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Introduction to
Quantum Information ProcessingQIC 710 / CS 768 / PH 767 / CO 681 / AM 871
Jon Yard
QNC 3126
http://math.uwaterloo.ca/~jyard/qic710
Lecture 20 (2017)
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Entanglement
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Separable statesA density matrix ๐๐ด๐ต is separable if there exist
probabilities ๐(๐ฅ) and density matrices ๐๐ฅ๐ด, ๐๐ฅ
๐ต such that
๐๐ด๐ต =
๐ฅ
๐ ๐ฅ ๐๐ฅ๐ด โ๐๐ฅ
๐ต .
If ๐๐ด๐ต is not separable, then it is called entangled.
Note: if ๐๐ด๐ต is separable, exists a decomposition with
๐๐ฅ๐ด = ๐๐ฅ โจ๐๐ฅศ
๐ด, ๐๐ฅ๐ต = ๐๐ฅ โจ๐๐ฅศ
๐ต.
Operational meaning: separable states can be prepared
starting with only classical correlations.
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Separable?
Theorem [Horodeckis โ96]: ๐๐ด๐ต is entangled iff there
exists a positive (but not completely positive) linear map
๐ on โ๐ร๐ such that (๐ โ ๐๐)(๐๐ด๐ต) is not positive
semidefinite.
We have already seen examples of positive-but-not-
completely positive maps, such asโฆ
Proof (Easy direction โ only if): Let ๐ be any positive map. If
๐๐ด๐ต =
๐ฅ
๐ ๐ฅ ๐๐ฅ๐ด โ๐๐ฅ
๐ต
is a separable density matrix, then
๐ฅ
๐ ๐ฅ ๐(๐๐ฅ๐ด) โ ๐๐ฅ
๐ต
is still positive semidefinite. Interpretation: every entangled
state is broken by some non-physical positive map.
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Separable?Example: The Werner state
๐๐ด๐ต = 1 โ ๐๐+ โจ๐+ศ + ศ๐โโฉโจ๐โศ + ๐+ โจ๐+ศ
3+ ๐ ๐โ โจ๐โศ
has a Positive Partial Transpose (PPT) ๐ โ ๐๐ ๐๐ด๐ต โฅ 0
iff ๐ โค1
2, where ๐ is the transpose map ๐ ๐ = ๐๐.
It turns out that the PPT test is sufficient to decide
entanglement, i.e. the Werner state is entangled iff ๐ > 1/2.
In fact, the PPT test is sufficient to decide whether an
arbitrary 2 ร 2 or 2 ร 3 density matrix is entangled.
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Separable?
Fundamental problem: Given a description of ๐๐ด๐ต, (i.e.
as a ๐2 ร ๐2 matrix), determine whether it is separable or
entangled.
Bad news: This problem is NP-hard [Gurvits โ02].
Good news: There exists [BCYโ12] an efficient
(quasipolynomial-time exp ๐โ2๐(log ๐ 2) algorithm for
deciding this given a promise that ๐๐ด๐ต is either separable
or a constant distance (in โ โ2-norm) from separable.
โ๐ โ ๐โ2 = Tr ๐ โ ๐ 2
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How entangled?(brief)
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Entanglement measures
Some nice properties for such a measure to satisfy:
1) Invariant under local unitaries
2) Non-increasing under Local Operations and Classical
Communication (LOCC)
3) Monogamous
4) Additive
5) Faithful
An entanglement measure is a function ๐ธ ๐๐ด๐ต on bipartite
density matrices ๐๐ด๐ต that quantifies, in one way or another,
the amount of bipartite entanglement in ๐๐ด๐ต.
Last time, we saw two examples for pure states:
โข Schmidt rank
โข Entanglement entropy
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Monogamy of entanglement
Many nice entanglement measures are monogamous:
The more ๐ด is entangled with ๐ต, the less it can be
entangled with ๐ถ.
๐ธ(๐๐ด๐ต1) + ๐ธ(๐๐ด๐ต2) โค ๐ธ(๐๐ด๐ต1๐ต2).Implies that quantum correlations cannot be shared.
Application of this idea: Quantum Key Distribution.
Extreme example: ๐๐ด๐ต1๐ต2 = ๐ โจ๐ศ๐ด๐ต1 โ๐๐ต2, where ๐ = 00 + 11 is a Bell state
1 + 0 โค 1
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Entanglement of formation
Entanglement of formation: How much entanglement
does it take, on average, to create a single copy of ๐๐ด๐ต?
๐ธ๐น(๐๐ด๐ต) = min
๐ ๐ฅ , ๐๐ฅ๐ด๐ต
๐ฅ
๐ ๐ฅ ๐ ๐๐ฅ๐ด :
๐ฅ
๐ ๐ฅ ๐๐ฅ ๐๐ฅ๐ด๐ต = ๐๐ด๐ต
Faithful, not monogamous, not additiveโฆ
๐ธ๐ถ ๐๐ด๐ต = lim๐โโ
1
๐๐ธ๐น ๐๐ด๐ต
โ๐ โค ๐ธ๐น(๐๐ด๐ต)
Entanglement cost: how much entanglement does it
take, per copy, to create many copies of ๐๐ด๐ต?
How much entanglement does it take to make ๐๐ด๐ต using
LOCC?
Shor โ01, Hastings โ08: Can have ๐ธ๐ถ < ๐ธ๐น (explicit example?).
Faithful, not monogamous. Additive?
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Distillable entanglementHow much entanglement can be extracted from ๐๐ด๐ต, in
the limit of many copies?does it take, on average, to
create a single copy of ๐๐ด๐ต?
๐ธ๐ท(๐๐ด๐ต) = the largest rate ๐ such that, by local operations
and classical communication, Alice and Bob can produce
๐๐ Bell states (ebits)
0โฉศ0 + 1 ศ1โฉ ๐๐ =
๐ฅโ 0,1 ๐๐
๐ฅ ศ๐ฅโฉ
from ๐๐ด๐ตโ๐
, with vanishing errors in the limit as ๐ โ โ.
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Bound entanglementThere exist โbound entangled statesโ with ๐ธ๐ท < ๐ธ๐น[Horodeckis โ97]
Analogous to bound energy in thermodynamics.
Has ๐ธ๐ท = 0 since it is PPT. But it is entangled.
So ๐ธ๐ท not faithful.
Big open question: do there exist NPT bound entangled states?
Would imply ๐ธ๐ท not additive.
0 < ๐ < 1
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Squashed entanglement๐ธ๐ ๐ ๐๐ด๐ต = inf
๐๐ด๐ต๐ถ๐ผ(๐ด; ๐ตศ๐ถ)
It is monogamous, additive and faithful!
Easy to show that ๐ธ๐ ๐ = 0 on separable states.
We donโt know how to compute itโฆ
๐ผ ๐ด; ๐ต ๐ถ
Conditional mutual information
๐ผ ๐ด; ๐ต ๐ถ = ๐ป ๐ด๐ถ + ๐ป ๐ต๐ถ โ ๐ป ๐ถ โ ๐ป(๐ด๐ต๐ถ)Satisfies strong subadditivity ๐ผ ๐ด; ๐ต ๐ถ โฅ 0 (not easy proof)
Generalizes mutual information
๐ผ ๐ด; ๐ต = ๐ ๐ด + ๐ ๐ต โ ๐(๐ด๐ต)
State redistribution problem
๐ ๐ด๐ต๐ถ๐ท
State redistribution problem
๐ ๐ด๐ต๐ถ๐ท
Cost of state redistribution
[Devetak & Y. โ PRLโ08]
[Y. & Devetak โ IEEE TIT โ09]
First known operational
interpretation of
quantum conditional
mutual information
๐ผ ๐ถ; ๐ท ๐ต = ๐ป ๐ต๐ถ + ๐ป ๐ต๐ทโ๐ป ๐ต๐ถ๐ท โ ๐ป(๐ต)
๐ผ ๐ถ; ๐ท ๐ต ๐ผ(๐ถ; ๐ต)
๐ป(๐ถศ๐ต)
Cost of state redistribution
[Devetak & Y. โ PRLโ08]
[Y. & Devetak โ IEEE TIT โ09]
๐ผ(๐ถ; ๐ต)
๐ป(๐ถศ๐ต)
First known operational
interpretation of
quantum conditional
mutual information
๐ผ ๐ถ; ๐ท ๐ต = ๐ป ๐ต๐ถ + ๐ป ๐ต๐ทโ๐ป ๐ต๐ถ๐ท โ ๐ป(๐ต)
๐ผ ๐ถ; ๐ท ๐ต
Cost of state redistribution
[Devetak & Y. โ PRLโ08]
[Y. & Devetak โ IEEE TIT โ09]
๐ผ(๐ถ; ๐ต)
๐ป(๐ถศ๐ต)
First known operational
interpretation of
quantum conditional
mutual information
๐ผ ๐ถ; ๐ท ๐ต = ๐ป ๐ต๐ถ + ๐ป ๐ต๐ทโ๐ป ๐ต๐ถ๐ท โ ๐ป(๐ต)
๐ผ ๐ถ; ๐ท ๐ต
Optimal protocol for state
redistribution
Explains the identity ๐
2๐ผ ๐ถ;๐ท ๐ต =๐
2๐ผ(๐ถ; ๐ทศ๐ด)
Simple proof: decoupling via random unitaries:[Oppenheim โ arXiv:0805.1065]
achieves different 1-shot quantities.
Applications:
โข Proof that ๐ธ๐ ๐ is faithful.
โข Proof of existence of quasipolynomial-time
algorithm for deciding separability.
โข Communication complexity
Letโs see how to prove a special case:
To emphasize the role of ๐ท as a reference
system, relabel ๐ท โ ๐
State merging
State merging
State merging
State merging
State merging