entangling without entanglement t. cubitt, f. verstraete, w. dür, j. i. cirac “separable state...
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EntanglingEntangling withoutwithout EntanglementEntanglement
T. Cubitt, F. Verstraete, W. Dür, J. I. Cirac
“Separable State can be used to Distribute Entanglement”(to appear in PRL vol. 91, issue 3)
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Send separable ancilla particle
Send entangled ancilla particle
Local Operations & Classical Communication
Entangling twoEntangling two distant particles distant particles
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C
A B
Define “separable”?Define “separable”?
What does “separable” mean for the messenger?
C
A B=
Implies separability tracing out one particle:
Choose strongest possible meaning:
B
C
A
For mixed states:
For pure states:
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Don’t Entangle the MessengerDon’t Entangle the Messenger
Alice and Bob prepare initial state:
Bob applies CNOT to B and a:
Alice applies CNOT to A and a:
10 0
3 a
21 1 1
3 AB a
1
3 GHZ GHZ
001 010 1011 1101
1
6
1
, , 0
, , , ,ijki j k
i j k i j k
3
0
1
0
1, ,0 , ,0
6
1, ,1 , ,1
6
k k k kk
j
j j j j
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If we think of rates of entanglement generation as ‘flows’…
Ba
A
How does entanglement ‘flow’?How does entanglement ‘flow’?
Chain with nearest neighbour interactions
BA
AaHBaH
a
…can entanglement ‘flow’ be 0 between & and & , yet be non-zero between and ??
AB
aa A B
B
Can and be entangled without entangling the ancilla ??
BAa
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PPhysical relevancehysical relevance
Interactions are often mediated by an ‘ancilla’ particle
Ion traps: interactions between ions mediated by phonons
Cavity QED: interactions between atoms mediated by photons in the cavity
Fundamentally, all interactions are mediated by the gauge bosons of particle physics
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Evolution can be discretized by Trotter formula
Continuous and discrete casesContinuous and discrete cases
Aa Bait H HU t e
Aa Ba Aa BaU U U U
/ 2 / / 2lim Aa Ba AanitH n itH n itH n
ne e e
BA
AaHBaH
a
Immediately gives a discrete procedure.
Continuous case is stronger than discrete case.
>
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So AB A B t A B t A B
Pure states:Pure states: impossibility proofimpossibility proof
Start with separable state BaA
Condition on separability of ancilla is then aBAaABa HHt 1 BaA
ABtBA ata
Multiplying by gives A B 0ABA B
A t A B t B
Evolve under for an infinitesimal time-step:
Aa BaH H H 2( )Aa aBU t i t H H O t 1
SeparableSeparable
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BaA000
As achieve effect for e.g. initial state . 0
Expand in perturbation theory:
Don’t entangle the mediatorDon’t entangle the mediator
BA
AaHBaH
a0
1
2
0
1
0
1
BxAxAB aaaaH ˆˆ1ˆˆ
2 40 AB aH H H O 1
Trivial?Trivial?
Want ancilla to really be separable, not just arbitrarily small entanglement as 0 t
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After evolution ( ): 2 40 AB aH H H O 1
Just add a dash of noiseJust add a dash of noise
4( )AB AB a ABat O t 1 1
Use mixed initial state:
BA
AaHBaH
a
Choose large enough to destroy all entanglement with . (States near maximally mixed state are separable).
a
Separable in ( )- aA B Entangled in ( )- aA B
Choose small enough such that mixing does not destroy - entanglement.BA
00 00 1a ABa 1
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EEntanglement properties of ntanglement properties of partitionspartitions
C
A B
C
A B
For mixed states, partitions are independent
C
A B& &
A
C
B
For pure states, entanglement properties of bipartite partitions are inter-dependent
C
A B
C
A B&
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C
AB
C
AB
TheoreticalTheoretical insightinsight
Alice and Bob prepare initial state:
Bob applies CNOT to B and a:
Alice applies CNOT to A and a:
C
AB
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ConclusionsConclusions “Wacky but Lovely” – Seth “Wacky but Lovely” – Seth LloydLloyd
Upsets notions of entanglement flow
Forces us to abandon any intuitive ideas of entanglement being sent through a quantum channel
(At least for general – i.e. mixed – states…)
Separable states can be used to distribute entanglement