How much entanglement is lost along a channel? TQC 2008, 30 Jan 2008

Transcription

1 How much entanglement is lost along a channel? Francesco Buscemi, TQC 2008, 30 Jan 2008

2 Overview Review of some known results about approximate quantum error correction Generalization of the theory to other entanglement measures Interesting by-product: new sufficient condition for distillability in terms of entanglement of formation

3 What is noise? ρ Q E Q ρ Q We say that an evolution is noisy, if some information is irreversibly lost.

4 What is noise? ρ Q E Q ρ Q We say that an evolution is noisy, if some information is irreversibly lost. Then we say that, a quantum channel is noisy on a subspace, if it cannot be inverted with fidelity 1 and probability 1 on such a subspace. Superpositions and entanglement have to be perfectly recoverable! (cf. decoherence processes, classically noiseless but non recoverable)

5 How to measure noisiness? Quantum information theory suggests us to purify the input, letting the noise act on one branch of the purification: Ψ RQ E Q } ρrq

6 How to measure noisiness? Quantum information theory suggests us to purify the input, letting the noise act on one branch of the purification: Ψ RQ E Q } ρrq Then we compare the pure bipartite input with the mixed bipartite output by calculating the so-called entanglement fidelity F e (ρ Q, E Q ) := Ψ RQ ρ RQ Ψ RQ

7 How to measure noisiness? Quantum information theory suggests us to purify the input, letting the noise act on one branch of the purification: Ψ RQ E Q } ρrq Then we compare the pure bipartite input with the mixed bipartite output by calculating the so-called entanglement fidelity F e (ρ Q, E Q ) := Ψ RQ ρ RQ Ψ RQ This quantity however does not tell us anything about noisiness, as related to irreversibility: it only quantifies how close the given channel E Q is with respect to the identity channel id Q

8 Approximate correction So, how noisy is a given channel? By using the idea of entanglement fidelity, we can define the corrected entanglement fidelity as F c (ρ Q, E Q ) := max R Q ΨRQ ρ RQ Ψ RQ } Ψ RQ E Q R Q ρ RQ

9 Approximate correction So, how noisy is a given channel? By using the idea of entanglement fidelity, we can define the corrected entanglement fidelity as F c (ρ Q, E Q ) := max R Q ΨRQ ρ RQ Ψ RQ } Ψ RQ E Q R Q ρ RQ Still, the maximization over all possible corrections is not handy at all... Can we overcome this technical problem?

10 Coherent information loss } Ψ RQ E Q ρ RQ I A B coh (ρ AB ) := S(ρ B ) S(ρ AB )

11 Coherent information loss } Ψ RQ E Q ρ RQ I R Q coh (Ψ RQ ) = S(ρ Q ) I R Q coh (ρ RQ ) I A B coh (ρ AB ) := S(ρ B ) S(ρ AB )

12 Coherent information loss } Ψ RQ E Q ρ RQ I R Q coh (Ψ RQ ) = S(ρ Q ) I R Q coh (ρ RQ ) I A B coh (ρ AB ) := S(ρ B ) S(ρ AB ) The following bound, due to Schumacher and Westmoreland (2002), is exactly what we are searching for: Let δ coh (ρ RQ ) := S(ρ Q ) I R Q coh (ρ RQ ) be the coherent information loss; then there exists a correction such that F c (ρ Q, E Q ) 1 2δ coh (ρ RQ )

13 Opposite direction: quantum Fano inequality On the other hand, we have that the converse is also true δ coh (ρ RQ ) f ( 1 F c (ρ Q, E Q ) ) 1.0 f x x

14 What is coherent information? It is NOT an entanglement monotone. If Icoh A B (ρ AB ) > 0 then the state is entangled; if the state is separable I A B. coh (ρ AB ) 0 It is related with the quantum capacity of channels. It is a lower bound on the one-way distillable entanglement (hashing ineq.) E A B D (ρ AB ) Icoh A B (ρ AB )

15 A natural question... A channel is approximately invertible if and only if the loss of coherent information is small. What s special about coherent information? For pure states, it is an entanglement measure. Why a generalization to other entanglement measures is not straightforward?

16 A natural question... A channel is approximately invertible if and only if the loss of coherent information is small. What s special about coherent information? For pure states, it is an entanglement measure. Why a generalization to other entanglement measures is not straightforward? The reason lies at the roots of entanglement theory...

17 Very quickly: some entanglement measures max{icoh A B (ρ AB ), 0} ED A B (ρ AB ) E (ρ AB ) E F (ρ AB ) min{s(ρ A ), S(ρ B )}

18 Very quickly: some entanglement measures max{icoh A B (ρ AB ), 0} ED A B (ρ AB ) E (ρ AB ) E F (ρ AB ) min{s(ρ A ), S(ρ B )} (one-way) distillable entanglement

19 Very quickly: some entanglement measures max{icoh A B (ρ AB ), 0} ED A B (ρ AB ) E (ρ AB ) E F (ρ AB ) min{s(ρ A ), S(ρ B )} (one-way) distillable entanglement entanglement of formation

20 Very quickly: some entanglement measures max{icoh A B (ρ AB ), 0} ED A B (ρ AB ) E (ρ AB ) E F (ρ AB ) min{s(ρ A ), S(ρ B )} (one-way) distillable entanglement entanglement of formation Notice: we are considering entanglement measures under LOCC restriction (distant laboratories paradigm)

21 Typical behavior for pure states, measures coincide I A B coh (Ψ AB ) = S(ρ B ) = S(ρ A ) for a randomly chosen mixed state, as dimension increases E A B D (ρ AB ) 0 E F (ρ AB ) min{s(ρ A ), S(ρ B )}

22 Typical behavior for pure states, measures coincide I A B coh (Ψ AB ) = S(ρ B ) = S(ρ A ) for a randomly chosen mixed state, as dimension increases E A B D (ρ AB ) 0 E F (ρ AB ) min{s(ρ A ), S(ρ B )} } 0 S(ρ A )} max{icoh A B (ρ AB ), 0} ED A B (ρ AB ) E (ρ AB ) E F (ρ AB ) min{s(ρ A ), S(ρ B )}

23 Typical behavior for pure states, measures coincide I A B coh (Ψ AB ) = S(ρ B ) = S(ρ A ) for a randomly chosen mixed state, as dimension increases E A B D (ρ AB ) 0 E F (ρ AB ) min{s(ρ A ), S(ρ B )} } 0 S(ρ A )} max{icoh A B (ρ AB ), 0} ED A B (ρ AB ) E (ρ AB ) E F (ρ AB ) min{s(ρ A ), S(ρ B )} irreversibility gap under LOCC!!

24 Other loss functions: Entanglement of Formation Ψ RQ E Q } ρrq F c (ρ Q, E Q ) 1 2(2d R d Q 1) 2 δ F (ρ RQ ) where formation loss δ F (ρ RQ ) := S(ρ Q ) E F (ρ RQ ) is the entanglement of

25 Other loss functions: Entanglement of Formation Ψ RQ E Q } ρrq F c (ρ Q, E Q ) 1 2(2d R d Q 1) 2 δ F (ρ RQ ) where formation loss δ F (ρ RQ ) := S(ρ Q ) E F (ρ RQ ) F c (ρ Q, E Q ) 1 is the entanglement of 2δ coh (ρ RQ )

26 Squashed measures loss Let us now consider another important threshold for entanglement measures, that is, half of the quantum mutual information. If a measure satisfies E (ρ AB ) IA:B (ρ AB ) 2 where I A:B (ρ AB ) := S(ρ A ) + S(ρ B ) S(ρ AB ), then F c (ρ Q, E Q ) 1 2 δ (ρ RQ ) where function δ (ρ RQ ) := S(ρ Q ) E (ρ RQ ) is the corresponding loss

27 Squashed measures loss Let us now consider another important threshold for entanglement measures, that is, half of the quantum mutual information. If a measure satisfies E (ρ AB ) IA:B (ρ AB ) 2 where I A:B (ρ AB ) := S(ρ A ) + S(ρ B ) S(ρ AB ), then F c (ρ Q, E Q ) 1 2 δ (ρ RQ ) where function δ (ρ RQ ) := S(ρ Q ) E (ρ RQ ) satisfied by squashed ent. and distillable ent. is the corresponding loss

28 Comparing the bounds Up to now, we dealt with three different bounds: F c (ρ Q, E Q ) 1 2δ coh (ρ RQ ) F c (ρ Q, E Q ) 1 2 δ (ρ RQ ) F c (ρ Q, E Q ) 1 2(2d R d Q 1) 2 δ F (ρ RQ )

29 Comparing the bounds Up to now, we dealt with three different bounds: F c (ρ Q, E Q ) 1 2δ coh (ρ RQ ) F c (ρ Q, E Q ) 1 2 δ (ρ RQ ) F c (ρ Q, E Q ) 1 2(2d R d Q 1) 2 δ F (ρ RQ ) On the other hand it holds that: δ coh (ρ RQ ) f ( 1 F c (ρ Q, E Q ) )

30 By putting pieces together ( ) δ coh (ρ AB ) f 2(2d A d B 1) 2 δ F (ρ AB ) ( ) δ coh (ρ AB ) f 4δ (ρ AB ) In a sense, we obtained an analytical lower bound on coherent information (and hence to distillable entanglement) in terms of the entanglement of formation. The strong dependence of the first inequality on the dimensions of the subsystems makes it possible the previously mentioned irreversibility gap.

31 By putting pieces together ( ) δ coh (ρ AB ) f Surprisingly, this is the REAL dimension of the set of bipartite states δ coh (ρ AB ) f 2(2d A d B 1) 2 δ F (ρ AB ) ( 4δ (ρ AB ) In a sense, we obtained an analytical lower bound on coherent information (and hence to distillable entanglement) in terms of the entanglement of formation. The strong dependence of the first inequality on the dimensions of the subsystems makes it possible the previously mentioned irreversibility gap. )

32 f 1.0 Example: Plot for twoqutrit states

33 Coherent information and distillable entanglement lie in the shaded area. f 1.0 Example: Plot for twoqutrit states f(e F ) I A B coh E A B D E F 0.2

34 Coherent information and distillable entanglement lie in the shaded area. f 1.0 Example: Plot for twoqutrit states f(e F ) I A B coh E A B D E F Slope exaggerated to remind that Ed<=Ef

35 Conclusions

36 Conclusions We proved that many inequivalent entanglement measures lead to equivalent conditions for approximate quantum error correction

37 Conclusions We proved that many inequivalent entanglement measures lead to equivalent conditions for approximate quantum error correction We found, as a by-product, a new sufficient condition for distillability in terms of entanglement of formation

38 Conclusions Thanks to JST and ERATO- SORST project We proved that many inequivalent entanglement measures lead to equivalent conditions for approximate quantum error correction We found, as a by-product, a new sufficient condition for distillability in terms of entanglement of formation