what is being plotted?
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What is being plotted?. Answer: Number of papers with “quantum entanglement” in title or abstract. N. D. Mermin, Phys. Rev. Lett. (1990). Entanglement is a physical resource : Bennett, DiVincenzo, Smolin and Wootters, Phys. Rev. A (November, 1996). Entanglement. Michael A. Nielsen - PowerPoint PPT PresentationTRANSCRIPT
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What is being plotted?
'90 '91 '92 '93 '94 '95 '96 '97 '98 '99 '00
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Answer: Number of paperswith “quantum entanglement”
in title or abstract
N. D. Mermin, Phys. Rev. Lett. (1990)
Entanglement is a physical resource:Bennett, DiVincenzo, Smolin and Wootters,Phys. Rev. A (November, 1996)
Michael A. Nielsen
University of Queensland
Entanglement
Goals: 1. To explain why we regard entanglement as a
physical resource, like energy or mass.2. To explain how entanglement can be quantified.3. To explain how the quantitative theory of
entanglement can be used to gain insight into quantum
information processing, and into other physical processes.
Entanglement revisitedAlice Bob
00 112
a b
Schroedinger (1935): “I would not call[entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”
Entanglement and classicalityBell (1964) and Aspect (1982): Entanglement can beused to show that no “locally realistic” (that is, classical) theory of the world is possible.
Further reading: Asher Peres, “Quantum theory: concepts and methods”, Kluwer (1993).
Using entanglement to do stuffentanglement-based quantum cryptographysuperdense coding
quantum teleportation
Entanglement is a useful resource that can be used to accomplish tasks that would otherwise bedifficult or impossible.
quantum computing
Given an information-processing goal, we canalways ask “What would I gain by throwing someentanglement into the problem?”
Representation independence of entanglement
Properties independent ofphysical representation
00 112
Electron spin: 2
HH VVPhoton polarization: 2
etcetera
Qualitative equivalence of different entangled states
1 1 12 22 equivalen2 copies of 00 11 22 is to
00 113 copies of
t
!2
Summary1. Entanglement is not classical.2. Entanglement is a resource that can be used to do interesting things.3. Entanglement has properties independent of physical representation.4. Different entangled states are qualitatively equivalent to one another.
Can we develop a quantitative theory of entanglement?
What might we get out of such a theory?
(Figure taken from Boston University’s 1999 PY105 class.)
Thermodynamics is a setof high-level principlesgoverning the behaviourof energy.
We hope that the theoryof entanglement will be asimilarly powerful set of high-level principles governingentanglement.
How massive is a given object?
How massive is a given object?
How massive is a given object?
How massive is a given object?
number of standard massesMass limnumber of copies of object
A standard unit for entanglementAlice Bob
00 112
Question: Why use the Bell state as the standard unit?Answer: “Because it’s there” – we’ll do so because it’sclearly an important state, and in the spirit of exploration.Answer: Later on, we’ll see that choosing the Bell stateleads to some interesting connections with other problems.
How can we “balance” entanglement?
m n
Entanglement limmn
What it means for one state to be“at least as entangled” as another
Alice Bob
Hello Bob … Hello Alice
Alice Bob
o 1o
Entanglement Entanglement
What it means for one state to be“at least as entangled” as another
can be converted to by an ("local operationsand classical communication") prot
LOCl.
Coco
Alice Bob
3 100 114 4
o 1o
00 11 (50% of the time)2
An example of an LOCC protocol
Such a protocol will let us copies of 3 100 11 into Bell pairs
dis
4 2
ll
4
i.
t nn
How the protocol works
Consider the circuit
3 100 114 4
0
0 1 U
measure m
'm
2
0
Show that2Pr 0 1 and
Exerc 0 1 .3
ise:
3
Find a circuit of controlled-nots and single-qubitunitaries to implEx
emercise:
ent .U
1 23300 00 11
01 01UU
Consider the circuit
3 100 114 4
0
0 1 U
measure m
'm
Distillation procedure:
0U
measure 03 100 114 4
03/ 4 1' 00 1143
00 11 1 w. p. 22
Thus copies of Bell pairs2nn
1 23300 00 11
01 01UU
How the protocol works
Back to balancing entanglement
m
n
m
n
Not possible in general!
m n
Entanglement limmn
LOCC LOCC
How to balance entanglement
m
n
(1 )m
n
For any 0 and suffi ciently large and :m n
Entanglement limmn
E( ) Bell statesn n
E( ) is the numberof Bell states
maximal
distille that can be
, per copy od f .
E( ) Bell statesn
n
Bell statesn k Show that by local operations
and classical communication, Alice and Bob can't increase the number of Bellpairs the
Exercis
y sh
e:
are.
How much entanglement?Alice Bob
trA B trAB
E A BS S
That is, Bell states.An nS
ExampleAlice Bob
cos 00 sin 11
2Suppose .3AS
BellBell
Schumacher
compressteleport
Bobcompletes
teleportation
0
Schumacherdecompress
How to go f rom Bell states to copies of , by LOCC
AnSn
Entanglement can only decrease underlocal operations and classical communication
n
n
E( ) Bell statesn
E( ) E( )
An entangled analogue to thesecond law of thermodynamics
E( ) is the numberof Bell states
maximal
distille that can be
, per copy od f .
Approximate teleportationAlice Bob
00 112
Approximate teleportationAlice Bob
01
The original teleportation protocolAlice Bob
01
Teleporting entanglementAlice Bob
Teleporting entanglementAlice Bob
01 01
The ability to teleport anarbitrary state implies the ability
to teleport entanglement
Approximate teleportationAlice Bob
E ebits (E < 1)1 ebit
Total initial entanglement between Alice and Bob at most E ebits.
If Alice and Bob only do local operations and classical communicationthen the final entanglement between their systems cannot be morethan when it started.
Approximate teleportationAlice Bob
At most E ebits
Since the final entanglement is not 1 ebit, some states must beimperfectly teleported.
Approximate teleportationAlice Bob
E ebits (E < 1)
min minF
min11 13F E
Back to the “Why Bell states?” question
Alice and Bob share a large number ofcopies of , and can do unlimited classical communication,as well as arbitrary operations on th
Phys
eir
ical resourc
local sys
:
.
e
tems
Alice wI nf ormat ants to ion processing t send qubits to ask: Bob. The qubit communication should take
place with fi deCriterion f o
lity approacr succ
hing ess:
one.How many copies of are needed to reliably communicatea qubit f rom Alice to Bob?
Teleportation: shared entanglement and classical communicationenables the communication of qubits.
max # of qubits that can be communicatedEntanglement copy of
Alice Bob