
Merton College
University of Oxford
A thesis submitted for the degree of
Doctor of Philosophy
Trinity 2005
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Abstract
Quantum entanglement is a fundamental concept both in quantum mechanics and in
quantum information science. It encapsulates the shift in paradigm, for the descrip-
tion of the physical reality, brought by quantum physics. It has therefore been a key
element in the debates surrounding the foundations of quantum theory. Entangle-
ment is also a physical resource of great practical importance, instrumental in the
computational advantages offered by quantum information processors. However, the
properties of entanglement are still to be completely understood. In particular, the
development of methods to efficiently identify entangled states, both theoretically
and experimentally, has proved to be very challenging. This dissertation addresses
this topic by investigating the detection of entanglement in physical systems.
Multipartite interferometry is used as a tool to directly estimate nonlinear properties
of quantum states. A quantum network where a qubit undergoes single-particle
interferometry and acts as a control on a swap operation between k copies of the
quantum state is presented. This network is then extended to a more general
quantum information scenario, known as LOCC. This scenario considers two distant
parties A and B that share several copies of a given bipartite quantum state.
The construction of entanglement criteria based on nonlinear properties of quantumstates is investigated. A method to implement these criteria in a simple, experimen-
tally feasible way is presented. The method is based of particle statistics effects
and its extension to the detection of multipartite entanglement is analyzed. Finally,
the experimental realization of the nonlinear entanglement test in photonic systems
is investigated. The realistic experimental scenario where the source of entangled
photons is imperfect is analyzed.
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Acknowledgements
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Contents
Abstract i
Acknowledgements ii
1 Introduction 1
1.1 Entanglement as a property of quantum systems . . . . . . . . . . . . . . . . . . 1
1.2 Entanglement as a physical resource . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Detection and characterization of entanglement . . . . . . . . . . . . . . . . . . . 2
1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Basic concepts 52.1 State Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Mathematical properties of density operators . . . . . . . . . . . . . . . . 8
2.2.2 Ensemble interpretation of density operators . . . . . . . . . . . . . . . . 9
2.3 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Superoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 Mathematical properties of superoperators . . . . . . . . . . . . . . . . . 10
2.4.2 Jamiolkowski isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Mathematical characterization of bipartite entanglement . . . . . . . . . . . . . . 11
2.5.1 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Experimental detection of entanglement . . . . . . . . . . . . . . . . . . . . . . . 13
2.6.1 Bells inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6.2 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Multipartite entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7.1 Maximally entangled state . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7.2 W State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7.3 Cluster state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Quantum networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8.1 Universal set of gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8.2 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
iii
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CONTENTS iv
3 Direct estimation of density operators 21
3.1 Modified interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Multiple target states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Spectrum estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Quantum communication . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Extremal eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.4 State estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.5 Arbitrary observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Quantum channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Direct estimation of density operators using LOCC 28
4.1 LOCC estimation of nonlinear functionals . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Structural Physical Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 SPA using only LOCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Entanglement detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Channel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Entanglement Detection in Bosons 33
5.1 Nonlinear entanglement inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Estimation of the purities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2.1 Bipartite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2.2 Multipartite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Realization of the entanglement detection network . . . . . . . . . . . . . . . . . 36
5.4 Detection of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5 Degree of macroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5.1 Determination of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Entropic inequalities 41
6.1 Entropic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.1.1 Graphical comparison between Bell-CHSH and entropic inequalities . . . 42
6.2 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2.1 Realistic sources of entangled photons . . . . . . . . . . . . . . . . . . . . 46
7 Conclusion 49
Bibliography 51
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List of Figures
2.1 The controlled-Ugate. The top line represents the control qubit and the bottom
line represents the target qubit. U acts on the target qubit iff the control qubit
is in the logical state|1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 The Mach-Zender interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 The quantum network corresponding to the Mach-Zender interferometer ( =
1 0). The visibility of the interference pattern associated withp0 varies as afunction of according to Eq.(2.70). . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 A modified Mach-Zender interferometer with coupling to an ancilla by a controlled-
Ugate. The interference pattern is modified by the factor vei = Tr [U ]. . . . 22
3.2 Quantum network for direct estimations of both linear and non-linear functionsof a quantum state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 A quantum channel acting on one of the subsystems of a bipartite maxi-
mally entangled state of the form|+ =
k |k|k/
d. The output state
= 1d
kl |kl| (|kl|), contains a complete information about the channel. 26
4.1 Network for remote estimation of non-linear functionals of bipartite density op-
erators. Since Tr[V(k)k] is real, Alice and Bob can omit their respective phaseshifters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1 Network of BS acting on pairs of identical bosons. The two rows of N atoms,
labelled I and IIrespectively, are identical, and the state of each of the rows is
123...N. The total state of the system is 123...N 123...N. . . . . . . . . . . . . . 345.2 In Fig. 4.2(a), we plot the violationVof the inequalities Eq. (5.2),V1= Tr(
2123)
Tr(212) (dashed),V2= Tr(212) Tr(21) (grey) and V3= Tr(212) Tr(22) (solid),
as a function of the phase , forN= 3 atoms. Whenever V >0, entanglement is
detected by our network. In Fig. 4.2(b) we plot different purities associated with
a cluster state of size N, as a function of . B is any one atom not at an end
(dotted), any two atoms not at ends and with at least two others between them
(dashed), any two or more consecutive atoms not including an end (dash-dotted),
any one or more consecutive atoms including one end (solid). The plotted purities
are independent ofN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v
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LIST OF FIGURES vi
5.3 Plot of the purity Nm for m = 1 (solid black), m= 7 (dashed black), m = 14(solid grey) andm = 20 (dashed grey), as a function of, for N = 300 atoms. . . 40
6.1 A graphical comparison of the Bell-CHSH inequalities with the entropic inequali-
ties (6.2). All points inside the ball satisfy the entropic inequalities and all pointswithin the Steinmetz solid satisfy all possible Bell-CHSH inequalities. NB not all
the points in the outlining cube represent quantum states. . . . . . . . . . . . . . 43
6.2 In a special case of locally depolarized states, represented by points within the
tetrahedron, the set of separable states can be characterized exactly as an octahe-
dron. All states in the ball but not in the octahedron are entangled states which
are not detectable by the entropic inequalities. . . . . . . . . . . . . . . . . . . . 44
6.3 An outline of our experimental set-up which allows to test for the violation of the
entropic inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 Possible emissions leading to four-photons coincidences. The central diagram
shows the desired emission of two independent entangled pairs one by sourceS1 and one by source S2. The top and the bottom diagrams show unwelcome
emissions of four photons by one of the two sources. . . . . . . . . . . . . . . . . 47
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CHAPTER1
Introduction
The subject of this dissertation is the detection of quantum entanglement in physical systems.
Quantum entanglement was singled out by Erwin Schrodinger as ...thecharacteristic trait of
quantum mechanics, the one that enforces its entire departure from classical lines of thought. [1].
Indeed, after playing a significant role in the development of the foundations of quantum me-
chanics [1, 2, 3], quantum entanglement has been recently rediscovered as a physical resource in
the context of quantum information science [4, 5, 6, 7]. This set of correlations, to which a clas-
sical counterpart does not exist, arises from the interaction between distinct quantum systems.
Entanglement is instrumental in the improvements of classical computation and classical com-
munication results, of which two particularly important examples are the exponential speedup
of certain classes of algorithms [8, 9] and physically secure cryptographic protocols [4].
1.1 Entanglement as a property of quantum systems
Entanglement was first used by Einstein, Podolski and Rosen (EPR) [2] to illustrate the con-
ceptual differences between quantum and classical physics. In their seminal paper published
in 1935, EPR argued that quantum mechanics is not a completetheory of Nature, i.e. it does
not include a full description of the physical reality, by presenting an example of an entangled
quantum state to which it was not possible to ascribe definite elements of reality. EPR defined
an element of reality as a physical property, the value of which can be predicted with certainty,
before the actual property measurement. This condition is straightforwardly obeyed in the con-text of classical physics, but not in the context of quantum mechanics. The predictive power
of quantum mechanics is limited to, given a quantum state and an observable, the probabilities
of the different measurement outcomes. This feature led EPR to deem quantum mechanics as
incomplete. The incompleteness of quantum mechanics, as understood by EPR, was to plague
physicists for decades.
On one hand the quantum mechanical formalism explained the behaviour of microscopical
systems to a great degree of accuracy. On the other hand, it was conceptually unsatisfactory
as a fundamental theory of Nature and the EPR argument seemed a valid one. It was not
until John Bell published his seminal paper in 1964 [3], where he discussed the validity of the
EPR assumptions, that light was shed into the matter. In his paper Bell does not make any
assumption about quantum mechanics. It does, however, assume that our classical common senseview of the world is true. He considered a thought experiment where two causally disconnected
1
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CHAPTER 1. INTRODUCTION 2
observers share many identical pairs of physical systems and are allowed to perform two different
types of measurements on their respective systems. The measurements performed in each pair
are chosen at random and correspond to elements of reality. The expectation values of these
observables depend of the probability associated with a given outcome and the actual value of
the outcome. Bell then derived a set of inequalities that bound the expectation value of a linearcombination of the observables. It turns out that certain entangled states theoretically violate
these inequalities, which means that either quantum mechanics is an incomplete description of
Nature or the EPR assumptions are incorrect. The only way to decide which is the case was
by performing an experimental test of Bells inequalities. This test was realized with entangled
pairs of photons in 1982 [10] and it shown the violation of the Bells inequalities, as predicted
by quantum mechanics. This type of experimental test has subsequently been used to detect
entanglement experimentally in physical systems [11].
1.2 Entanglement as a physical resource
Fundamental quantum effects, such as quantum tunnelling or stimulated emission, have yieldedover the last century important technological breakthroughs, of which semiconductors or lasers
are two examples. Entanglement too has proved to be a physical resource capable of revolution-
izing the theories of computation and information. Within quantum information science, the
logical unit of information is the qubit, a two-level quantum system. The qubit differs from the
bit in that is can be any superposition of 0 and 1. In particular, a set of qubits can be in
an entangled state. The possibility of exploiting these quantum correlations between qubits, for
realizing computations faster than it would be possible classically, was first realized by Deutsch
in 1985 [12]. The development of quantum algorithms that ensued culminated with a result by
Shor for the efficient factoring the primes of a number [8]. The best classical algorithms for
this task scale exponentially with the size of the number to be factored, which means that it iseffectively impossible to factor large numbers. However, Shors algorithm can factor the primes
in a time that scales polynomially with the number size, i.e. efficiently. This result is particularly
relevant since the security of currently used cryptographic protocols is based on the difficulty of
factoring large numbers. Therefore a quantum factoring machine would render these protocols
useless.
Ironically, entanglement turns out to be the key resource in one of the possible solutions
to the security of cryptographic protocols. This solution, proposed by Ekert in 1991 [4], uses
entangled states as the carrier of protected information. The security of the protocol comes from
the fact that any attempt to gain access to the encrypted information, via a measurement on the
state, will necessarily disturb the quantum correlations. As mentioned earlier, the amount of
entanglement in a given state can be measured by checking for the violation of Bells inequalities.Therefore, any tampering of the carriers of information can be detected and the protocol aborted.
1.3 Detection and characterization of entanglement
We have seen how entanglement is not only a key concept in quantum mechanics, but also
a physical resource of great practical importance. It is therefore no wonder that it has been
extensively researched, both as a mathematical concept and as a property of physical systems.
In particular the experimental detection of entanglement is of paramount relevance for both
probing the limits of validity of quantum mechanics, as a physical theory, and for the monitoring
of quantum information processes. Its success is intimately related to the successful development
of theoretical tools that not only help us to further understand the properties of entanglement,
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CHAPTER 1. INTRODUCTION 3
but also provide practical experimental methods of detection.
There have been so far two different approaches to investigating the concept of entanglement.
One approach, the mathematical one, treats quantum states as mathematical objects and tries to
define entanglement as a mathematical property. It considers the density matrix representation
of quantum states and attempts to derive conditions that the matrices must obey in order torepresent an entangled state. This approach enabled the derivation of necessary and sufficient
conditions for entanglement in systems of two or three qubits. These results were obtained by
Peres [13] and the Horodeckis [14]. They pointed the way to a more general strategy of identifying
the mathematical properties of entanglement, based on the theory of positive maps. I will return
to this statement in more detail in the next chapter. However, a full characterization of the set
of entangled states for high-dimensional bipartite systems is yet to be found. In particular the
understanding of entanglement between more than two systems, multipartite entanglement, is
at present quite limited. Here, additional problems arise in the classification of entanglement,
since it is possible for states to exhibit multipartite entanglement while being separable with
respect to some of the subsystems. A general framework for the classification of entanglement
is yet to be developed and researchers have so far concentrated in studying specific classes ofmultipartite entangled states. I will present some examples of these classes in the next chapter
that we believe illustrate simultaneously the complexity of multipartite entanglement and its
great potential for quantum information processing.
The second approach to entanglement research, the physical one, treats quantum states as
properties of physical systems, that either exist in Nature or can be experimentally generated in
the laboratory. This approach differs fundamentally from the mathematical one in that it focuses
on the types of states actually generated in a given physical setting. The characterization or
detection of entanglement in this case is accomplished by via tests that are tailored for the specific
class of states considered. In the next chapter we will present the two most commonly used
experimental entanglement tests. Rather than aiming at a full characterization of entanglement,
this approach aims at developing techniques and methods for entanglement detection that are
experimentally accessible. In particular, it tries to identify which properties of a given quantum
system are relevant for entanglement detection. Providing a solution for this question will have
important consequences on the realization of experiments in quantum information processing,
since it will direct the experimentalists to a more efficient, and possibly easier, detection of
entanglement in the laboratory.
Despite all the effort devoted in recent years to the characterization of entanglement, the full
understanding of entanglements properties still eludes researchers. My doctoral research aimed
to contribute to our knowledge about entanglement by pursuing the physical approach. I have
developed new methods for not only the detection of both bipartite and multipartite entangle-
ment but also the characterization of certain properties of quantum states. These methods areexperimentally realistic and one of them was in particular realized experimentally.
1.4 Outline of thesis
When writing this thesis, I was faced with the difficult choice of which of my doctoral research
results to include. I decided to include the results that were not only the most directly relevant
to the subject of the dissertation, entanglement detection, but also the results that formed
the most chronologically coherent set. It will become apparent that these results were obtained
sequentially and that they are different instances of one research program. This program started
from a rather abstract setting of quantum networks, specifically designed to measure state
properties, and ended in the development of tailor-made experimental methods for the detection
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CHAPTER 1. INTRODUCTION 4
of entanglement in photons. However, I also pursued other research projects, such as the study
of the computational complexity of quantum languages [15], the development of methods to
generate classes of bound entangled states [16] and the investigation of methods to efficiently
generate graph states [17].
1.5 Chapter outline
I will now present the outline of remainder chapters of the thesis. Chapter 2 introduces the basic
concepts underlying the research results of the thesis. In particular it provides a mathematical
description of entanglement and discusses in more detail the general methods to detect and
characterize entanglement. Chapter 3 addresses the problem of estimating nonlinear functionals
Trk, k= 1, 2,... of a general density operator. The estimation method we proposed allows the
direct estimation of these nonlinear functionals. Our method uses an interferometric network
where a qubit undergoes single-particle interferometry and acts as a control on a swap operation
betweenkcopies of. Chapter 4 extends the above result to a more general quantum information
scenario, known as LOCC. In this scenario we consider two distant parties A andB that shareseveral copies of a given bipartite quantum state AB and are only allowed to perform local
operations and communicate classically. Chapter 5 investigates entanglement criteria based
on nonlinear functionals of that could be implemented in a simple, experimentally feasible
way. Our method is based of particle statistics effects and uses the fact that measuring the
purity of is tantamount to measuring the probability of projecting the state of two copies of
in its symmetric or antisymmetric subspaces. We extend of the nonlinear inequalities to the
detection of multipartite entanglement. Chapter 6 investigates the experimental realization of
the nonlinear entanglement test. We consider two copies of a polarization entangled pair of
photons AB. We also analyze the realistic experimental scenario where the source of entangled
photons is imperfect. Chapter 7 presents a conclusion to the thesis, with a summary of the mainresearch results presented.
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CHAPTER2
Basic concepts
2.1 State Vectors
Statistical predictions of quantum mechanics are based on two main concepts, quantum states
and quantum observables. With everyisolatedphysical system S, we associate a complex HilbertspaceHS of a suitable dimension, so that quantum states are represented by time-dependentunit vectors | (t) HS, and quantum observables by Hermitian operators acting in this space.Given a observable represented by the operatorA, there is a set of vectors{|i} such that
A
|i
=ai
|i
, ai
R. (2.1)
The vectors|i are called the eigenvectors of A, with respective eigenvalues ai. The set ofvalues{ai}is called the spectrum ofA. The time evolution of state vectors is unitary, i.e.
| (t) =U(t, t0) | (t0) , (2.2)whereU(t, t0) is a unitary operator, U U
= 11.Given a quantum system described by a state vector | and any observable A, represented
by a Hermitian operator, we can calculate all statistical properties ofA from the relation
A = | A | , (2.3)
whereA stands for the average value of A. In particular, when A is a projection operator,projecting on a one dimensional subspace spanned by vector |, A =||. In this caseA =|||2 represents the probability, for a system in state|, to pass a test for being inthe state|.
Quantum states can be equally well represented by projectors on the state vectors. Namely,
if instead of states|we consider the corresponding projectors ||, then the time evolutionof the state of the system will be given by
| (t) (t) | =U | (t0) (t0) | U, (2.4)and the average value of observable observable A will be written as
A = Tr A, (2.5)
5
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CHAPTER 2. BASIC CONCEPTS 6
where =|| and the trace Tr A stands for the sum of the diagonal elements ofA. Thetrace operation is linear, Tr (A+B) =Tr A+Tr B, and is basis-independent. The operator
is called density operator.
2.1.1 SubsystemsConsider a quantum systemScomposed of two subsystemsA andB. The Hilbert space asso-ciated with systemSis the tensor product of the Hilbert spaces of sub-systemAandB
HS= HA HB. (2.6)The dimension ofHS is dim HS= dim HA dim HB and any state|S of the system Scan beexpressed as a linear superposition of elements of the type |a|b, where |a HAand |b HB .Whenever convenient, well also write |a |b as |a|b or as |a, b. If we introduce orthonormalbases, i.e. maximal sets of vectors{|ak} inHA and{|bm} inHB, such thatak| al = kl,
bm
|bn
=mn, then any vector in
HScan be written as,
|S =k,l
ckl|ak|bl ,kl
|ckl|2 = 1. (2.7)
A particular subset of the states inHS can be written as a tensor product of state vectorsofHA andHB,
|S = |A |B =
k
k|ak
l
l|bl
(2.8)
= kl
ll
|ak
|bl
, (2.9)
where
k |k|2 =
l |l|2 = 1. This requires (comparing Eq.(2.8) and Eq.(2.7)) that
ckl = kl. (2.10)
The states for which this holds are called separable states. Note that this decomposition is
basis-independent. Thus, if |S is separable, we can associate state |A with the subsystem Aand state |B with the subsystem B. Otherwise we need to resort to density operators in orderto represent quantum states in subsystemsAandB.
2.2 Density OperatorsAny linear operator Sacting inHScan be written as a superposition of operators of the typeA B, whereA acts onHA andB acts onHB. We can choose operators bases,{Ak}acting onHA,{Bk} acting onHB, such that
S=k,l
SklAk Bl. (2.11)
The most common operator bases are formed from operators of the type |i j |. In our casewe have|ak al |, for operators acting onHA, and|bm bn |, for operators acting onHB (recallthat
|ai
and
|bj
are, respectively, orthonormal bases in
HA and
HB). This means that S can
be expressed as
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CHAPTER 2. BASIC CONCEPTS 7
S=
k,l,m,n
Skmln | ak al | | bm bn | . (2.12)
Any operatorApertaining only to sub-system
Acan be trivially extended to systemSthrough 1
A A 11. (2.13)The average value of an observable S= A B acting onSis given by
S | S| S = S | (A B) | S (2.14)=
k,l,m,n
clnckm al | bn | (A B) | ak | bm
=
k,l,m,nclnckm(al | A | ak)(bn | B | bm).
In the special case of an observable pertaining to one of the subsystems, i.e. if either A = 11 or
B = 11, we obtain (we choose B = 11),
S|S|S =
k,l,m,n
clnckm(al | A | ak)(bn | 11 | bm),
=
k,l,m,n
clnckmal|A|aknm,
=
k,l,mclmckmal|A|ak,
=Tr
k,l,m
clmckm|akal|A, (2.15)
=TrAA, (2.16)
whereA=
k,l,m clmckm | ak am | is called the reduced density operator and is associatedonly
with sub-systemA. Recall that the density operator associated with the total system is
AB = |SS| =
k,l,m,n
ckmcln(|akal|) (|bmbn|) . (2.17)
Given AB, the density operator of a bipartite system, we obtain A, the reduced densityoperator of the subsystem A, by taking the partial trace over the subsystem B. Mathematicallythe partial trace operation
AB A, (2.18)is defined as
Tr B(A B) =ATr B. (2.19)Thus,
1The procedure for sub-system B is analogous.
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CHAPTER 2. BASIC CONCEPTS 8
Tr B(AB) =
k,l,m,n
ckmcln | ak al | Tr| bm bn | (2.20)
=
k,l,m,nckmcln | ak al | mn
=k,l,m
ckmclm | ak al |
=A.
2.2.1 Mathematical properties of density operators
Density operators provide a description of quantum states. They can be defined as such without
any reference to state vectors. LetH be a finite-dimensional Hilbert space. A density operator, on
H, is a linear operator such that
is positive semi-definite, that is|| 0, for any| H. Tr= 1.
Any linear positive semi-definite operator X onH is always Hermitian, with non-negativeeigenvalues, and can be written as X=YY for some Y[18]. Many inequalities regarding pos-itive operators can be derived directly from|X| 0 by special choices of|. In particular,if| has only two non-zero components, labelled by i and j, then the submatrix of X withthe elements labelled by the indices i and j is also positive semi-definite. More generally, any
submatrix of a positive semi-definite matrix, obtained by keeping only the rows and columns
labelled by a subset of the original indices, is itself a positive semi-definite matrix and as suchmust have a nonnegative determinant (because all its eigenvalues are nonnegative).
To make a connection with the state vectors, let us consider a particular state (a pure state)
which can be described by a state vector| H. The density operator of any pure statecorresponds to a projection operator on that particular state, defined as
= ||, (2.21)which, like any projection operator, is idempotent:
2 =. (2.22)
For example, the state of a qubit|0 + |1 is described by the density operator= (|0 + |1) (1|+ 0|) = ||2 |00| + |01| + |10| + ||2 |11|, (2.23)
or, in the matrix form,
=
||2 ||2
. (2.24)
The diagonal elements 00 =||2 and 11 =||2 correspond, respectively, to the expectationvalues0||0 and1||1, giving the probabilities of observing bit values 0 and 1 respectively.
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CHAPTER 2. BASIC CONCEPTS 9
2.2.2 Ensemble interpretation of density operators
Consider a quantum source which emits particles in states|1, |2...|n with a priori prob-abilities p1,p2...pn. We will write it as an ensemble{pi,|i} . In this case
S = ni=1
pii|S|i = ni=1
piTrS|ii| = TrS
ni=1
pi|ii|
= TrS. (2.25)
The result depends on the observable Sand on the quantum state, which appears in the expres-
sion above only as the combination
=n
i=1
pi|ii|. (2.26)
We call this operator the density operator that describes a mixture of pure states|1, |2...
|n
with weights p1,p2...pn. The operator is not a projector any more,
2
= , but it has
all the properties we require for density operators (self-adjoint, semi-positive, unit-trace). If werefer to a single particle, we are uncertain as to which particular pure state|i it is preparedin. However, it makes perfect sense to say that the particle is in the state . Please note that
many different mixtures may lead to the same density operator:
=n
i=1
pi|ii| =n
i=1
qi|ii|. (2.27)
Note the sets of pure states{|i, |},{|i, |} are not in general orthonormal. In fact, unlessthere is any degeneracy in the values pi, only one such set can be orthonormal.
Now take, for example, this particular density operator of a qubit:
=
34 00 14
. (2.28)
It can be viewed as the mixtures of|0 and|1 with the probabilities 34 and 14 , or as a mixtureof|1 =
32|0 + 12 |1 and|2 =
32|0 12 |1 with probabilities p1 = 12 and p2 = 12 . Even
though states|0and|1are clearly different from states |1and|2, according to Eq.(2.25),these mixtures behave identically under any any physical investigation, i.e. we are not able to
distinguish between different mixtures described by the same density operator.
2.3 EntanglementWe have previously introduced the concept of separable sates. However, there are states inHSwhich are not separable, i.e. they cannot be written as a simple tensor product of two states
|A and|B (states for which ckl= kl). These states are referred to as entangled states.Entanglement is a set of quantum correlations arising from the interaction between two or more
quantum systems that does not have a classical counterpart. An example of an entangled state
is the singlet state of two spin-half particles
| = 12
(||||) , (2.29)
where | and | denote respectively spin up and spin down with respect to a chosen quantizationaxis.
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CHAPTER 2. BASIC CONCEPTS 10
As we mentioned in the previous chapter, entanglement is a very important physical resource
in quantum information science and both its mathematical characterization and experimental
detection have been subjected to extensive research. Unfortunately, the only general mathe-
matical definition of an entangled state is a negative one. A state is entangled if it cannot be
written as a convex sum of product states [19]
123...N =
C1 2 3 . . . N, (2.30)
where j is a state of subsystem j, and
C = 1. This fact means that in order to test
whether a given unknown state is entangled, we have in principle to check whether the state
can be decomposed in any of all the possible convex sums of product states. We will discuss
in a later section the most important results concerning the characterization and detection of
entanglement. But first, we will introduce the concept of superoperators, since they have proved
particularly relevant in the construction of entanglement criteria.
2.4 Superoperators
As we pointed out before, the time evolution of a state of systemS is unitary and obeysEq.(2.4). Suppose now thatS is composed of two sub-systems,A andB , and that we areinterested in the time evolution of sub-system A only. We can, without loss of generality, choosethe state ofA to be A and the state ofB to be the pure state|0. The time evolution of thestate of systemS, A |00|, is given by
= U A |00|U, (2.31)which is still a density operator describing systemS. The time evolution of the state of sub-systemA is then obtained by performing the partial trace, on sub-systemB, of the state ofsystemS:
A= Tr B(U A |00|U). (2.32)If we now consider an orthonormal basis |i,i = 0, 1,..., for sub-system B, Eq.(2.32) becomes
A=i
i|U|0A0|U|i i
EiAEi , (2.33)
whereEi=
i
|U
|0
are operators, acting on sub-system
A, and are trace-preserving:
i
Ei Ei=i
0|U|ii|U|0 = 0|UU|0 =11. (2.34)
Eq.(2.33) defines a linear mapL that takes linear operatorsA to linear operators A. Such
a map, if the property in Eq.(2.34) is satisfied, is called a superoperator. The representation of
the superoperator given in Eq.(2.33) is called the operator-sum representation.
2.4.1 Mathematical properties of superoperators
A superoperator L : that takes density operators to density operators has the followingproperties [18]:
L is trace-preserving, that is Tr= TrL() = 1.
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CHAPTER 2. BASIC CONCEPTS 11
L is linear, that is L(11+ 22) =1L(1) + 2L(2), 1+ 2= 1. L is a is completely positive map, that is, if is positive, then = L() is positive and
the extension ofL to a larger sub-system (11 L) is also positive.
All the mathematical properties originate from physical requirements. The first and thirdproperties originate from the requirement that, assuming to be a density operator, will alsobe a density operator. The second property originates from our desire to reconcile the density
operator time evolution and its ensemble interpretation.
The first property of superoperators is quite straightforward to accept, since any density
operator has, by definition, trace equal to one. The third property is perhaps less obvious.
Clearly, L must be a positive map to assure that will be a positive operator (necessarycondition for to be a density operator). But why must L be completely positive? The answeris: in order to assure that, if we decide to consider the action of the superoperator on an extended
system,ext , the resulting operator = ext L() will still be a density operator.
2.4.2 Jamiolkowski isomorphism
The Jamiolkowski isomorphism [20] establishes an equivalence between quantum states and
superoperators. Consider the action of a superoperator on half of the maximally entangled
state|J = 1NN
i |i|i:
1 |JJ| ij
|ij| (|ij|) =. (2.35)
The bipartite state encodes all the properties of the superoperator , as from it we can learnhow each density matrix element is transformed by
|ij| (|ij|) . (2.36)This establishes the equivalence between a completely positive map acting on density operators
pertaining to a Hilbert spaceH of dimension d2 1 and a density operator pertaining to aHilbert spaceH H of dimension 4d2 1.
2.5 Mathematical characterization of bipartite entanglement
When studying the existence of entanglement in bipartite states, it is very useful to distinguish
between pure states of the form Eq.(2.7) and mixed states. Pure bipartite states are entangled iff
the number number of terms of their Schmidt decomposition is greater than one. The Schmidtdecomposition of|Sis defined as:
|S =k,l
ckl|ak|bl =i
i|ai|bi, (2.37)
where|aiand|biare orthonormal bases forHA andHB, respectively, and i are non-negativereal coefficients such that
i
2i = 1. Any state of the form Eq.(2.7) admits a Schmidt de-
composition [18]. Hence, given a pure bipartite state, the computation of the coefficients i in
the Schmidt decomposition is sufficient for entanglement detection. However, there are not any
known efficient methods to determine experimentally the Schmidt coefficients of an unknown
state Eq.(2.7). Therefore, other more accessible entanglement criteria were developed. An ex-
ample is the entropic inequalities. Entropy measures uncertainty or our lack of information
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CHAPTER 2. BASIC CONCEPTS 12
about a particular physical property. Entropic inequalities, which quantify relations between
the information content of a composite quantum system and its parts, are of the form
S(A) S(AB) , S(B) S(AB), (2.38)
whereAB is a density operator of a composite quantum system and A andB are the reduceddensity operators pertaining to individual subsystems. They indicate that no matter which
physical property is measured there is more uncertainty in the composite system than in any
of its parts. Here S stands for several different types of entropies, including the regular von
Neumann entropy S() =Tr log and the Renyi entropy S() = log Tr2 [21]. Theseinequalities depend on the spectrum of both the state of the composite system and the states
of each individual subsystem, and provide necessary conditions for separability of bipartite pure
states. We will introduce in a later chapter of the thesis an efficient method for the determination
of the spectrum of unknown density operators.
2.5.1 Mixed statesHowever, not all bipartite states are of the form Eq.(2.7). In fact, for more general bipartite
states such as Eq.(2.26), the Schmidt decomposition is no longer valid [18]. Therefore new
methods to identify entangled states were developed. These methods are based on the theory
of positive maps.
Positive, but not completely positive maps are the most powerful tool in the detection of
entanglement. These maps are not physical, that is, they cannot be directly implemented in the
laboratory, but they provide the best mathematical criteria for the existence of entanglement
in a given state. In fact, they provide a necessary and sufficient condition for the existence of
entanglement [22]: a bipartite state AB HA HB is entangled iff (11 L)AB 0, for allL HB
2. Unfortunately, very little is known about the structure of positive maps, even for
small dimensional spaces like C. It is therefore very difficult to extract practical entanglementcriteria from the above condition.
Still, Peres [13] and the Horodeckis [14] have shown that the positive partial-transposition
map provides a necessary and sufficient condition for systems of two or three qubits. This map
preserves the eigenvalues of , so its clearly positive and trace preserving. For example, let
consider a generic density operator of a qubit. This is a 2 2 matrix of the form
, (2.39)
where the coefficients ,, are chosen such that Eq.(2.39) is a valid density operator. It is
sometimes convenient to represent the density operators of qubits as
= 1 + r
2 =
1 +
i=x,y,zrii
2 , (2.40)
where 1 is the identity operator,r is a three dimensional vector of length smaller or equal toone and
x=
0 11 0
, y =
0 i
i 0
, z =
1 00 1
, (2.41)
are the Pauli operators. The action of the transposition map on the density operator of the
qubit is
2Or conversely, (L 11)AB 0.
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CHAPTER 2. BASIC CONCEPTS 13
T
. (2.42)
Suppose now that we consider a qubit, part of a larger system in the entangled state
|+ = 12
(|0|0 + |1|1) . (2.43)
If we now apply the transposition map to the second qubit, which corresponds to a situation in
which we consider the extension of transposition to a larger system (11T), the density operatorwill suffer a partial transpose of its matrix elements:
1
2
1 0 0 10 0 0 00 0 0 01 0 0 1
T 12
1 0 0 00 0 1 00 1 0 00 0 0 1
. (2.44)
The resulting density matrix has eigenvalues 12 , 12 , 12 and 12 , so its not a valid density operator.The negativity under partial transposition is a signature of entanglement, even for more general
cases. It is in fact a sufficient condition for the existence of entanglement.
2.6 Experimental detection of entanglement
Entanglement tests based on positive maps are not physical, since positive maps cannot be
directly implemented in the laboratory. While this problem can be circumvented, by mathemat-
ically constructing completely positive maps out of the positive maps relevant for entanglement
detection [23], the actual implementation of these tests in the laboratory is yet to be achieved.
Instead researchers have focussed on experimental tests that, albeit less powerful than positivemaps, are within reach of current technology.
2.6.1 Bells inequalities
Bells inequalities [3] were introduced as an attempt to encapsulate the non-locality of quantum
mechanics. While this is a completely different goal from the detection of entanglement, the
fact that they were designed to capture the quantum essence of physical systems meant that
they were also an entanglement test. In fact, they are the most widely used experimental
entanglement test. We will next briefly present the derivation of the Bell-CHSH inequality [24]
and show that it is violated by the maximally entangled singlet state introduced in Eq.(2.29).
If we remember the thought experiment mentioned in the introduction, we have the followingscenario: two distant observers A and B share many identical pairs of particles; A and B can
perform two different types of measurements on their respective particles, XA, YA and XB , YB ,
respectively; Each measurement is chosen randomly and has two possible outcomes: +1 and 1.Let us consider the quantity Q = XAXB+ YAXB+ YAYB XAYB. Note that
XAXB+ YAXB+ YAYB XAYB = (XA+ YA)XB+ (XA YA)YB. (2.45)Since XA, YA =1, it follows that either XA+YA = 0 or XA YA = 0, which in turn meansXAXB+ YAXB+ YAYB XAYB = 2. Hence, the expectation value ofQ is
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CHAPTER 2. BASIC CONCEPTS 14
E(Q) =
xAyAxByB
p(xA, yA, xB, yB)(xAxB+ yAxB+ yAyB xAyB) (2.46)
xAyAxByBp(xA, yA, xB, yB) 2 = 2, (2.47)where p(xA, yA, xB, yB) is the probability that, before the measurements are performed, XA =
xA, YA = yA, XB = xB, YB = yB. If we further notice that E(Q) = E(XAXB) + E(YAXB) +
E(YAYB) E(XAYB), we obtain the Bell inequality
E(XAXB) + E(YAXB) + E(YAYB) E(XAYB) 2. (2.48)However, if we now compute the expectation value ofQ, with
XA = Az, (2.49)
YA = Ax , (2.50)
XB = Bz +
Bx
2, (2.51)
YB = Bz Bx
2, (2.52)
on the singlet state|, we obtain that
XAXB| = YAXB| = YAYB| = XAYB| = 1
2. (2.53)
Thus,Q| = 22, which is in clear violation of Eq.(2.47) and implies that the state isentangled.
The violation of this and other Bells inequalities has been extensively observed experimen-
tally [10, 11], mostly in systems of photons. While being a very convenient entanglement test,
that requires only the computation of expectation values of linear operators on the state of the
composite system, these inequalities fail to detect many entangled states currently produced in
the laboratory. Hence, researchers have actively looked for other types of experimental entan-
glement tests.
2.6.2 Entanglement witnesses
Entanglement witnessesWwere recently introduced as a tool for experimental entanglement de-
tection [25, 26]. They are particularly well suited to the experimental detection of entanglement,
where quite often the type of entangled state generated is known. They are linear operators
acting on the composite Hilbert spaceHA HB that obey the following properties:
W is Hermitian, that is W = W. Tr(W|a, ba, b|) 0, for all states|a, b inHA HB, that is, the expectation value ofW
on any separable state is greater or equal to zero.
Wis not a positive operator, that is, it has at least one negative eigenvalue. Tr(W)=1.
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CHAPTER 2. BASIC CONCEPTS 15
Thus, if we have Tr(W ) < 0 for some , then is entangled. In that case we say that W
detects. Every entanglement witness detects something [26], since it detects in particular the
projector on the subspace corresponding to the negative eigenvalues of W. We will next give an
example of an entanglement witness that detects bipartite entangled states.
Consider an experimental setup that, due to the imperfections, produces the mixed ratherthan pure bipartite state of two qubits [27]
= p|| + (1 p) 14
, (2.54)
where || is the pure state generated under ideal experimental circumstances, 0 p 1 and1/4 is the completely mixed state (white noise).
The witness is constructed by first computing the eigenvector corresponding to the negative
eigenvalue of the partially transposed density operator TB . The witness is given by the partially
transposed projector onto this eigenvector. If the Schmidt decomposition of | is | =a|01 +b|10, with a, b 0, the spectrum ofTB is given by
{1 p4
+pa2,1 p
4 +pb2,
1 p4
+pab,1 p
4 pab}. (2.55)
Therefore is entangled iff p > 1/(1 + 4ab). The eigenvector corresponding to the minimal
eigenvalue is given by
| = 12
(|00 |11). (2.56)
Hence the witness W is given by
W = ||TB =1
2
1 0 0 00 0
1 0
0 1 0 00 0 0 1
. (2.57)Note that this witness does neither depend on p, nor on the Schmidt coefficients a, b. It detects
iff it is entangled, since we have that
Tr(||TB) = Tr(||TB) =. (2.58)Note also that in this particular case we just considered, if Tr(W ) 0, is separable. This isnot a general property of witnesses, and indeed if the noise is not white this is not true anymore.
2.7 Multipartite entanglement
Multipartite entanglement, as a set of quantum correlations, is much more complex than bi-
partite entanglement. Hence, we know considerably less about its mathematical structure and
experimental detection. Still, the general approach of the methods described in the previous sec-
tion is equally suited to detect multipartite entanglement. In fact, Bells inequalities have been
derived for multipartite entangled states [28] and so have entanglement witnesses [29]. How-
ever their experimental implementation has proved to be too challenging so far. The approach
to multipartite entanglement detection is similar to the bipartite case. Therefore we will use
this section to try to capture the complexity of multipartite entanglement by presenting three
examples of multipartite entangled states. These states were all introduced in the context of
quantum information and have proved to be useful resources for quantum information tasks.
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CHAPTER 2. BASIC CONCEPTS 16
The classification of multipartite entanglement differs from the bipartite case in that it
is difficult to compare the different types of multipartite entanglement that are possible in a
given composite system. For example, multipartite states ofNsubsystems can be biseparable,
i.e. admit the decomposition
=i
ciiA iB , (2.59)
where A, B are two disjunct partitions of the composite system. How does one compare this
type of state with a state that is triseparable or non-separable with respect to any partition?
This question is still open and considerable research is being currently devoted to it.
We will next present three classes of states that are representative of different features of
multipartite entanglement. We will also briefly discuss their application to quantum information.
2.7.1 Maximally entangled state
Just as we introduced the concept of maximally entangled state for the case of two qubits, wewill equally define the maximally entangled state ofN qubits:
|N = 12
(|0000....0N+ |1111....1N) , (2.60)
where|iiii....iN =|iN, i = 0, 1. In this case all the qubits are entangled with one another,but the state of any subset m of qubits is separable
m= TrNm(|NN|) =12
(|00...0m00...0|m+ |11...1m11...1|m) . (2.61)These states are particularly useful for multi-party quantum communication protocols, such
as multiparty quantum coin flipping [30].
2.7.2 W State
This class of symmetric states is, after the maximally entangled state, the most widely used
example of multipartite entanglement. Unfortunately, a practical application in the context of
quantum information is yet to be found. The W state is defined as
|WN = 1N
(|1000....0N+ |0100....0N+ |0010....0N+ ... + |0000....1N) . (2.62)
In this case all the qubits are again entangled with one another, but interestingly enough thestate of any subsetm of qubits is not separable. In fact, for the case of three qubits, theW state
retains maximally bipartite entanglement when any one of the three qubits is traced out [31].
2.7.3 Cluster state
The cluster state is perhaps the best example of the computational advantage of multipartite
over bipartite entanglement. This class of pure states is represented by a connected subset of a
simple cubic lattice of qubits [32]. The cluster state is defined as the set of states|{k}C thatobey the set of eigenvalue equations
K
(a)
|{k}C= (1)ka
|{k}C, (2.63)with the correlation operators
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CHAPTER 2. BASIC CONCEPTS 17
K(a) =(a)x
bnghb(a)(b)z . (2.64)
Therein,
{ka
{0, 1
}|a
C
} is a set of binary parameters which specify the cluster state and
nghb(a) is the set of all neighboring lattice sites ofa. This class of states in cubic lattices withtwo or more dimensions is, together with single qubit measurements, sufficient for universal
quantum computation [32]. It is remarkable how a multipartite entangled state is alone the
computational resource required for quantum computation.
2.8 Quantum networks
A quantum computation is nothing but changing the logical values of a set of qubits through a
series of operations, such that the final result has logical meaning. Similarly to classical com-
putations, quantum computations are described through quantum circuits or networks. These
networks are a sequence of quantum gates, unitary operations that change the logical values ofthe qubits, acting on one or more qubits at a time. They are a very useful paradigm to describe
the dynamical evolution of systems of qubits, where the emphasis is on the state of the system
after the implementation of the quantum gate, rather than on the actual physical interaction
that realizes the gate. Deutsch [33] showed the existence of a universal set of quantum gates,
i.e. a set of gates that can approximate any unitary evolution of a set of qubits with arbitrary
accuracy. It was later shown that this set is finite [34].
2.8.1 Universal set of gates
The universal set of quantum gates is constituted by the set of all possible single qubit unitaries
plus an entangling two-qubit gate [18]. Any single qubit unitary operator can be written in the
form
U= exp(i)Rn() = exp(i)exp(in ), (2.65)where , are real numbers and Rn() denotes a rotation by about the n axis. However,
the actual implementation of arbitrary rotations in a given physical qubit can be experimentally
very challenging. Therefore, researchers have instead concentrated in finding a finite set of single
qubit gates that can approximate an any unitary operation Uto arbitrary accuracy , i.e.
(U, V) = max|
||(U V)|| | , (2.66)
whereV is the unitary implemented instead ofU,(U, V) is the unitary error and the maximumis taken over all normalized states|. A possible such set of gates is constituted by theHadamard, /4 and /8 gates [18]:
H= 1
2
1 11 1
, /4 =
1 00 i
, /8 =
1 0
0 ei/4
. (2.67)
As for the two-qubit gate, it is a controlled operation, i.e. it is a quantum gate where the
inputs have different roles. One of the inputs is the control qubit while the other is the target
qubit. The gate acts on the target qubit iff the control qubit is in state |1. A generic controlled-Ugate is depicted in Fig. 2.1. The prototypical example of the entangling two-qubit gate is the
controlled-NOT gate. It has the following matrix representation in the|control, targetbasis:
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CHAPTER 2. BASIC CONCEPTS 18
U
Truth tableC T
0 0 0 00 1 0 11 0 1 U(0)1 1 1 U(1)
C TC
T
Figure 2.1: The controlled-Ugate. The top line represents the control qubit and the bottomline represents the target qubit. Uacts on the target qubit iff the control qubit is in the logicalstate|1.
C N OT =
1 0 0 00 1 0 00 0 0 10 0 1 0
. (2.68)
TheC N OTgate flips the target qubit iff the control qubit is in state |1, otherwise it acts asthe identity gate. Let us now understand why is it that this gate is an entangling gate. Consider
the case where the control qubit is in state (
|0
+
|1
)/
2 and the target qubit is in state
|0
.
The composite state of the two qubits is clearly separable. After the C N OT,
12
(|0 + |1)|0 = 12
(|00 + |10) CNOT 12
(|00 + |11), (2.69)
which is no longer separable and is in fact the maximally entangled state |+ mentioned earlier.
2.8.2 Interferometry
As we mentioned earlier, the quantum network formalism provides us with a very useful set of
tools to describe the dynamical evolution of physical systems of qubits. A particularly simple
and relevant example that of quantum interferometry. Consider a single particle going througha Mach-Zender interferometer (Fig. 2.2).
The incoming particle enters the interferometer from the lower left, in the path labelled
|0. It encounters a 50:50 beam-splitter that deflects the particle into arm|1 with probabilityp = 0.5. If the particle goes through arm|0, it acquires a phase 0, while if it goes througharm|1 it acquires the phase 1. The two paths are then recombined in a second 50:50 beam-splitter. If we place particle detectors at each of the output ports of the second beam-splitter,
and repeated this experiment many times, we would observe that the probability of the particle
being detected in port|0or|1 after the interferometer is given by
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CHAPTER 2. BASIC CONCEPTS 19
1
0
0
1
2 1 0
0 cos
2P
2 1 0
1 sin
2P
50/50 Beam Splitter
50/50 Beam Splitter
Figure 2.2: The Mach-Zender interferometer.
p0 = cos2
1 0
2
, (2.70)
p1 = sin2
1 0
2
. (2.71)
This result can be easily understood if we translate the Mach-Zender interferometer into thelanguage of quantum networks (Fig. 2.3). Let us encode our qubit in the two arms of the
interferometer. The qubit is initially is state|0. After the beam-splitter, which is nothing buta Hadamard gate, the state of the qubit becomes (|0 + |1)/2. The qubit then acquires thephases0, 1: (e
i0 |0+ ei1 |1)/2, which is equivalent the the action of a phase gate 2(10).After the second beam-splitter the state of the qubit is
|0 BS|0 + |12
|0 + ei(10)|1
2BS (1 + e
i(10))|0 + (1 ei(10))|12
, (2.72)
hence the probability of finding the qubit in state
|0
or
|1
after the interferometer is simply
given by Eq.(2.70) and Eq.(2.71), respectively.
2.9 Summary
We have now presented the basic concepts underlying this thesis: mixed states, superoperators,
entanglement and quantum circuits. We have discussed the ambiguity in the definition of any
mixed quantum state, which is due to the indistinguishability of state preparations. We have
introduced superoperators, completely positive maps acting on quantum states, as the most
general evolution of quantum systems. We have mentioned the Jamiolkowski isomorphism be-
tween superoperators and quantum states. Entanglement, and in particular its detection, is the
main object of research of this thesis. We gave an overview of the main results concerning the
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CHAPTER 2. BASIC CONCEPTS 20
VISIB
ILITY
Figure 2.3: The quantum network corresponding to the Mach-Zender interferometer (= 1 0). The visibility of the interference pattern associated with p0 varies as a function of according to Eq.(2.70).
mathematical characterization and experimental detection of entanglement. Finally, we intro-duced the circuit model of quantum computation and we shown its suitability to describe the
dynamical evolution of quantum systems, and in particular interferometric effects.
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CHAPTER3
Direct estimation of density operators
This chapter presents the results that were published in an article written in collaboration with
A. K. Ekert, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek: A. K. Ekert, C. Moura
Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek, Phys. Rev. Lett. 88, 217901
(2002).
Certain properties of a quantum state , such as its purity, degree of entanglement, or its
spectrum, are of significant importance in quantum information science. They can be quantified
in terms of linear or non-linear functionals of. Linear functionals, such as average values of
observables{A}, given by TrA, are quite common as they correspond to directly measurablequantities. Non-linear functionals of state, such as the von Neumann entropyTr ln , eigen-values, or a measure of purity Tr2, are usually extracted from by classical means i.e. is first
estimated and once a sufficiently precise classical description of is available, classical evalu-
ations of the required functionals can be made. However, if only a limited supply of physical
objects in state is available, then a direct estimation of a specific quantity may be both more
efficient and more desirable [35]. For example, the estimation of purity of does not require
knowledge of all matrix elements of , thus any prior state estimation procedure followed by
classical calculations is, in this case, inefficient. However, in order to bypass tomography and to
estimate non-linear functionals ofmore directly, we need quantum networks [33, 36] performing
quantum computations that supersede classical evaluations.
In this chapter, we shall present and examine a simple quantum network that can be used as
a basic building block for direct quantum estimations of both linear and non-linear functionalsof any . The network can be realized as multiparticle interferometry. While conventional
quantum measurements, represented as quantum networks or otherwise, allow the estimation of
TrA for some Hermitian operator A, our network can also provide a direct estimation of the
overlap of any two unknown quantum states a and b, i.e. Trab.
3.1 Modified interferometry
In order to explain how the network works, let us start with a general observation related to
modifications of visibility in interferometry. Consider a typical interferometric set-up for a single
qubit: Hadamard gate, phase shift
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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 22
Figure 3.1: A modified Mach-Zender interferometer with coupling to an ancilla by a controlled-Ugate. The interference pattern is modified by the factor vei = Tr [U ].
=
1 00 ei
, (3.1)
Hadamard gate, followed by a measurement in the computational basis. We modify the interfer-
ometer by inserting a controlled-Uoperation between the Hadamard gates, with its control on
the qubit and withU acting on a quantum system described by some unknown density operator
, as shown in Fig. 3.1. The action of the quantum network is given by
|0| H 12
(|0 + |1) |cU 1
2(|0| + |1U(|))
12
(|0| + ei|1U(|))H 1
2
|0| + eiU(|)
+ |1
| eiU(|)
. (3.2)
The action of the controlled-U on modifies the interference pattern:
P0() = 14
(1 + veiei + veiei + 1) =12
(1 + v cos( + )) , (3.3)
by the factor Tr(||U) = TrU = vei [37], where v is the new visibility and is the shiftof the interference fringes, also known as the Pancharatnam phase [38]. Thus, the observed
modification of the visibility gives an estimate of TrU , i.e. the average value of the unitary
operator U in state . Let us mention in passing that this property, among other things, allows
the estimation of an unknown quantum state as long as we can estimate TrUk for a set of
unitary operators Uk which form a basis in the vector space of density operators.
Let us now consider a quantum state of two separable subsystems, such that = a b.We choose our controlled-U to be the controlled-V, where V is the swap operator, defined as,
V|A|B = |A|B, for any pure states|A and|B . In this case, the modification of theinterference pattern given by Eq. (3.3) can be written as,
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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 23
Va
b
a b
Figure 3.2: Quantum network for direct estimations of both linear and non-linear functions of
a quantum state.
v = TrV (a b)=
ij
rs
rsfj|ei| (|fs|erfs|er|) |ei|fj
=ij
rs
rsjsir ei | fs fj | er
= ij
ij
|ei
|fj
|2
= Trab. (3.4)
Since Trab is real, we can fix = 0 and the probability of finding the qubit in state |0at theoutput, P0, is related to the visibility by v = 2 P0 1. This construction, shown in Fig. 3.2,provides a direct way to measure Trab (c.f. [39] for a related idea).
3.2 Multiple target states
There are many possible ways of utilizing this result. For pure statesa= || andb = ||the formula above gives Tr
ab
=|
|
|2 i.e. a direct measure of orthogonality of
|
and
|. If we put a = b = then we obtain an estimation of Tr2. In the single qubit case,this measurement allows us to estimate the length of the Bloch vector, leaving its direction
completely undetermined. For qubits Tr2 gives the sum of squares of the two eigenvalues which
allows to estimate the spectrum of.
3.2.1 Spectrum estimation
In general, the evaluation of the spectrum of any d ddensity matrix requires the estimationofd 1 parameters Tr2, Tr3,... Trd. We can do so directly via the controlled-shift operation,which is a generalization of the controlled-swap gate. Givenk systems of dimensiond we define
the shift V(k) as
V(k)|1|2...|k = |k|1...|k1, (3.5)
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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 24
for any pure states|. Such an operation can be easily constructed by cascadingk 1 swapsV. If we extend the network and prepare = k at the input then the interference will bemodified by the visibility factor,
v= Tr V(k)k = Tr k =k
i=1
ik. (3.6)
Thus measuring the average values ofV(k) for k = 2, 3...d allows us to evaluate the spectrum
of [35]. Although we have not eliminated classical evaluations, we have reduced them by a
significant amount. The average values ofV(k) for k = 2, 3...d provide enough information to
evaluate the spectrum of, but certainly not enough to estimate the whole density matrix.
It should be mentioned that other spectrum estimation methods, relying on single collective
measurements of several copies of, have been proposed [40]. These methods essentially project
the initial state = n, which forms an operator on the n-fold tensor product space, ontoorthogonal subspaces corresponding to irreducible representations of the permutation group of
n points. This decomposition is labelled by Young frames, the arrangement ofn boxes into drows of decreasing length. The normalized row lengths of each tableau are taken as estimates
of the ordered sequence of eigenvalues of . The probability that the error in the spectrum
estimation is greater than some fixed decreases exponentially withn [40].
3.2.2 Quantum communication
So far we have treated the two inputs a andb in a symmetric way. However, there are several
interesting applications in which one of the inputs, say a, is predetermined and the other is
unknown. For example, projections on a prescribed vector |, or on the subspace perpendicularto it, can be implemented by choosinga=
|
|. By changing the input state
|
we effectively
reprogram the action of the network which then performs different projections. This propertycan be used for quantum communication, in a scenario where one carrier of information, in state
|, determines the type of detection measurement performed on the second carrier. Note thatas the state|of a single carrier cannot be determined, the information about the type of themeasurement to be performed by the detector remains secret until the moment of detection.
3.2.3 Extremal eigenvalues
Another interesting application is the estimation of the extremal eigenvalues and eigenvectors
ofb without reconstructing the entire spectrum. In this case, the input states are of the form
||band we vary | searching for the minimum and the maximum ofv = |b|. This,at first sight, seems to be a complicated task as it involves scanning 2(d 1) parameters of.The visibility is related to the overlap of the reference state|andb by,
v = Tr
||
i
i|ii|
=i
i | | i |2 =i
ipi, (3.7)
where
ipi = 1. This is a convex sum of the eigenvalues ofb and is minimized (maximized)
when| =|min (|max). For any| =|min (|max), there exists a state,|, in theneighbourhood of
|
such that v < v (v > v). Thus this global optimization problem
can be solved using standard iterative methods, of which steepest decent [41] is an example.
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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 25
Estimation of extremal eigenvalues plays a significant role in the direct detection [35] and
distillation [22] of quantum entanglement. As an example, consider two qubits described by the
density operatorb, such that the reduced density operator of one of the qubits is maximally
mixed. We can test for the separability ofb by checking whether the maximal eigenvalue ofb
does not exceed 1
2 [42].
3.2.4 State estimation
Finally, we may want to estimate an unknown state, say a d d density operator, b. Such anoperator is determined byd21 real parameters. In order to estimate matrix elements |b|,we run the network as many times as possible (limited by the number of copies of b at our
disposal) on the input|| b, where| is a pure state of our choice. For a fixed|, afterseveral runs, we obtain an estimation of,
v= |b|. (3.8)
In some chosen basis{|n} the diagonal elementsn|b|n can be determined using the inputstates |nn|b. The real part of the off-diagonal element n|b|k can be estimated by choosing| = (|n + |k)/2, and the imaginary part by choosing| = (|n + i|k)/2. In particular,if we want to estimate the density operator of a qubit, we can choose the pure states,|0 (spin+z), (|0 + |1) /2 (spin +x) and (|0 + i|1) /2 (spin +y), i.e. the three components of theBloch vector.
Quantum tomography can also be performed in many other ways, the practicalities of which
depend on technologies involved. However, it is worth stressing that the strength of our scheme is
the use of a fixed architecture network, controlled only by input data, to perform the estimation
of properties of.
3.2.5 Arbitrary observables
We can extend the procedure above to cover estimations of expectation values of arbitrary
observables A. This can be done with the network shown in Fig. 3.2, since estimations of mean
values ofanyobservable can always be reduced to estimations of a binary two-output POVMs.
We shall apply the technique developed in Refs. [23, 35]. As A= 1 + Ais positive ifis theminimum negative eigenvalue ofA, we can construct the state a= A =
A
Tr(A)and apply our
interference scheme to the pair A b. The visibility gives us the mean value of V,
v= VAb = Tr A
Tr(A)b , (3.9)
which leads us to the desired value,
Ab Tr(bA) =vTrA + (vd 1), (3.10)
where Tr1= d.
3.3 Quantum channel estimation
Any technique that allows direct estimations of properties of quantum states can be also used
to estimate certain properties of quantum channels. Recall that, from a mathematical point
of view, a quantum channel is a superoperator, (), which maps density operators intodensity operators, and whose trivial extensions, 1k do the same, i.e. is a completely positive
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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 26
Figure 3.3: A quantum channel acting on one of the subsystems of a bipartite maximallyentangled state of the form |+ =
k |k|k/
d. The output state =
1d
kl |kl| (|kl|),
contains a complete information about the channel.
map. In a chosen basis the action of the channel on a density operator =
kl kl|kl| can bewritten as
() =
kl
kl|kl|
=kl
kl (|kl|) . (3.11)
Thus the channel is completely characterized by operators (|kl|). In fact, with every channel we can associate a quantum state that provides a complete characterization of the channel.
If we prepare a maximally entangled states of two particles, described by the density operator
P+ = 1d
kl |kl| |kl|, and we send only one particle through the channel, as shown in
Fig. 3.3, we obtain
P+ [1 ] P+= , (3.12)where
= 1
d
kl
|kl| (|kl|) . (3.13)
We may interpret this as mapping the|kl|th-element of an input density matrix to theoutput matrix, (|kl|). Thus, knowledge of allows us to determine the action of onan arbitrary state, (). If we perform a state tomography on we effectively performa quantum channel tomography. If we choose to estimate directly some functions of thenwe gain some knowledge about specific properties of the channel without performing the full
tomography of the channel.
For example, consider a single qubit channel. Suppose we are interested in the maximal
rate of a reliable transmission of qubits per use of the channel, which can be quantified as the
channel capacity. Unlike in the classical case, quantum channels admit several capacities [43, 44],
because users of quantum channels can also exchange classical information. We have then the
capacities QC where C = , , , , stands for zero way, one way and two way classicalcommunication. In general, it is very difficult to calculate the capacity of a given channel.
However, our extremal eigenvalue estimation scheme provides a simple necessary and sufficient
condition for a one qubit channel to have non-zero two-way capacity. Namely, Q > 0 iff
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CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 27
has maximal eigenvalue greater than 12 . Note that this condition is also necessary for the other
three capacities to be non-zero.
This result becomes apparent by noticing that if we traceover the qubit that went through
the channel (particle 2 in Fig. 3.3), we obtain the maximally mixed state. Furthermore, the
two qubit state, , is two-way distillable iff the operator 1
2 1 has a negative eigenvalue(see [42] for details), or equivalently when has the maximal eigenvalue greater than 12 . Thisimplies Q()> 0 because two-way distillable entanglement, which is non-zero iff given stateis two way distillable, is the lower bound for Q() [44].
3.4 Summary
In summary, we have described a simple quantum network which can be used as a basic building
block for direct quantum estimations of both linear and non-linear functionals of any density
operator . It provides a direct estimation of the overlap of any two unknown quantum states
a and b, i.e. Trab. Its straightforward extension can be employed to estimate functionals
of any powers of density operators. The network has many potential applications ranging frompurity tests and eigenvalue estimations to direct characterization of some properties of quantum
channels.
Finally let us also mention that the controlled-SWAP operation is a direct generalization
of a Fredkin gate [45] and can be constructed out of simple quantum logic gates [36]. This
means that experimental realizations of the proposed network are within the reach of quantum
technology that is currently being developed (for an overview see, for example, [46]).
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CHAPTER4
Direct estimation of density operators using LOCC
This chapter presents the results that were published in an article written in collaboration with
D. K. L Oi, P. Horodecki, A. K. Ekert, L. C. Kwek: C. Moura Alves, D. K. L Oi, P. Horodecki,
A. K. Ekert, L. C. Kwek, Phys. Rev. A 68, 32306 (2003).
In the previous chapter we presented a family of quantum networks that directly estimate
multi-copy observables, Tr[k], of an unknown state [47, 35, 23]. As mentioned before, these
nonlinear functionals quantify important properties of, such as the degree of entanglement or
the spectrum. Therefore it would be very useful to be able to estimate them even when is a
bipartite stateAB shared by two distant parties, Alice and Bob, who can perform only local
operations and communicate classically (LOCC). In this chapter we show that the estimationof non-linear functionals of quantum states admit LOCC implementation. We also show that
Structural Physical Approximations [35, 23], an important tool for entanglement detection, can
be constructed locally. This opens the possibility of the direct estimation of entanglement and
some channel capacities using only LOCC.
As a general remark, let us recall that a quantum operation can be implemented using
LOCC if it can be written as a convex sum
=k
pk Ak Bk, (4.1)
whereAk acts on the subsystem at Alices location and Bk on the subsystem at Bobs location,
andpk represent the respective probabilities.
4.1 LOCC estimation of nonlinear functionals
The direct estimation method is extended to the LOCC scenario by constructing two local
networks, one for Alice and one for Bob, in such a way that the global network is similar to
the network with the controlled-shift. Unfortunately, the global shift operation V(k) cannot
be implemented directly using only LOCC, since it does not admit local decomposition (4.1).
Hence, we will implement it indirectly, using the global network shown in Fig. 4.1. Alice and
Bob share a number of copies of the state AB
Hd. They group them respectively into sets
ofk elements, and run the local interferometric network on their respective halves of the stateAB =
kAB. For each run of the experiment, they record and communicate their result.
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CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 29
Figure 4.1: Network for remote estimation of non-linear functionals of bipartite density opera-tors. Since Tr[V(k)k] is real, Alice and Bob can omit their respective phase shifters.
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CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 30
The individual interference patterns Alice and Bob record will depend only on their respective
reduced density operators. Alice will observe the visibility vA = Tr[kA] and Bob will observe
the visibility vB = Tr[kB]. However, if they compare their individual observations, they will be
able to extract information about the global density operator AB, e.g. about
Tr[kAB] = Tr
kAB
V(k)A V(k)B . (4.2)This is because Alice and Bob can estimate the probabilities Pij that in the measurement Alicesinterfering qubit is found in state |iA and Bobs in state|jA for i, j = 0, 1. These probabilitiescan be conveniently expressed as
Pij = 1
4Tr
kAB
1 + (1)iV(k)A 1 + (1)jV(k)B , (4.3)
hence the formula for the basic non-linear functional ofAB reads
Tr[kAB] = P00 P01 P10+ P11. (4.4)In fact, the expression above is the expectation valuez z, measured on Alices and Bobsqubits (the two qubits that undergo interference). Given that we are able to directly estimateTr[kAB] for any integer value ofk , we can estimate the spectrum ofAB without resorting to a
full state tomography.
4.2 Structural Physical Approximations
We next show how to implement Structural Physical Approximations within the LOCC scenario.
Structural Physical Approximations (SPAs) were introduced recently as tools for determining
relevant parameters of density operators (see [23, 35] for more details). Basically the SPA of a
mathematical operation , denoted as , is a physical operation, a process that can be carried
out in a laboratory, that emulates the character of . More precisely, suppose :
Hd
Hd is a
trace preserving map which does not represent any physical process, for example, an anti-unitary
operation such as transposition. Then a convex sum
=D + (1 ), (4.5)whereD is the depolarizing map which sends any density operator into the maximally mixedstate, represents a physical process, i.e. a completely positive map, as long as is sufficiently
large. On top of thisD, with its trivial structure, does not mask the structure of . TheStructural Physical Approximation to is obtained by selecting, in the expression above, the
threshold value = (d2)/(d2 + 1), whereis the lowest eigenvalue of (1 )P(d)+ andP(d)+is a maximally entangled state of ad dsystem 1. Note that we impose the positivity condition
on the map1
to ensure that is a completely positive map.Please note that the physical implementation of SPAs is not a trivial problem as the for-
mula (4.5), which explicitly contains the physically impossible map , is of little guidance here.
Let us also mention in passing that if is not trace preserving then may be implementable
but only in a probabilistic sense e.g. using experimental post-selection.
There are many examples of mathematical operations which, though important in the quan-
tum information contexts, do not represent a physical process. For example, mathematical cri-
teria for entanglement involve positive but not completely positive maps [21] and as such they
are not directly implementable in a laboratory they tacitly assume that a precise description
of a quantum state of a physical system is given and that such operations are mathematical
transformations on the matrix describing the quantum state.
1The threshold value for is obtained from the requirement of complete positivity of, which in this case canbe reduced to P
(d)+ 0
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CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 31
4.2.1 SPA using only LOCC
If does not represent any physical process then its trivial extension to a bipartite case, 1 ,does not represent a physical process either. Still, its SPA, 1 , does describe a physi