bell measurement and quantum teleportation using entangled single...
TRANSCRIPT
“Bell Measurement and QuantumTeleportation using Entangled Single Spins in
Nitrogen–Vacancy Diamond Defects”
by
Johannes Nikolaus Greiner
in Partial Fulfilment of the Requirements for the Degree of
Bachelor of Science in Physics
at the University of Stuttgart
Faculty of Mathematics and Physics
September 22, 2011
Thesis Supervisor:
Prof. Dr. Jorg Wrachtrup
Main Advisor:
Dipl. Phys. Florian Rempp
Abstract
This work investigates the possibility of quantum teleportation in nitrogen-vacancy cen-ters (NVs) in diamond. The theoretical background is examined as well as experimentalimplementations.The main challenges towards a realization lie in the possibilities of Bell measurementand a proper readout of the teleported state.
A variation of the standard Bell measurement is proposed to be realized by subject-ing the system to decoherence after transforming into the Bell basis by standard gateoperations (CNOT, Hadamard).As a result, the system dephases towards a superposition of Bell states which can thenbe measured.
Subselection of Bell states after dephasing allows to avoid the controlled gates usedin the standard teleportation scheme. One of the four states is selected while measure-ments showing a different state must be dismissed.This is necessary since performing those controlled gate operations would require a se-lective pulse between the two NVs which could not be realized with a present pair. Also,these operations would greatly increase the time needed to finish the whole teleportation.
The qubits used for the scheme are a 15N nuclear spin and the electron spin of oneNV (Alice) as well as the electron spin of a second (Bob).
The requirement to read out both the nitrogen spin and the electron spin of the first NVcan be avoided by the subselection. Nevertheless, to realize the teleportation schemeproposed for two entangled NV electron spins, it is mandatory to at least read out bothelectron spins of the entangled pair within a short interval of time.To that end, the current experimental technologies will need to be improved. It willbe necessary to do a parallel readout on both NVs or to develop a method of indirectmeasurement.Another possibility is to change the method of generating entanglement from the recentusage of dipole-dipole interaction to optical entanglement.
Furthermore, the proposed variation of a Bell measurement can potentially be usedto carry out a teleportation from the NV’s nitrogen nuclear spin to a single photon. Thenecessary entanglement between NV spins and single photons has been demonstratedbefore.
When quantum teleportation in nitrogen-vacancy centers can be performed, the possi-bilities for quantum algorithms and the transfer of information will be greatly increased.Ultimately, the realization of this proposal could present a meaningful step towardsaccomplishing quantum cryptography with the use of diamond defects.
i
Zusammenfassung
In dieser Arbeit wird die Moglichkeit von Quantenteleportation in Stickstoff-FehlstellenZentren (NVs) in Diamant untersucht. Der theoretische Hintergrund wird ebenso wieexperimenteller Implementationen beleuchtet.Das Hauptproblem fur eine Realisierung liegt in der Moglichkeit einer Bell-Messung unddem richtigen Auslesen des teleportierten Zustands.Eine Variation der normalen Bell-Messung wird vorgeschlagen, indem das System derDekoharenz uberlassen wird, nachdem es durch normale Quanten-Gateoperationen(CNOT, Hadamard) in die Bell Basis transformiert wurde.Daher dephasiert das System auf eine Uberlagerung von Bell Zustanden hin, welchedann ausgelesen werden konnen.Eine Subselektion von Bell Zustanden nach dem Dephasieren fuhrt dazu, dass die imnormalen Teleportationsschema ausgefuhrten kontrollierten Quanten-Gates ausbleibenkonnen. Einer der vier Zustande wird ausgewahlt, wahrend Messungen, die einen an-deren Zustand anzeigen, verworfen werden mussen.Dies ist notwendig, da zum Ausfuhren dieser kontrollierten Quanten-Gates ein selektiverPuls zwischen den beiden NVs vonnoten ist, der mit einem vorliegenden Paar nicht re-alisiert werden konnte. Zudem wurden diese Gates die Zeit, die zum Durchfuhren dergesamten Teleportation notig ist, signifikant verlangern.
Die Qubits, die fur das Schema benutzt werden, sind ein 15N Kernspin und der Elektro-nenspin eines NVs (Alice) sowie der Elektronenspin eines zweiten NVs (Bob).Die Notwendigkeit, sowohl den Kernspin des Stickstoff als auch den Elektronenspin desersten NVs auszulesen, kann durch die Subselektion verhindert werden. Nichtsdestotrotzist es unvermeidbar, zumindest die Elektronenspins beider NVs wahrend eines kurzenZeitintervalls auszulesen.Dazu mussen die aktuellen experimentellen Technologien verbessert werden. Es istnotwendig, beide NVs parallel auszulesen oder eine Methode indirekter Messung zuentwickeln.Eine andere Moglichkeit ist, die Methode der Erzeugung von Verschrankung vom jetzi-gen Nutzen der Dipol-Dipol Kopplung hin zu optischer Verschrankung zu andern.
Die vorgeschlagene Variation der Bell-Messung kann weiterhin potentiell dazu genutztwerden, eine Teleportation vom Stickstoff-Kernspin eines NVs auf ein einzelnes Photondurchzufuhren. Die dazu notwendige Verschrankung zwischen NV-Spins und einzelnenPhotonen wurde bereits realisiert.Wenn Quantenteleportation mit Stickstoff-Fehlstellen Zentren durchgefuhrt werden kann,werden die Moglichkeit fur Quantenalgorithmen und fur den Transfer von Informationenverbessert.Die experimentelle Durchfuhrung der in dieser Arbeit vorgeschlagenen Konzepte konnteletztlich einen signifikanten Schritt hin zur Realisierung von Quantenkryptographie mitDiamantenfehlstellen bedeuten.
ii
Declaration
I declare that this thesis was composed by myself, that the work contained herein is myown except where explicitly stated otherwise in the text, and that this work has notbeen submitted for any other degree or professional qualification except as specified.
Stuttgart, September 22, 2011Johannes Greiner
iii
Preface
On the first encounter, the word ‘teleportation’ seems to have a very mysterious touch. Onemay think about matter being beamed from one place to another or even recall sci-fi storiesor the claims of Buddhist monks to instantly travel long distances in space.
In the context of quantum physics however teleportation simply means the transfer of in-formation. To be more precise, it means that the information conserved in a certain state of aquantum particle is passed on to a different quantum object.Still, there does remain a mystery about it since this transfer can be done over distancesof arbitrary length. So, it would be possible to pass information to another galaxy or anyother far away corner of the universe in very short time. It is limited though by the fact thatclassical information needs to be interchanged as well as by the result of causality and gen-eral relativity that information may not be transmitted faster than the speed of light [1, p.225].
Teleportation nowadays even has industrial applications since it can be used for quantumcryptography, transferring information with a very high level of security since any measure-ment on the systems causes the wave functions to collapse.
Since nitrogen-vacancy centers (NVs) in diamond have properties that allow their usage forthis kind of extremely secure communication, it is very interesting to investigate whether thesediamond defects can be used for the purposes of teleportation.
All the necessary ingredients are basically there. The nitrogen nuclear spin of one NV canbe used to prepare the information to be sent and entanglement between electron spins of twoNVs presents a ‘quantum channel’ to send this information through.
There is just one problem: In teleportation, it is not enough to just measure what has beenteleported. There are four different possibilities of states the whole system can be in and so,a second measurement is necessary to find out which one it is.
Otherwise the transferred ‘information’ is useless since the whole procedure would not pro-duce much more than what a usual die (or the same as a tetrahedral, 4-sided die) would do.
Unluckily, this so-called ‘Bell measurement’ can not easily be performed on a system of twoNVs. That is why the central proposal in this work is to give the system some time to dephasetowards one of the four possible states (or a superposition) and to subselect one particularstate rather than just measuring them.A variation of a Bell measurement can then be carried out by reading out the NV’s electronspin.
iv
Beforehand, one recent achievement necessary for teleportation between two NVs, the en-tanglement of their electron spins, is described in chapter 1.A part on basics of quantum computing (2.1) is followed by the description of quantum tele-portation (2.2) with an analysis of different initial Bell states and transformation into the Bellbasis (2.2.2) which is crucial for the proposed dephasing process.
Subsequently, a realization of teleportation with NVs (2.3) is discussed theoretically, includingthe proposed variation of the Bell measurement through dephasing in the Bell basis (2.2.2) aswell as a subselection of Bell states (2.3.2).Further attention is given to benefits like the analysis of the quality of entanglement and ofgate errors (3). The possibilities of experimental realizations and measurements (3.2) are thenportrayed until the closing chapter (4) provides an outlook on new methods of entanglementand single photons as qubits.
v
Contents
1 Entanglement of two different NV electron spins 11.1 Description and general properties of the NV center . . . . . . . . . . . . . . . 11.2 On pulse-operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Construction of pulse matrices . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 The method of driven transitions . . . . . . . . . . . . . . . . . . . . . . 3
1.3 General description of entanglement . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Coupling and Pulse-sequences used to entangle NV centers . . . . . . . . . . . 61.5 Detection of NV entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.1 Approaches currently used for detection . . . . . . . . . . . . . . . . . . 71.5.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Quantum teleportation using entangled NVs 82.1 Quantum computing Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Quantum circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Quantum teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 On transformation into the Bell basis . . . . . . . . . . . . . . . . . . . 142.2.3 Standard quantum circuit scheme . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Specific teleportation scheme proposed for NV centers . . . . . . . . . . . . . . 182.3.1 Dephasing in the Bell basis . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Subselection of Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Experimental benefits and possible errors 233.1 Analysis of an entangled state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Separation of gate errors and Bell state impurities . . . . . . . . . . . . 253.1.2 Decoherence caused by transversal relaxation (T2) . . . . . . . . . . . . 263.1.3 Spin-lattice relaxation (T1) errors . . . . . . . . . . . . . . . . . . . . . . 273.1.4 Subselection - effects on Bell state analysis . . . . . . . . . . . . . . . . 28
3.2 Measurement of the teleported state . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Parallel NV pair readout . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 The quest for a herald . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Outlook 304.1 A different method of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Optical entanglement and single-photon qubits . . . . . . . . . . . . . . . . . . 30
Bibliography 32
vi
List of Figures
1.1 NV center illustration used with permission by J. Podevin . . . . . . . . . . . . 1
2.1 Example of a quantum circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 General teleportation protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 First part of the teleportation quantum circuit . . . . . . . . . . . . . . . . . . 142.4 General teleportation quantum circuit. . . . . . . . . . . . . . . . . . . . . . . . 162.5 Variation of the general teleportation scheme . . . . . . . . . . . . . . . . . . . 172.6 Teleportation between two NVs using entangled electron spins as quantum chan-
nel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Proposed teleportation scheme for two NVs . . . . . . . . . . . . . . . . . . . . 192.8 Level scheme of NVA ⊗ NA after change of basis towards the Bell basis . . . . 21
4.1 Teleportation from an NV’s nitrogen nuclear spin to a single photon. . . . . . 31
List of Tables
2.1 Resulting states after Bell measurements for a∣∣∣Φ+
2,3
⟩input state . . . . . . . . 12
2.2 Resulting states on qubit 3 for∣∣∣Ψ+
2,3
⟩,∣∣∣Φ−
2,3
⟩and
∣∣∣Ψ−2,3
⟩as input Bell states . 13
2.3 Resulting states after Bell measurements for a mixture of Bell states (2.10) . . 14
vii
Chapter 1
Entanglement of two different NVelectron spins
1.1 Description and general properties of the NVcenter
A nitrogen-vacancy (NV) center is a point defect in the tetrahedral lattice of carbons in thediamond structure. It is composed of a substitutional nitrogen atom (N) associated with anadjacent vacancy (V).Six electrons occupy the center, where in a normal carbon structure, eight electrons are present.The resulting two holes cause the NV to possess an electron spin (S=1).About 11 % of the spin density is distributed over the nearest-neighbour carbons [2]. Thus,hyperfine and dipolar couplings [3] can be used to control a nuclear spin close to the vacancyindividually [4] by applying radio frequency pulses. Pulses in the microwave regime allow co-herent manipulation of the electron spin.
Figure 1.1: NV center illustration used with permission by J. Podevin [5]
A great advantage of the NV center is that it can be used under ambient conditions [6]. Also,since the NV’s spin is hardly affected by lattice phonons and other sources of decoherenceusually encountered in solid-state systems, its coherence times are exceptionally long.
1
Chapter 1 Entanglement of two different NV electron spins
The significant temperature dependency of the relaxation times causing decoherence is com-pensated by a high-Debye temperature and small spin-orbit interaction in diamond [7].So the electron spin of the combined NV system displays a relaxation time for spontaneoustransition between pure states (T1) in the order of milliseconds [8], while the phase memorytime (T2) is about 0.6 ms [9].Single NV spins can be addressed using optical and microwave radiation and can be read outwithout limitations of diffraction [10]. Multipartite entanglement between electron spin andclose by nuclear spins is realizable [4], providing the possibility to use the nuclear spins asqubits to store a quantum state for some ms.
Because of these unique properties the NV center is a promising candidate for processingquantum information [11, 12] yielding a possible application in quantum computing in the fu-ture [13–15]. NVs have already been used in the function of a quantum repeater [16] and aquantum register [17, 18]. As a recent achievement, a quantum device composed of an NVcoupled to a nano-wire has been constructed [19].Furthermore, the NV center is a reliable single-photon source and a quantum cryptographyprotocol [20] using NVs has even been implemented.
Teleportation between NVs and especially teleportation onto single photons from an NV isthus of particular interest in furthering the technology of quantum cryptography. If the detec-tion of emitted photons from an NV center can be improved, NV centers could become usefulsenders or storage devices in cryptography.
The fact that teleportation between light and matter has been realized on a large scale [21]shows this is a realistic vision.Further support is given since entanglement between an NV electron spin and a single photonpolarization has been demonstrated experimentally [22].
2
Chapter 1 Entanglement of two different NV electron spins
1.2 On pulse-operations
In an NV center, transitions between the energy levels of nuclear spins can be caused byapplying AC magnetic field radio frequency pulses while electron spins are manipulated throughmicrowave pulses.These experimental operations can be simulated through unitary transformations applied tothe wave function or the density matrix of the given spin system. In the following, the matrixrepresentation of these operations will be discussed.
1.2.1 Construction of pulse matrices
An oscillating magnetic and electric field causes Rabi oscillations of a single quantum state [23].Pulses of a certain angle are then implied by stopping the application of the field and conse-quently stopping the evolution of the Rabi oscillation at the corresponding time. In the pictureof the Bloch sphere [24], such a pulse represents a rotation of the state vector. Hence, it can becreated by rotation generating matrices, namely the spin matrices composed of Pauli matricesor their generalisations in higher dimensions (e.g. (1.3)).
To understand the construction of the pulse matrices, a three-level-system will be analysed.It is to be noted that these operations are carried out in a rotating frame i.e. a coordinatesystem rotating with the frequency of the applied rotating magnetic field [23].The Hamiltonian that generates the rotations is then time independent (cf. [25, p.16]):
H =~2~Ω · ~σ , (1.1)
where ~Ω is the rotation axis and ~σ represents a tensor of the appropriate Pauli matrices.The corresponding time evolution operator is given by
exp
[− i~Ht
]= exp
[− i
2~Ω · ~σ t
]= exp
[− i
2~ϕ · ~σ
], (1.2)
in which ~ϕ contains the different angles of rotation with Rabi frequency |~Ω| and |~ϕ| = |~Ω|t.Thus, given a Rabi frequency, a rotation of any angle can easily be constructed through thismethod.
1.2.2 The method of driven transitions
To generate the matrix representation of an actual pulse operating between two energy levels,it is possible to neglect the elements of the Pauli matrix that do not correspond with thetransition.
The spin operator for a three dimensional system is a generator of two transitions:
Jx =1√2
0 1 01 0 10 1 0
(1.3)
3
Chapter 1 Entanglement of two different NV electron spins
Let us interpret this matrix as a display of a single spin system (S=1) with rows and columns inthe order |−1〉 , |0〉 , |1〉. Consequently, it allows transitions from |−1〉 to |0〉 in both directionsas well as |0〉 ↔ |1〉.If we are to carry out a pulse operation only between |0〉 ↔ |1〉, it is possible to reduce of thethree-level-system to a two-level-system. To this end we shall use an approximation:
Let the transitions |0〉 ↔ |1〉 and |−1〉 ↔ |0〉 correspond to frequencies ω10 and ω−10, re-spectively. As we focus on the first transition rather than on the latter, the probability ofoccupation of |1〉 is given by [26]:
p1(t) =Ω2
Ω2 + ∆2Sin
(√Ω2 + ∆2 · t
2
). (1.4)
In this formula, Ω is the Rabi frequency and ∆ is a detuning frequency of a non-resonanttransition driving oscillation.Detuning (∆) for the desired transition |0〉 ↔ |1〉 is given to be small. However, this correspondsto a large detuning visible in the probability of occupation of state |−1〉:
∆′ = ω10 − ω−10 ±∆ . (1.5)
The condition (ω10 − ω−10) Ω is mandatory for the use of this approximation:
p−1(t) ∝ 1
1 + (∆′/Ω)2
∆′Ω−−−−→ 0 . (1.6)
This results shows there is no effect on the transition that is not desired. A pulse generatingmatrix can therefore be composed by taking all matrix elements not corresponding to the de-sired transition to be 0. The matrix generating the |0〉 ↔ |1〉 transition is:
S01 =
0 0 00 0 10 1 0
. (1.7)
S01 can then be used instead of the standard Pauli matrices in (1.2) with ~Ω usually being oneof the standard Euclidian basis vectors, here in x-direction. The computation then yields:
exp
[− i
2S01 ϕx
]=
1 0 00 Cos
[ϕ2
]−iSin
[ϕ2
]0 −iSin
[ϕ2
]Cos
[ϕ2
] (1.8)
If we let be ϕ = π we get the matrix for a π-pulse: 1 0 00 0 −i0 −i 0
. (1.9)
The pulse operations for the subselection of Bell states (sec 2.3.2) and the NV teleportationscheme as calculated in chapter 3 are constructed by this method.
4
Chapter 1 Entanglement of two different NV electron spins
1.3 General description of entanglement
Entanglement, once described by Einstein as a “telepathically” [27, p.85] working interactionand used to criticise quantum mechanics for its counter-intuitiveness [28] can today be ob-served in experiments. In quantum computing, entanglement is a source of exponential speedup of algorithms [29–31] and its existence is a necessary condition for teleportation [32] andquantum cryptography [33–35].The debate of an underlying mechanism that causes the long distant relation and how it shouldbe interpreted however has not proceeded much since the times of Einstein and Bell [36, p.142].
The effect can take place between any two quantum objects. If two objects are entangled,their state is best described by one common wave function. Upon measurement of one of theobject’s properties (e.g. spin), the other one will show either the complementary value (e.g.clockwise spin if counter-clockwise has been measured) or the same value, depending on whichBell state [32] the system is in. The main point is that the state of the second object is exactlydetermined once the first has been measured.In theory, entanglement does not weaken if the distance between the two quantum objectsincreases but remains the same at arbitrary distances [37].
An entangled system is represented by one of the following states, named ‘Bell states’ afterNorthern Irish physicist John Stewart Bell.∣∣Φ+
⟩=
1√2
(|0〉A ⊗ |0〉B + |1〉A ⊗ |1〉B) (1.10)∣∣Ψ+⟩
=1√2
(|0〉A ⊗ |1〉B + |1〉A ⊗ |0〉B) (1.11)∣∣Φ−⟩ =1√2
(|0〉A ⊗ |0〉B − |1〉A ⊗ |1〉B) (1.12)∣∣Ψ−⟩ =1√2
(|0〉A ⊗ |1〉B − |1〉A ⊗ |0〉B) . (1.13)
Both quantum objects A and B are defined as two dimensional quantum systems which cantake on two explicit values of a given quantum number, e.g. 0 or 1.So, if the entangled quantum objects are for example in state |Φ+〉 and a |0〉A is measured insubsystem A, subsystem B will be known to be in state |0〉B.Now it would still be possible that the two systems exchanged a “hidden variable” [37] beforebeing measured. This variable would determine which measurement outcome to show depend-ing on the result of the partnering system.However it was shown by Bell [38] that the effect of entanglement is independent of the basisin which the measurement is carried out and that a hidden variable theory is inconsistent withprobability theory.Thus, it became unavoidable to state that entanglement is indeed a phenomenon which canneither be explained by classical means nor limited relativistically by the speed of light, c.In recent years, it has been shown experimentally that the effects of entanglement are trans-mitted at a speed at least of the order c · 103 [39, 40].As we shall see in chapter 2.2, the actual transfer of information is still limited [1] since it canonly happen by the means of quantum teleportation.
5
Chapter 1 Entanglement of two different NV electron spins
1.4 Coupling and Pulse-sequences used to entangle NVcenters
While entanglement between the nuclear spins and the electron spin of one NV center has beendemonstrated before [4], cf. [41], it is a very recent achievement to entangle the electron spinsof two different NV centers.The exact details of the procedures involved in this realization will be described in a yet un-published paper ( [42], cf. [17]) however a summary shall be given below.To entangle two quantum systems, naturally, there needs to be an interaction between them.As for two NVs, it is a dipolar coupling that has been used so far, with its strength being inthe order of kilohertz [17].The main experimental challenge is caused by the scaling of dipolar coupling with 1/r3 wherer is the distance between two NV centers. So, it is necessary to create and detect a pair of NVcenters with less than 30 nm distance. Recent implantation techniques, namely the so calledmica mask implantation [43, 44] have been used to increase the chances for finding a suitablepair.
The usual method to produce the actual entanglement is to transform the combined system ofthe two NV spins into the Bell basis by applying a Hadamard and a CNOT gate in succession(see sec. 2.1.3 for quantum gates and 2.2.2 for the transformation).A different pulse sequence (1.14) has been chosen since the given relaxation time of one of thetwo NVs in the setup does not allow controlled operations (e.g. CNOT) as a selective pulse cannot be produced. For the only suitable pair which has been found so far [42], T ∗2 (relaxationtime of the free induction decay (FID) signal) is in the range of 10 − 15µs. This time is tooshort compared to the frequency of the dipole-dipole interaction: T ∗2 1/νdip.
Since the equation T ∗2 ≤ T2 always holds [45, sec 11.9.2], the minimum coupling frequencywould get lower if the relaxation time could be limited by T2 rather than T ∗2 and consequently,the minimum distance between the NVs would increase. This can indeed be accomplished byperforming a Hahn echo sequence [46] on both NVs simultaneously, the sequence being
π
2− τ − π − τ − π
2, (1.14)
where π andπ
2denote the angles of a pulse and τ is a free evolution time.
This variation of the double electron electron resonance sequence (DEER) [47] represents aphase gate on the two NVs. It generates an entangled state between the two defects if τ ischosen appropriately (in this case τ ≈ 13µs) [42].
Usually, the low energy transition |ms = 0〉 ↔ |−1〉 is used to perform pulse sequences. Togenerate entanglement, it is much more effective (in the given setup even necessary) to use thedouble quantum transition |−1〉 ↔ |1〉 which provides a speed-up of factor four [17].
6
Chapter 1 Entanglement of two different NV electron spins
1.5 Detection of NV entanglement
Since entanglement between two NVs has only been established very recently [42], it has beenof particular interest to find methods to prove the fact that this has been accomplished.
1.5.1 Approaches currently used for detection
As for entanglement between the NV’s electron spin and the spins of the surrounding nuclei,detection of entanglement has been well established in recent years.The technique normally used for detection is “density matrix tomography” [3, 4]. Applied toentangled systems this method includes the transfer of coherences existing if the system is inone of the Bell states (or even in so called GHZ and W states in tripartite entanglement) ontoa difference in population which can be read out directly.The disadvantage of this method for the given entanglement between two NVs is that the pulsesequence generating a phase gate necessary for the transfer uses up a long time (≈ 25µs). Thiswould allow the system to dephase and the coherences to vanish.
So, the method of phase measurement has been used.In brief, this method reduces the detection of entanglement to a measurement of a globalphase difference. If entanglement exists, the phases of the single NVs, Φ1 and Φ2, cannot bedistinguished. The phases can only be measured in combination Φ1 ± Φ2 since the subspacescan no longer be addressed individually in an entangled state [41].Contrarily, if both phases are detected separately, no entanglement is given.In essence, this method uses the fact that an entangled state is described by one common wavefunction rather than two individual ones.
1.5.2 Indirect detection
Before the technique of phase measurement was used, other possibilities were investigated.Since measurements of photon antibunching [48] failed to show the desired results, the use ofentanglement witnesses [49] and direct detection of entanglement [50] was examined as well asgeneral criteria for entanglement measurements [51–53].
It was then proposed by the author of this work to detect entanglement indirectly. Theexistence of entangled states is a necessary condition for the performance of both superdensecoding [54] and quantum teleportation.
So, if a teleportation could be carried out successfully, this would imply a proof of entan-glement. As we shall see later, a working teleportation scheme could also be used to analyse agiven Bell state (see chap. 3).
Beforehand, we will have a look at the basics of quantum teleportation in the following.
7
Chapter 2
Quantum teleportation using entangledNVs
To understand quantum teleportation, familiarity with basic quantum computing is necessary.Information is stored in ‘qubits’ and ‘quantum circuits’ portray the use of ‘quantum gates’used to transform a given state.
2.1 Quantum computing Basics
2.1.1 Qubits
The analogue of a classical bit is called qubit (quantum bit) in quantum information. It is theelementary unit of all quantum information processes. Unlike a classical bit it does not onlytake on states 0 or 1 but any superposition of two given values of a quantum number since itrepresents a two-level quantum mechanical system.The state of a qubit is thus fully described by the wave function
|Ψ〉 = a |0〉+ b |1〉 (2.1)
The term “qubit” has been coined by U.S. theoretical physicist Benjamin Schumacher in his1995 paper “Quantum coding” [55]
An NV electron spin even represents a “qutrit”, meaning a three-level quantum system, sinceit possesses a total electron spin S=1 with three possible quantum numbers. As any otherqutrit it can be reduced to a qubit if necessary.
Also, as described before, the spins of the surrounding nuclei can be used as qubits or evenqutrits (14N).
2.1.2 Quantum circuits
As we use qubits instead of bits the analogy to classical computing is furthered by the use ofquantum circuits replacing classical electrical circuits. [56, p.22]Here, a horizontal wire represents a qubit and any symbol on it represents a quantum gate i.e.
8
Chapter 2 Quantum teleportation using entangled NVs
a unitary operation performed on the qubit or several qubits.Controlled operations are marked by a dot on the control qubit. They will be carried out ifand only if the control qubit is in state |1〉.In this work, a qubit subspace is marked by a hat on top of the qubit number or name e.g.“1, 2, NVA”.The circuit is read from left to right. Note that if the circuit is translated to matrices operatingon states, the order of course needs to be reversed with the quantum gate on the far left beingthe first matrix operating on the state. An example of a quantum circuit follows below.
Figure 2.1: Example of a quantum circuit
After two single-qubit quantum gates on qubits 2 and 3 a controlled NOT operation (CNOT)is carried out with 3 being the controlling qubit and 1 being the target. A two-qubit unitaryoperation U being controlled by 1 ends the circuit also represented by
C1[U2,3
]· C3
[NOT1
]· H2 · X3 (2.2)
2.1.3 Quantum gates
The controlled NOT gate maps |0〉 7→ |1〉 if active. Its matrix representation is
C1[NOT2
]=
1 0 0 00 1 0 00 0 0 10 0 1 0
(2.3)
Even though this gate seems rather unspectacular, it is the most essential of all quantum gates,unparalleled in universality since “Any multiple qubit logic gate may be composed from CNOTand single qubit gates” [56, p.191]. It is thus comparable in functionality with the NAND gateof classical electronics.
9
Chapter 2 Quantum teleportation using entangled NVs
The next gate to investigate is the Hadamard gate, represented by
H =1√2
(1 11 −1
)(2.4)
It maps pure states to superpositions e.g. H(|0〉) = 1√2|0〉 + 1√
2|1〉. If two Hadamard gates
appear in direct succession they annihilate each other, ergo H2 = 1 as the reader can easilyverify. We will also see the use of the single-qubit gates X and Z:
X = σx =
(0 11 0
)(2.5) Z = σz =
(1 00 −1
)(2.6)
X is basically a NOT operation without control on a single qubit. It can also be described asthe rotation of the Bloch sphere around the x-axis by π which gives the operation its name.
The same is true for the Z-gate except the rotation here is around the z-axis, leaving theEigenstate |0〉 invariant and changing the sign of |1〉.
All these gates will be used in the standard qubit circuit scheme of teleportation (sec. 2.2.3).Before this will be discussed, however, it is necessary to gain some understanding of howteleportation actually works.
10
Chapter 2 Quantum teleportation using entangled NVs
2.2 Quantum teleportation
2.2.1 General description
Quantum teleportation was first described by Bennett et al. [32] and has since been realizedin different experiments. Mostly, pairs of entangled photons have been used [57, 58], butteleportation has also been realized with entangled squeezed states [59], quantum dot singlephoton sources [60], running wave fields [61], and wave fields trapped inside cavities [62].Recent developments include the use of trapped ions [63, 64] and teleportation from a lightpulse to a macroscopic object [21].It is common to name the sender Alice and the receiver of information Bob. Alice is inpossession of two qubits, one containing the information to be transferred, the other one beingentangled to the one qubit on Bob’s side. The standard protocol for quantum teleportation isthe following (cf. [65]):
1. Preparation of an initial state to be teleported
2. Prepare a nonlocal quantum channel i.e. entangle two qubits belonging to Alice and Bob
3. Perform a Bell measurement on Alice’s two qubits
4. Communicate the measured result (to Bob) through a classical channel
5. Correct the teleported state (Bob) according to the result of the measurement
Figure 2.2: General teleportation protocol. Illustration taken from [32]
The main challenge on the experimental side is the Bell state measurement [32] of step three.It is carried out between Alice’s qubits with the measurement taking place in the basis ofthe Bell states (section 1.3). Their wave functions collapse to form one of these four states,swapping the entanglement from Alice’s and Bob’s qubit to Alice’s two qubits.Note that the Bell measurement creates entanglement if it was non-existent before.
11
Chapter 2 Quantum teleportation using entangled NVs
For Alice to actually measure the resulting Bell state and to communicate the two bits necessaryto determine the state she needs to have a closer look at the qubits in the computational basis.Let the initial state be
|Ψ1〉 = a∣∣01
⟩+ b
∣∣11
⟩(2.7)
with |a|2 + |b|2 = 1 since⟨Ψ1
∣∣Ψ1〉 = 1
Alice and Bob’s entangled qubits shall be in the state∣∣∣Φ+
2,3
⟩yielding a state of all three
qubits∣∣Ψ1
⟩⊗∣∣∣Φ+
2,3
⟩=
1√2
(a(∣∣01, 02, 03
⟩+∣∣01, 12, 13
⟩)+ b
(∣∣11, 02, 03
⟩+∣∣11, 12, 13
⟩))(2.8)
Using the definition of the Bell states (sec. 1.3) this is equivalent to
1
2
(∣∣∣Φ+1,2
⟩⊗(a∣∣03
⟩+ b
∣∣13
⟩)+∣∣∣Ψ+
1,2
⟩⊗(b∣∣03
⟩+ a
∣∣13
⟩)+∣∣∣Φ−
1,2
⟩⊗(a∣∣03
⟩− b
∣∣13
⟩)+∣∣∣Ψ−
1,2
⟩⊗(−b∣∣03
⟩+ a
∣∣13
⟩)). (2.9)
So, if the result of Alice’s Bell measurement is∣∣∣Φ+
1,2
⟩Bob has to do nothing. The initial
state (2.7) has been recovered. If one of the other Bell states results from the measurement,appropriate operations have to be carried out to map these states to the actual initial state.
It is important to note that these operations do depend on the initial Bell state between qubits2 and 3.The following tables show the results for different input Bell states (top left corner). The Bellstates following the initial state in the first row are those measured in the Bell measurement.Compare (2.9) and Table 2.1 for understanding the notation.
∣∣∣Φ+
2,3
⟩7→
∣∣∣Φ+
1,2
⟩ ∣∣∣Ψ+
1,2
⟩ ∣∣∣Φ−1,2
⟩ ∣∣∣Ψ−1,2
⟩1
2|03〉 a b a -b
1
2|13〉 b a -b a
Table 2.1: Resulting states after Bell measurements for a∣∣∣Φ+
2,3
⟩input state
12
Chapter 2 Quantum teleportation using entangled NVs
∣∣∣Ψ+
2,3
⟩7→
∣∣∣Φ+
1,2
⟩ ∣∣∣Ψ+
1,2
⟩ ∣∣∣Φ−1,2
⟩ ∣∣∣Ψ−1,2
⟩1
2|03〉 b a -b a
1
2|13〉 a b a -b
∣∣∣Φ−2,3
⟩7→
∣∣∣Φ+
1,2
⟩ ∣∣∣Ψ+
1,2
⟩ ∣∣∣Φ−1,2
⟩ ∣∣∣Ψ−1,2
⟩1
2|03〉 a b a -b
1
2|13〉 -b -a b -a
∣∣∣Ψ−2,3
⟩7→
∣∣∣Φ+
1,2
⟩ ∣∣∣Ψ+
1,2
⟩ ∣∣∣Φ−1,2
⟩ ∣∣∣Ψ−1,2
⟩1
2|03〉 -b -a b -a
1
2|13〉 a b a -b
Table 2.2: Resulting states on qubit 3 for∣∣∣Ψ+
2,3
⟩,∣∣∣Φ−
2,3
⟩and
∣∣∣Ψ−2,3
⟩as input Bell states
Thus, there is a correlation between the initial Bell state and the operations that are to becarried out after the Bell measurement.
This is very important if we consider the analysis of the purity of a given initial Bell state(as described in sec 3.1.4).
13
Chapter 2 Quantum teleportation using entangled NVs
The purity can be analysed directly through a Bell measurement since if the state∣∣∣Bmix,2,3⟩ = u∣∣∣Φ+
2,3
⟩+ v
∣∣∣Ψ+2,3
⟩+ w
∣∣∣Φ−2,3
⟩+ x
∣∣∣Ψ−2,3
⟩(2.10)
is used, the result allows to determine the parameters by variation of a and b in the initialstate (2.7) as can be seen in the final table
∣∣Bmix,2,3
⟩7→
∣∣∣Φ+
1,2
⟩ ∣∣∣Ψ+
1,2
⟩ ∣∣∣Φ−1,2
⟩ ∣∣∣Ψ−1,2
⟩1
2|03〉 a (u+w)+b (v-x) a (v-x)+b (u+w) a (u+w)+b (x-v) a (v-x)-b (u+w)
1
2|13〉 a (v+x)+b (u-w) a (u-w)+b (v+x) a (v+x)+b (w-u) a (x-v)+b (u+w)
Table 2.3: Resulting states after Bell measurements for a mixture of Bell states (2.10)
Now however we will restrict ourselves to a pure Bell state∣∣∣Φ+
2,3
⟩to examine the standard
scheme for teleportation realized with a quantum circuit.
2.2.2 On transformation into the Bell basis
Let us first have a closer look at the first two gates of the general teleportation scheme, namelythe Hadamard and the CNOT gate.The following is to show that these two gates represent a change of basis. This is crucial tounderstand why the dephasing process as proposed in sec. 2.3.1 can evolve towards Bell states.
Figure 2.3: First part of the teleportation quantum circuit
We remember the matrix representations of the individual gates (see section 2.1.3) and com-bine
(H1 ⊗ 12
)· C1
[NOT2
]= T =
1√2
1 0 0 10 1 1 01 0 0 −10 1 −1 0
. (2.11)
14
Chapter 2 Quantum teleportation using entangled NVs
To demonstrate that this is actually a transformation into the Bell basis [66] we will nowconstruct a transformation matrix by standard methods of quantum mechanics.In general, the transformation matrix into another basis is defined piecewise by [67, p.115]
Tij =⟨Ψ′i∣∣Ψj〉 (2.12)
where 〈Ψ′i| is the bra in the new basis and |Ψj〉 the ket in the original basis.We thus use the four Bell states (see section 1.3) as bras in the following order:
〈Ψ′1| =⟨
Φ+1,2
∣∣∣ ...⟨
Ψ+1,2
∣∣∣ ,⟨Φ−1,2
∣∣∣ ,⟨Ψ−1,2
∣∣∣and the original basis set is the standard computational basis
∣∣01, 02
⟩,∣∣01, 12
⟩,∣∣11, 02
⟩,∣∣11, 12
⟩A piecewise computation of the transformation matrix elements according to equation (2.12)then actually yields exactly the same matrix as in eq. (2.11) which is the result we need.
The operation thus maps Bell states to energy Eigenstates∣∣∣Φ+1,2
⟩⟨Φ+
1,2
∣∣∣ 7→ ∣∣01, 02
⟩ ⟨01, 02
∣∣ (2.13)∣∣∣Φ−1,2
⟩⟨Φ−
1,2
∣∣∣ 7→ ∣∣11, 02
⟩ ⟨11, 02
∣∣ (2.14)∣∣∣Ψ+1,2
⟩⟨Ψ+
1,2
∣∣∣ 7→ ∣∣01, 12
⟩ ⟨01, 12
∣∣ (2.15)∣∣∣Ψ−1,2
⟩⟨Ψ−
1,2
∣∣∣ 7→ ∣∣11, 12
⟩ ⟨11, 12
∣∣ (2.16)
which can then be addressed by manipulating the qubits.
Conclusively, the operation represents a basis change since it has the desired effect on anydensity matrix:
T · ρ · T † := ρ (2.17)
and on any subsequent unitary operation that may follow in the quantum circuit:
(TU) · ρ · (TU)† = T · (UρU †) · T † := Tρ′T † = ρ′ . (2.18)
The concept of basis change using common quantum gates is also established in the literature[68–70].
15
Chapter 2 Quantum teleportation using entangled NVs
2.2.3 Standard quantum circuit scheme
In the standard quantum circuit scheme, a Bell measurement is realized through first turningAlice’s two qubits (1, 2) into the Bell basis and subsequently performing a measurement oneach one of them. The final state on Bobs qubit (3) is corrected by a controlled X (= CNOT)and a controlled Z gate.
Let me emphasize again that this scheme, as proposed in [56, p.27] only works for a∣∣∣Φ+
2,3
⟩as
initial Bell state.
Figure 2.4: General teleportation quantum circuit.
After transforming in the Bell basis by the CNOT and Hadamard gate as described in theprevious section, equation (2.9) becomes
1
2
(∣∣01, 02
⟩⊗(a∣∣03
⟩+ b
∣∣13
⟩)+∣∣01, 12
⟩⊗(b∣∣03
⟩+ a
∣∣13
⟩)+∣∣11, 02
⟩⊗(a∣∣03
⟩− b
∣∣13
⟩)+∣∣11, 12
⟩⊗(−b∣∣03
⟩+ a
∣∣13
⟩)). (2.19)
A Bell measurement can thus be easily carried out by simply measuring the state of the firsttwo qubits. The third qubit (Bob’s) will then be in the state linked to the measurement out-come i.e. if the result is
∣∣01, 12
⟩it will be in state b
∣∣03
⟩+ a
∣∣13
⟩.
To reproduce the initial state, Bob will now have to apply according operations. If the mea-surement yields
∣∣01, 02
⟩, there is nothing to do. In the case of
∣∣01, 12
⟩, a X gate needs to be
applied, in case of∣∣11, 02
⟩the Z gate is the right operation and if
∣∣11, 12
⟩is the measurement
outcome, both gates will need to operate.
This is why the controlled operations as seen in the scheme above are appropriate.
16
Chapter 2 Quantum teleportation using entangled NVs
An allowed variation of the scheme, according to [56, p.186] is to perform the measurementafter the controlled gates.
Figure 2.5: Variation of the general teleportation scheme
Even though in this circuit some of the interpretation of it performing a teleportation is lostsince classical information is not transmitted prior to Bob’s operating on his qubit, the resultof both quantum circuits is exactly the same.
As we investigate the possibility of teleportation through NVs this is an important step tonotice. A measurement and thus a polarization of the NV can be done at the end of alloperations, avoiding the experimental problems that result once a read-out laser has beenapplied.
17
Chapter 2 Quantum teleportation using entangled NVs
2.3 Specific teleportation scheme proposed for NVcenters
For the proposed experimental teleportation between two NVs, the electron spins of two dif-ferent NVs and the nitrogen nuclear spin of one NV will be used as qubits as described in thefollowing figure.
Figure 2.6: Teleportation between two NVs using entangled electron spins as quantumchannel. Single NV illustration used with permission [5], modified.
An initial state prepared on the nitrogen nuclear spin of NVA (qubit 1) is to be teleportedto the electron spin of NVB (qubit 3). The curved line between the two NVs symbolizes thenecessary entanglement between their electron spins.Readout requirements (single shot on NVA, standard readout on NVB) will be discussed insection 3.2.
As we aim to realize this teleportation, two main points guide our considerations:Firstly, the scheme will have to be carried out as quick as possible since the entangled statebetween the two NVs is unstable. Secondly, a ‘classical’ Bell measurement is very difficultto realize. Addressing both the nuclear spin of the nitrogen and the NV’s spin at the sametime has been demonstrated using stimulated emission depletion microscopy (STED) [71, 72].However, the implementation of this technology is a laborious undertaking, requiring abouttwo months for the preparation of the set-up only.
As a solution, in this work a variation of the Bell measurement is proposed as illustratedon the succeeding page.
18
Chapter 2 Quantum teleportation using entangled NVs
NA is the first NV’s nitrogen while NVA represents the electron spin of the first NV and in thesame way NVB that of the second one.
Figure 2.7: Proposed teleportation scheme for two NVs
The goal is to teleport a state |Ψ〉 from NA to NVB. This can indeed be achieved through thisscheme if the resulting Bell state between NA and NVA is the one that does not require furthercorrection on the receiving qubit (NVB). In the given example this we select |Φ+〉 and thecontrolled operations between the two NVs that would be necessary in the standard schemecan be left out.Remember this is only true if the initial state prepared between the two NVs is a |Φ+〉 as wellsince otherwise, either a different state needs to be subselected or the measurement outcomevaries according to tables 2.1 and 2.2.
Successful (approximate) teleportation has even been demonstrated before in several experi-ments without the use of any Bell measurement [65,73–76]. This is a very meaningful point toconsider for any different teleportation scheme using NVs that may be developed in the futuresince the Bell measurement is indeed the main experimental challenge.
Here however, the Bell measurement is performed indirectly as we allow Alice’s qubits (NA,NVA) to dephase in the Bell basis and subselect the desired Bell state to make sure the tele-portation has been carried out correctly.
Both of these procedures will be described in greater detail in the following.
19
Chapter 2 Quantum teleportation using entangled NVs
2.3.1 Dephasing in the Bell basis
The general idea for a Bell measurement brought forth in this work is prepared here.As described in section 2.2.2 the first two gates effectively perform a transformation into theBell basis. Being in this basis the pure Bell states are found on the diagonal of the densitymatrix. This is the central point for the desired effect to take place during dephasing.
If we allow the system to evolve freely for a certain time, it will interact with its environ-ment, the diamond structure that it is located in. Since this environment has many degrees offreedom, the system will be subject to decoherence [77–80].It is a characteristic of this process that the off-diagonal elements of the density matrix vanishin the given basis [81,82].So, the resulting state after dephasing is either a pure Bell state or, more likely, a mixture ofdifferent Bell states.
Consequently, this procedure would allow to measure the resulting Bell state(s) by readingout the first NV’s electron spin as well as its nitrogen nuclear spin.Here, however, lays the difficulty for actual realizations. As of now, the nitrogen spin can onlybe read out individually by being mapped to the electron spin [83], making it impossible toread out both the electron spin and the nuclear spin at once.This procedure would require even more than the parallel readout as discussed in section 3.2.1.Instead, a parallel single-shot readout would be necessary reading out the spins of the nitrogennucleus as well as both NV electron spins.Since the electron spins of both NVs are polarized by the readout laser [84], it is currently notpossible to measure the Bell state and still carry out a successful teleportation.Hence, it is our goal to postselect the resulting Bell state. In the example used here the exper-iment should be repeated until a |Φ+〉 state results on Alice’s quibts. Subsequently, the finalCNOT and CZ gates of the standard scheme will be without effect and can thus be skipped(see sec 2.2.3).These gate operations would not only greatly delay the procedure because of the necessarycommunication between the two systems but they are also hard to be realized since the timeof the free induction decay T ∗2 of the two NVs has to be long enough to carry out a selectivepulse (cf. sec. 1.4, [42]).
Naturally, the question arises whether dephasing can be performed quick enough as well. Ad-ditionally, the coherence time corresponding to the dephasing is much longer on the nitrogenthan on the NV’s electron spin (T ∗2,N T ∗2,NV ).However, the process of decoherence can be accelerated by applying a readout laser [84, 85]which has to be done in any case since the NV electron spin needs to be measured.Further acceleration is possible by applying a resonant magnetic field [86].
The possibility of selecting the resulting Bell state on qubits 1 and 2 will now be examined.
20
Chapter 2 Quantum teleportation using entangled NVs
2.3.2 Subselection of Bell states
Subselecting one of the four Bell states is proposed to be realized by correlating one of thestates to a single NV electron spin state.
Bell states effectively occupy the different energy levels of the nitrogen spin - NV electronspin system (Alice) since the change of basis (sec 2.2.2) yields∣∣∣Φ+
1,2
⟩⟨Φ+
1,2
∣∣∣ 7→ ∣∣01, 02
⟩ ⟨01, 02
∣∣ (2.20)∣∣∣Φ−1,2
⟩⟨Φ−
1,2
∣∣∣ 7→ ∣∣11, 02
⟩ ⟨11, 02
∣∣ (2.21)∣∣∣Ψ+1,2
⟩⟨Ψ+
1,2
∣∣∣ 7→ ∣∣01, 12
⟩ ⟨01, 12
∣∣ (2.22)∣∣∣Ψ−1,2
⟩⟨Ψ−
1,2
∣∣∣ 7→ ∣∣11, 12
⟩ ⟨11, 12
∣∣ (2.23)
Now however, we shall move away from the computational basis towards the physical situationin the NV. We remember that our first qubit (1) is given by the nitrogen spin and qubit 2 bythe electron spin.Since the NV’s spin is a qutrit, we need to restrict ourselves to two of three levels and chose|ms = −1〉 to represent
∣∣12
⟩. We leave |ms = 1〉 unused since it is the state of highest energy.
The level scheme of the NV and its nitrogen-15 is as portrayed below (fig 2.8).For the sake of good illustration of the energy levels and understanding of the subselection,the order of notation used for the physical states is |ms,NV ,mI,N 〉.E.g. | −1︸︷︷︸
ms,NV
, 0〉 =∣∣12, 01
⟩where ↑ denotes as 1 and ↓ denotes as 0 for the nuclear spin.
Figure 2.8: Level scheme of NVA ⊗ NA after change of basis towards the Bell basis
To select one of the Bell states it is now enough to exchange the populations between theseenergy levels in such a way that only the NV’s electron spin needs to be measured. This meansto isolate the desired state in the given ms manifold.
21
Chapter 2 Quantum teleportation using entangled NVs
In particular, selecting the∣∣∣Φ+
1,2
⟩state is realized by applying a π-pulse between |0, 0〉 and
|1, 0〉 to exchange the populations and altogether map∣∣∣Φ−
1,2
⟩to the |1, 0〉 state.
Eventually, the∣∣∣Φ+
1,2
⟩state is the only one remaining in ms,NV = 0.
Thus, if the experimental properties allow to do so, the first NV’s electron spin could be read
out. If the measurement outcome is merely ms=0 it is sure that the∣∣∣Φ+
1,2
⟩state is the one
that resulted after dephasing in the Bell basis.
The problem of reading out both the nitrogen nuclear spin and the electron spin of the NV atthe same time can consequently be solved through this subselection.
Effect on a superposition of Bell states
In an actual experiment, it is however unlikely for the system to dephase towards a pure Bellstate. A very general and more realistic state is constructed as follows:
ρ1,2,dephased = γ∣∣∣Φ+
1,2
⟩⟨Φ+
1,2
∣∣∣+ µ∣∣∣Φ−
1,2
⟩⟨Φ−
1,2
∣∣∣+ ν∣∣∣Ψ+
1,2
⟩⟨Ψ+
1,2
∣∣∣+ ξ∣∣∣Ψ−
1,2
⟩⟨Ψ−
1,2
∣∣∣ . (2.24)
The matrix of the necessary π pulse can be generated according to section 1.2.
Uπ = exp
(−i2· π · S0010
)(2.25)
the numbers indexing the transition generating matrix are to be understood like this: Sabcd isthe matrix for a transition from |a, b〉 to |c, d〉. This operation performs:
|−10〉 |−11〉 |00〉 |01〉 |10〉 |11〉|−10〉 ξ 0 0 0 0 0|−11〉 0 ν 0 0 0 0|00〉 0 0 µ 0 0 0|01〉 0 0 0 γ 0 0|10〉 0 0 0 0 0 0|11〉 0 0 0 0 0 0
Uπ−−→
ξ 0 0 0 0 00 ν 0 0 0 00 0 0 0 0 00 0 0 γ 0 00 0 0 0 µ 00 0 0 0 0 0
. (2.26)
We see, since the Bell state after dephasing is not a pure∣∣∣Φ+
1,2
⟩, it is not enough to simply
determine whether the NV is in the state ms,NV = 0 or not. It is necessary to read out alldiagonal elements of the density matrix. As we trace out the nitrogen subspace we get the result ν + ξ 0 0
0 γ 00 0 µ
(2.27)
The parameters can be read out, unsurprisingly limited by one degree of freedom (ν+ ξ) since
the trace is preserved. Because∣∣∣Φ+
1,2
⟩⟨Φ+
1,2
∣∣∣ is the part of (2.24) that is to be subselected, it is
desired that γ should get close to one. This will have to be examined experimentally, repeatingthe experiment until the factor γ is large enough.
22
Chapter 3
Experimental benefits and possibleerrors
3.1 Analysis of an entangled state
Quantum teleportation can be used to analyse the purity of a given Bell state. Specifically,after entangling two NV centers one can use a teleportation to check how much a state differsbetween a mixture of all Bell states (totally mixed state) and a certain pure Bell state.
As an example we will check the entangled state∣∣∣Φ+
2,3
⟩on the two NV centers and use an
initial state |Ψ1〉 = a∣∣01
⟩+ b
∣∣11
⟩on the nitrogen-15 of the first NV.
The following will show that it is possible to determine the parameter α if the initial entan-glement density matrix is not actually given by a pure Bell state but by the superposition
ρinit,2,3 = (3.1)
α∣∣∣Φ+
2,3
⟩⟨Φ+
2,3
∣∣∣+1
4(1− α)
(∣∣02, 03
⟩ ⟨02, 03
∣∣+∣∣02, 13
⟩ ⟨02, 13
∣∣+∣∣12, 03
⟩ ⟨12, 03
∣∣+∣∣12, 13
⟩ ⟨12, 13
∣∣)Where the second part of the sum represents the totally mixed state of all Bell states.
In the qubit subspace of the two NVs the matrix representation of this Dirac notation yields1+α
4 0 0 α2
0 1−α4 0 0
0 0 1−α4 0
α2 0 0 1+α
4
. (3.2)
As we now generalize this initial matrix to the actual 18 dimensional Hilbert space of the twoNVs and the 15N of the first NV we can run the teleportation scheme in this space. It is setup by the following subspaces:
1 : 15NA(2D) ⊗ 2 : NVA(3D) ⊗ 3 : NVB(3D).
Note that for this calculation, the standard teleportation scheme has been used (see 2.2.3)with controlled gates present at the end to demonstrate the general benefits of a teleportation.The proposed scheme for NVs (section 2.3) should yield the same results. This is becausethe subselection is implemented by a mere unitary pulse matrix which is traced out without
23
Chapter 3 Experimental benefits and possible errors
effect. Since the one Bell state that requires no corrections on the final state is subselected,the controlled X and Z gates are effect-less as well.The effects of subselection however need to be verified in experiments.
Since the matrix representations of the common gates are known and rather voluminous in the18-d space, let us go straight to the resulting density matrix after performing the teleportationscheme without measurement.Effects of the proposed subselection of Bell states will be discussed in section 3.1.4. We traceout the first two subspaces and get:
Tr1,2 (ρfinal) =
(12 + α
2 − αbb∗ −iαab∗
iαba∗ 12 −
α2 + αbb∗
)(3.3)
If we now manage to prepare an initial state |Ψ1〉 =∣∣01
⟩on the 15N with the parameter a=1
and b vanishing we could easily read out the parameter α since (3.3) then becomes:
Tr1,2 (ρfinal)∣∣∣a=1b=0
=
(12 + α
2 00 1
2 −α2
)(3.4)
Thus we are able to determine the quality of our initial entangled state. Its fidelity [87] being
Tr( ∣∣∣Φ+
2,3
⟩⟨Φ+
2,3
∣∣∣ · ρinit,2,3 ) =1
4(1 + 3α) (3.5)
with ρinit,2,3 defined in equation (3.1).
This demonstrates the possibility of analyzing a Bell state through a successful teleporta-tion.
To verify the fact that the scheme of teleportation has indeed been calculated correctly, thefinal matrix will be subjected to further analysis.
For a pure∣∣∣Φ+
2,3
⟩⟨Φ+
2,3
∣∣∣ with α = 1, (3.3) is varied to
Tr1,2 (ρfinal)∣∣∣α=1
=
(aa∗ −iab∗iba∗ bb∗
). (3.6)
Apart from a phase i obtained through the construction of the pulse gates, this is exactly thedensity matrix of the initial state |Ψ1〉 = a
∣∣01
⟩+ b
∣∣11
⟩, transferred to subspace 3 (NVB).
So, in the common language of teleportation, Bob has received Alice’s quantum state anddoes not have to perform further corrections.
24
Chapter 3 Experimental benefits and possible errors
3.1.1 Separation of gate errors and Bell state impurities
One useful benefit of a successful teleportation is the possibility to distinguish between phaseerrors in the quantum gate pulses and the impurities of the initial Bell state described in thepreceding section.The calculation of pulse errors can be done by constructing the pulse matrix according tosection 1.2 except for adding an additional phase.This means the Hadamard gate is constructed by
UH = e−i
2
(π2
+δ
) 0 −ii 0
. (3.7)
δ is the additional phase added to the standardπ
2phase.
In the same way an additional phase error ε is included in the construction of the CNOT gatewhere S is the generating matrix for the specific transition (see subsection 1.2.2)
UCNOT = e−i
2(π+ε)S1−110
(3.8)
The transition used here is the one between |1,−1〉 and |1, 0〉. The first number representsthe nitrogen nuclear spin where ↑ denotes as 1 and ↓ denotes as 0. The second number is thequantum number ms of the NV’s electron spin.This transition equates to
∣∣11, 02
⟩↔∣∣11, 12
⟩in the computational basis which is the driven
transition in the CNOT unitary matrix.It can be seen that the resulting matrix is a clear variation of the CNOT gate:
1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 −Sin
(ε2
)−iCos
(ε2
)0
0 0 0 −iCos(ε2
)−Sin
(ε2
)0
0 0 0 0 0 1
(3.9)
So these unitary operations are used instead of those of perfect quantum gates. Now, thesame initial state as in (3.1) is used and the teleportation scheme is carried out. The resultingdensity matrix on the subspace of the second NV’s electron spin is
Tr1,2 (ρfin,errors) =
(12 (1 + α− αbb∗(1 + Cos ε)) −iαb∗Cos ε2
(aCos δ + bSin δ Sin ε
2
)ibαCos ε2
(a∗Cos δ + b∗Sin δ Sin ε
2
)12 (1− α+ αbb∗(1 + Cos ε))
)(3.10)
A very appealing part of this result is the fact that the parameter α can be determined apartfrom the pulse errors.
25
Chapter 3 Experimental benefits and possible errors
Since if we set b=0 in the initial nitrogen state |Ψ1〉 = a∣∣01
⟩+ b
∣∣11
⟩the resulting matrix is
again (12 + α
2 00 1
2 −α2
)(3.11)
It is then possible to determine the CNOT gate error ε by varying a and b and reading outthe diagonal elements of the final density matrix.
The Hadamard phase gate error δ can be found through the off-diagonal elements in (3.10)when α and ε are known.
3.1.2 Decoherence caused by transversal relaxation (T2)
We will now examine a different, possibly more realistic ansatz for the initial state of the twoentangled electron spins of the NVs, namely
ρ2init,2,3 = α∣∣∣Φ+
2,3
⟩·⟨
Φ+2,3
∣∣∣+1
2(1− α)
(∣∣02, 03
⟩·⟨02, 03
∣∣+∣∣12, 13
⟩·⟨12, 13
∣∣) . (3.12)
In this density matrix the diagonal stays unaffected by the parameter α and it is only thecoherences that vary:
12 0 0 α
20 0 0 00 0 0 0α2 0 0 1
2
(3.13)
This matrix was chosen since a variation in the off-diagonal elements only shows the effectsof transversal relaxation. After performing the teleportation scheme as described above, thisresults in the final matrix
Tr1,2 (ρ2final) =
(aa∗ −iαab∗iαba∗ bb∗
). (3.14)
Again, we can easily determine α by varying the parameters a and b of the initial state.For example, if a=1 and b=0, α can be read out directly. This can be used to determine theeffects of transversal relaxation as well. The fidelity changes to
Tr( ∣∣∣Φ+
2,3
⟩⟨Φ+
2,3
∣∣∣ · ρ2init,2,3
)=
1 + α
2(3.15)
26
Chapter 3 Experimental benefits and possible errors
3.1.3 Spin-lattice relaxation (T1) errors
In addition to the gate errors and impurities discussed above, we can also vary the initialdensity matrix of the electron spins of NVA and NVB by adding errors on the diagonal. Theseare caused by spin-lattice relaxation corresponding to the relaxation time T1.
The errors β1...3 are limited by one degree of freedom since the trace of the density matrixstill is to be equal to one.
1+α4 + β1 0 0 α
2 − β4
0 1−α4 + β2 0 0
0 0 1−α4 + β3 0
α2 − β
∗4 0 0 1+α
4 − (β1 + β2 + β3)
(3.16)
After performing the teleportation scheme with gate errors as described before, the resultingmatrix on the electron spin of the second NV is composed of(
12 + (aa∗ − bb∗Cos[ε])
(α2 − (β2 + β3)
)−i (α− 2 Re [β4]) · f(a, b, ε, δ)
i (α− 2 Re [β4]) · f∗(a, b, ε, δ) 12 − (aa∗ − bb∗Cos[ε])
(α2 − (β2 + β3)
) ) (3.17)
with f(a, b, ε, δ) = b∗Cos(ε
2
)(aCos(δ) + bSin(δ) Sin
(ε2
)). (3.18)
It is interesting to observe that β2 and β3 cannot be distinguished since they only appear insummation with each other. Also, the clear determination of α is much more difficult sincethe factor only appears in combination with the spin-lattice errors.
If (α − 2 Re[β4]) and β2 + β3 can be seen as constant it is however still possible to deter-mine ε in the diagonal elements of (3.17) by varying a and b. As a result, δ can be found inthe off-diagonals and consequently, the determination of α is possible.
Effects of the spin-lattice process on the off-diagonal of the density matrix (β4) can oftenbe neglected when T2 << T1 (cf. [88]).An estimation of β4 can however be given by two general conditions on density matrices.Namely, it is non-negativity of the elements and the condition proven in Ref. [89]:
|ρij | ≤√ρii · ρjj (3.19)
Application of these criteria to the initial density matrix in (3.17), series expansion and equatingcoefficients yields:
3
4+β2 + β3
2≤ 2 Re[β4] ≤ α (3.20)
The off-diagonal elements only allow to find the factor (α − 2 Re[β4]) in a simple way if thegate errors are small enough. Without gate errors we simply have(
12 + (aa∗ − bb∗)
(α2 − β2 − β3
)−i (α− 2Re [β4]) ab∗
i (α− 2Re [β4]) a∗b 12 − (aa∗ − bb∗)
(α2 − β2 − β3
) ) . (3.21)
27
Chapter 3 Experimental benefits and possible errors
Nonetheless, the analysis of the Bell state does get a lot more difficult if both spin-latticeerrors and gate errors play a significant role.
3.1.4 Subselection - effects on Bell state analysis
If the impurities of the initial Bell state (i.e. factor α) are not negligible, another effect maygain significance. The subselection of Bell states constructed in section 2.3.2 is designed topostselect a pure Bell state. By maximizing factor γ in (2.27) its purity can at least be maxi-mized as well experimentally.
The key point here is the correlation that exists between the initial Bell state on the firsttwo qubits (Alice) and the resulting Bell state on qubits two and three (Bob). It determinesthe outcome of the final measurement as observed in the general discussion of teleportation(section 2.2) and more precisely in table 2.3, p. 14. In this table, the resulting wave functionsare listed for the case of an initial superposition of Bell states (on Alice’s qubits) and a mea-sured pure state (on Bob’s).
If the results of the previous section do not comply with experimental data that might begenerated, the results would have to be corrected according to this table.
To sum up, the analysis of the purity of a Bell state through teleportation in NVs is in-deed possible and can be done apart from phase errors in the gates. Even spin-lattice errorscan be dealt with even though the accuracy decreases with the rise of their significance.A further correction due to significant Bell state impurities is necessary according to thissection.
3.2 Measurement of the teleported state
Measurement of the teleported state is crucial to prove that a teleportation has been carriedout correctly.With the discussed method, it is not a problem to read out the state that has been teleportedindividually. Yet, the challenge is to determine at the same time which Bell state the remainingtwo-qubit system of the first NV and its nitrogen nuclear spin is in.
To prove that teleportation has taken place it is neccesary to either know if the subselec-tion of a Bell state (see sec 2.3.2) has been carried out properly or a direct Bell measurementwould need to take place.
28
Chapter 3 Experimental benefits and possible errors
3.2.1 Parallel NV pair readout
The experimental difficulty results from the limitation of distance between two NVs to a fewnm [42] when dipole-dipole interaction is used to generate entanglement.Therefore, it is not possible to address one NV spin individually by standard application ofthe read-out laser without the other one being affected.As mentioned before, it is however possible to use the STED-technology [71,72]. Notwithstand-ing the possibility of a readout at short distances, it has the disadvantage of being extensiveand expensive and is to be avoided.
Normally, when an NV electron spin is read out, several measures of the spin quantum numberare carried out while the readout-laser is switched on. However, for the present proposal asingle-shot readout [83] would be necessary (on NVA). It is not enough to apply filters to astandard measurement since entanglement needs to be restored after each measuring opera-tion.By performing a Bell measurement, entanglement is swapped from Alice’s and Bob’s qubits(2, 3) to Alice’s qubits (1, 2) only. A repetition of the entanglement generating sequence (on2, 3) thus needs to be performed.
The second NV can in contrast be read out by a standard measurement. However, the difficultyof distance remains.
In summary, if dipole-dipole entanglement should be used, readouts would have two be opti-mised in such a way that they could be performed on both NVs even with very short intervalsof time and distance.
3.2.2 The quest for a herald
With standard readout methods, it would also be possible to detect a successful teleportationif it could be made noticeable that a subselection was carried out properly. Then, the first NVwould not have to be measured and the teleportation scheme could be repeated. This wouldbe a do-until-success procedure.
Essentially the “herald” that is searched would be an indirect measurement of the NV state orthe final entangled state (15N-NV). It could be realized if a specific event (e.g. an entanglementwitness detection [49]) would be caused and detected once the NV center is in a given state.Ideally, it should cause the measurement of NVB to take place if and only if NVA is in theright state, namely the state which does not require further corrections on NVB.
The herald would thus represent an indirect Bell Measurement and could be used for a ‘con-trolled measurement’ on NVB.
29
Chapter 4
Outlook
4.1 A different method of entanglement
To increase the possibilities of metrology on both NVs separately after teleportation a differentway of entanglement could be investigated.
Namely, it is of particular interest to research the possibilities of optical entanglement be-tween two defects. A central idea here is the use of NV centers in optical cavities [90,91].It could even be possible to entangle two different NVs located in one cavitiy [92,93].
If entanglement can be generated in this way, a clear benefit on the experimental side isthat the time needed to search for a suitable pair would be greatly reduced. While it can takeseveral weeks to find two NVs close enough to each other to show a decent dipole-dipole cou-pling, optical entanglement might even be realized with NV centers in two different diamonds.
NV centers could then be used for quantum cryptography as well as integrated elements incavity networks. The diamond defects would then be part of algorithms in quantum computingand the simulation of complex interaction systems [94].
4.2 Optical entanglement and single-photon qubits
The variation of Bell measurement proposed to be realized by dephasing and subselection(chap. 2.3) can also be applied to different qubit situations.
For example, instead of teleporting to a second NV, it is of particular interest to use a trans-ferable target qubit. The most flexible and most commonly used of these is a single photon.Entanglement between an NV electron spin and a single photon has been realized recently [22].Therefore, it is realistic to envision the teleportation of a quantum state prepared on an NV’snitrogen nuclear spin to a single photon polarisation.
So, a quantum state initialized on a diamond defect could be transferred optically over largedistances. An illustration of this teleportation scheme is given on the following page.
30
Chapter 4 Outlook
Figure 4.1: Teleportation from an NV’s nitrogen nuclear spin to a single photon.NV illustration used with permission [5], modified
Teleporting onto the polarization state of a single photon has clear advantages to using an-other NV as receiver. Information can be transferred at the speed of light, for example throughoptical fibers.
A scheme with reversed roles is subject to current research, with the photon being the senderof information and the NV providing a receiving quibt system.
Thus, NV centers could eventually be used both as receivers and as senders in the extremlysecure transfer of information through optical quantum crytography.
31
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38
Acknowledgements
I would like to use the opportunity to thank Professor Wrachtrup for allowing me to workat the 3rd Institute of Physics in Stuttgart as a scientific assistant in the fascinating area ofquantum information before his professional supervision of the work leading to this thesis.In the same way, I’d like to acknowledge the contributions, discussions and corrections of Flo-rian Rempp who was my main advisor and taught me - among many other things - what itmeans to do research as a theoretician at an experimental institute.
Furthermore, I express great thanks to Dr. Friedemann Reinhardt, Philipp Neumann, Chris-tian Burk, Julia Michl, Konstantin Schukraft, Seyed Ali Momenzadeh, Dr Roman Kolesov,Florian Dolde, Ingmar Jacobi, Steffen Steinert, Matthias Nitsche, Gerald Waldherr, Dr PhilipHemmer (Texas A&M University) and Dr Almut Beige (University of Leeds) for fruitful dis-cussions and teaching.
The institute is not only a place of excellent research but the atmosphere has always beenvery encouraging and inspiring and I particularly recognize anyone who spent time with mein deep or shallow conversation or while playing soccer and all the members of the amazingfoosball crew.
My appreciation also goes to Jean Francois Podevin for allowing me to use his illustrations aswell as to Jose Luis Gomez and Francisco Delgado whose software “Quantum” - a Mathematicaadd-on - was very useful for this work.
I am also very grateful to my family for their continuous support in many areas of my life.Moreover, I thank Debbie, my awesome girlfriend, for her encouragement and love.
Finally, I wish to thank my Creator whose work I believe to study.
A