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Towards a one-shot entanglement theory
Francesco Buscemi and Nilanjana Datta
Beyond i.i.d. in information theory,"
Univers...
Part One:
Introduction
Resource theory of (bipartite) entanglement
Entanglement is useful (quantum information processing) but expensive
(difficult...
study of entanglement as a resource
raw resources: bipartite quantum systems (in pure and/or mixed
states)
processing: loc...
Asymptotic manipulation of (bipartite) quantum
correlations
ρAB ⊗ ρAB ⊗ · · · ⊗ ρAB
Min
L∈LOCC
−−−−−−→ σA B ⊗ · · · σA B
N...
Asymptotic entanglement distillation and dilution
Entanglement distillation
ρAB ⊗ · · · ⊗ ρAB
Min
L∈LOCC
−−−−−−→ Ψ−
A B ⊗ ...
Criticisms to this approach
The asymptotic framework is operational but not practical, for two reasons:
asymptotic achieva...
The one-shot case
One-shot entanglement distillation:
ρAB
L∈LOCC
−−−−−−→ Ψ−
A B ⊗ · · · Ψ−
A B
Nmax(ρAB)
.
One-shot entang...
Allowing for finite accuracy
Again, with an eye to practical implementations:
One-shot entanglement ε-distillation:
ρAB
L∈L...
Outline of the talk
one-shot distillable entanglement (pure state case)
generalized entropies: Smin and Smax
one-shot enta...
Part Two:
The Strange Case of Pure States
Case study: pure bipartite states
|ψAB
L∈LOCC
−−−−−−→ |φA B
True in this case (but grossly false in general):
all the prop...
One-shot zero-error distillable entanglement: E
(1)
D (ψAB; 0)
Nielsen: given an initial pure state ψAB, a maximally entan...
A maximally entangled state of rank R = 1
λmax
ψ
can always be
distilled exactly, i.e.,
E
(1)
D (ψAB; 0) log2
1
λmax
ψ
.
Finite accuracy: E
(1)
D (ψAB; ε)
Consider the set of pure states B∗
ε (ψAB) := | ¯ψAB : ¯ψAB
ε
≈ ψAB
A maximally entangled state of rank ¯R = 1
λmax
¯ψ
can always be
distilled up to an ε-error, i.e.,
E
(1)
D (ψAB; ε) max
¯ψ...
Getting the right smoothing
With B∗
ε (ψAB) := | ¯ψAB : ¯ψAB
ε
≈ ψAB :
E
(1)
D (ψAB; ε) max
¯ψ∈B∗
ε (ψ)
log2
1
λmax
¯ψ
f(ψ...
Smin is the one-shot distillable entanglement
A converse also holds:
Sε
min(ψA) E
(1)
D (ψAB; ε) Sε
min(ψA) − log2(1 − 2
√...
min-entropy of the reduced state ≈ one-shot distillable entanglement of
a pure bipartite state.
Beside the inequality E
(1...
Smax is the one-shot entanglement cost
Vidal, Jonathan, and Nielsen: a pure bipartite state ψAB can be
obtained by LOCC fr...
Summary of the pure state case
E
(1)
D (ψAB; ε) Sε
min(ψA) Sε
max(ψA) E
(1)
C (ψAB; ε)
↓ ↓
E∞
D (ψAB) = S(ψA) = E∞
C (ψAB)...
asymptotic reversibility holds for pure states
One-shot irreversibility gap for pure states
Reversibility only holds asymptotically. Define the one-shot irreversibility
g...
“Increasing irreversibility requires communication.”
Part Three:
The Complicated Case of Mixed
States
(an overview)
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Towards a one shot entanglement theory

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Towards a one shot entanglement theory

  1. 1. Towards a one-shot entanglement theory Francesco Buscemi and Nilanjana Datta Beyond i.i.d. in information theory," University of Cambridge, 9 January 2013
  2. 2. Part One: Introduction
  3. 3. Resource theory of (bipartite) entanglement Entanglement is useful (quantum information processing) but expensive (difficult to establish and fragile to preserve)
  4. 4. study of entanglement as a resource raw resources: bipartite quantum systems (in pure and/or mixed states) processing: local operations and classical communication (LOCC). (Why? Operational paradigm of “distant laboratories.”) standard currency: the singlet state |Ψ− = |01 −|10 √ 2 . (Why? Perhaps because teleportation and superdense coding both use the singlet.) basic tasks: distillation (extraction of singlets from raw resources) and dilution (creation of generic bipartite states from singlets) by LOCC
  5. 5. Asymptotic manipulation of (bipartite) quantum correlations ρAB ⊗ ρAB ⊗ · · · ⊗ ρAB Min L∈LOCC −−−−−−→ σA B ⊗ · · · σA B Nout where A-systems belong to Alice, B-systems belong to Bob, and the transformation L is LOCC between Alice and Bob Jargon: Min copies of the initial state ρAB are diluted into Nout copies of the target state σA B ; equivalently, Nout copies of the target state σA B are distilled from Min copies of the initial state ρAB Task is optimized with respect to the resources created (optimal distillation, N = N(Min)) or those consumed (optimal dilution, M = M(Nout)) Optimal rates are computed as limMin→∞ N(Min)/Min (optimal distillation rate) and limNout→∞ M(Nout)/Nout (optimal dilution rate)
  6. 6. Asymptotic entanglement distillation and dilution Entanglement distillation ρAB ⊗ · · · ⊗ ρAB Min L∈LOCC −−−−−−→ Ψ− A B ⊗ · · · Ψ− A B N(Min) distillable entanglement: E∞ D (ρAB) = limMin→∞ N(Min)/Min Entanglement dilution Ψ− AB ⊗ · · · ⊗ Ψ− AB M(Nout) L∈LOCC −−−−−−→ σA B ⊗ · · · σA B Nout entanglement cost: E∞ C (σA B ) = limNout→∞ M(Nout)/Nout
  7. 7. Criticisms to this approach The asymptotic framework is operational but not practical, for two reasons: asymptotic achievability (and often without knowing how fast the limit is approached) i.i.d. assumption: hardly satisfied in practical scenarios A third remark: the asymptotic i.i.d. argument mixes information theory and probability theory. As noticed by Han and Verd´u, we’d like to distinguish what is information theory from what is probability theory.
  8. 8. The one-shot case One-shot entanglement distillation: ρAB L∈LOCC −−−−−−→ Ψ− A B ⊗ · · · Ψ− A B Nmax(ρAB) . One-shot entanglement dilution: Ψ− AB ⊗ · · · ⊗ Ψ− AB Mmin(σA B ) L∈LOCC −−−−−−→ σA B . Correspondingly, one-shot distillable entanglement: E (1) D (ρAB) = Nmax(ρAB); one-shot entanglement cost: E (1) C (σA B ) = Mmin(σA B )
  9. 9. Allowing for finite accuracy Again, with an eye to practical implementations: One-shot entanglement ε-distillation: ρAB L∈LOCC −−−−−−→ ˜ρA B ε ≈ Ψ− A B ⊗ · · · Ψ− A B Nmax(ρAB;ε) . One-shot entanglement ε-dilution Ψ− AB ⊗ · · · ⊗ Ψ− AB Mmin(σA B ;ε) L∈LOCC −−−−−−→ ˜σA B ε ≈ σA B . Correspondingly, one-shot ε-distillable entanglement: E (1) D (ρAB; ε) = Nmax(ρAB; ε); one-shot entanglement ε-cost: E (1) C (σA B ; ε) = Mmin(σA B ; ε)
  10. 10. Outline of the talk one-shot distillable entanglement (pure state case) generalized entropies: Smin and Smax one-shot entanglement cost (pure state case) overview of the mixed state case: asymptotic results relative R´enyi entropies and derived quantities mixed state case: one-shot results comparison and discussion
  11. 11. Part Two: The Strange Case of Pure States
  12. 12. Case study: pure bipartite states |ψAB L∈LOCC −−−−−−→ |φA B True in this case (but grossly false in general): all the properties of a pure bipartite state ψAB are determined by the list of eigenvalues λψ of the reduced density matrix ψA = TrB[ψAB]; Lo and Popescu: the action of a general LOCC map on a pure state can be also obtained by another one-way, one-round LOCC map; Nielsen: there exists an LOCC transformation mapping ψAB into φA B if and only if ψA φA , i.e., k i=1 λ↓i ψ k i=1 λ↓i φ , for all k; asymptotic reversibility (total ordering): E∞ D (ψAB) = E∞ C (ψAB) = S(ψA).
  13. 13. One-shot zero-error distillable entanglement: E (1) D (ψAB; 0) Nielsen: given an initial pure state ψAB, a maximally entangled state of rank R, i.e. R−1/2 R i=1 |i |i , can be distilled if and only if λmax ψ ≡ λ↓1 ψ R−1, λ↓1 ψ + λ↓2 ψ 2R−1, and so on.
  14. 14. A maximally entangled state of rank R = 1 λmax ψ can always be distilled exactly, i.e., E (1) D (ψAB; 0) log2 1 λmax ψ .
  15. 15. Finite accuracy: E (1) D (ψAB; ε) Consider the set of pure states B∗ ε (ψAB) := | ¯ψAB : ¯ψAB ε ≈ ψAB
  16. 16. A maximally entangled state of rank ¯R = 1 λmax ¯ψ can always be distilled up to an ε-error, i.e., E (1) D (ψAB; ε) max ¯ψ∈B∗ ε (ψ) log2 1 λmax ¯ψ .
  17. 17. Getting the right smoothing With B∗ ε (ψAB) := | ¯ψAB : ¯ψAB ε ≈ ψAB : E (1) D (ψAB; ε) max ¯ψ∈B∗ ε (ψ) log2 1 λmax ¯ψ f(ψAB,ε) ≡ Sε min(ψA) Given a (mixed) state ρ, define the set of (mixed) states Bε(ρ) := ¯ρ : ¯ρ ε ≈ ρ . The smoothed min-entropy of ρ is defined as (Renner) Sε min(ρ) := max¯ρ∈Bε(ρ) [− log2 λmax(¯ρ)].
  18. 18. Smin is the one-shot distillable entanglement A converse also holds: Sε min(ψA) E (1) D (ψAB; ε) Sε min(ψA) − log2(1 − 2 √ ε). ε = 2 5 4 ε 1 8
  19. 19. min-entropy of the reduced state ≈ one-shot distillable entanglement of a pure bipartite state. Beside the inequality E (1) D (ψAB; ε) Sε min(ψA), a converse can also proved: E (1) D (ψAB; ε) Sε min(ψA) − log2(1 − 2 √ ε). This corroborates the idea that the min-entropy of the reduced state is the natural quantity measuring the one-shot distillable entangleme a pure bipartite state. Smin(ψA) (α=∞) · · · S(ψA) (α=1) · · · Smax(ψA) (α=0) Figure: is Smax associated with anything?
  20. 20. Smax is the one-shot entanglement cost Vidal, Jonathan, and Nielsen: a pure bipartite state ψAB can be obtained by LOCC from a maximally entangled state of rank R with a minimum error of ε = 1 − R i=1 λ↓i ψ . As a consequence, E (1) C (ψAB; 0) = log2 rank ψA = Smax(ψA). With finite accuracy: E (1) C (ψAB; ε) Sε max(ψA), where Sε max(ρ) = min¯ρ∈Bε(ρ) Smax(¯ρ).
  21. 21. Summary of the pure state case E (1) D (ψAB; ε) Sε min(ψA) Sε max(ψA) E (1) C (ψAB; ε) ↓ ↓ E∞ D (ψAB) = S(ψA) = E∞ C (ψAB) where “F(ρ; ε) → G(ρ)” means limε→0 limn→∞ 1 n F(ρ⊗n; ε) = G(ρ)
  22. 22. asymptotic reversibility holds for pure states
  23. 23. One-shot irreversibility gap for pure states Reversibility only holds asymptotically. Define the one-shot irreversibility gap as ∆(ψAB; ε) : = E (1) C (ψAB; ε) − E (1) D (ψAB; ε) Sε max(ψA) − Sε min(ψA) This quantity is related with the communication cost C of transforming an initial pure state ψi AB into a final state ψf A B (Hayden and Winter, 2003): 2C ∆(ψf A B ; 0) − ∆(ψi AB; 0).
  24. 24. “Increasing irreversibility requires communication.”
  25. 25. Part Three: The Complicated Case of Mixed States (an overview)
  26. 26. Mixed state case: asymptotic i.i.d. results Distillable entanglement and entanglement cost are naturally quantified by different functions of ρAB (Hayden, Horodecki, Terhal, 2001; Devetak, Winter, 2005): E∞ D (ρAB) E∞ C (ρAB) IA→B c (ρAB) pure states −−−−−−→ S(ρA) pure states ←−−−−−− minE i piS(ψi A) where: IA→B c (ρAB) = S(ρB) − S(ρAB) = −H(ρAB|B): coherent information minE i piS(ψi A) is done over all pure-state ensemble decompositions ρAB = i piψi AB: entanglement of formation EF (ρAB)
  27. 27. Relative entropies and derived quantities All such entropic quantities are originated from a common parent Relative entropy: S(ρ σ) = Tr [ρ log2 ρ − ρ log2 σ] 1 S(ρ) := − Tr[ρ log2 ρ] = −S(ρ 1) 2 H(ρAB|B) := S(ρAB) − S(ρB) = − minσB S(ρAB 1A ⊗ σB) 3 IA→B c (ρAB) := −H(ρAB|B) Relative R´enyi entropy of order zero: S0(ρ σ) = − log2 Tr [Πρ σ] 1 S0(ρ) := −S0(ρ 1) = Smax(ρ) 2 H0(ρAB|B) := − minσB S0(ρAB 1A ⊗ σB) 3 IA→B 0 (ρAB) := −H0(ρAB|B)
  28. 28. Technical remark: quasi-entropies In our proofs we employed the notion of quasi-entropies (Petz, 1986) SP α (ρ σ) = 1 α − 1 log2 Tr √ Pρα √ P σ1−α , defined for ρ, σ 0, 0 P 1, and α ∈ (0, ∞)/{1}. In particular, we enjoyed working with SP 0 (ρ σ) = lim α 0 SP α (ρ σ) = − log2 Tr √ PΠρ √ P σ , smoothing w.r.t. ρ or P, depending on the problem at hand.
  29. 29. Mixed state case: one-shot results Keeping in ming the asymptotic i.i.d. case: E∞ D (ρAB) E∞ C (ρAB) IA→B c (ρAB) pure −−→ S(ρA) pure ←−− minE i piS(ψi A) minE H(ρRA|R) Here are the one-shot analogues: E (1) D (ρAB; ε) E (1) C (ρAB; ε) IA→B 0,ε (ρAB) pure −−→ Sε min(ρA) Sε max(ρA) pure ←−− minE Hε 0(ρRA|R) where minE Hε 0(ρRA|R) is done over all cq-extensions ρRAB = i pi|i i|R ⊗ ψi AB, such that TrR[ρRAB] = ρAB
  30. 30. A by-product worth noticing Since E∞ C (ρAB) = limε→0 limn→∞ 1 nE (1) C (ρ⊗n AB; ε), from the previous slide: min E Hε 0(ρRA|R) limε→0 1 n limn→∞ −−−−−−−−−−−→ min E H(ρRA|R) Both well-known guests of the zoo of entanglement measures: minE H(ρRA|R) is the entanglement of formation (Bennett et al, 1996) EF (ρAB) = minE i piS(ψi A) minE H0(ρRA|R) is the logarithm of the generalized Schmidt rank (Terhal, Horodecki, 2000) Esr(ρAB) = log2 minE maxi rank ψi A By introducing a smoothed Schmidt rank as follows: Eε sr(ρAB) := min ¯ρAB∈Bε(ρAB) Esr(¯ρAB), we have implicitly proved that lim ε→0 lim n→∞ 1 n Eε sr(ρ⊗n AB) = lim n→∞ 1 n EF (ρ⊗n AB).
  31. 31. Conclusions and open questions mix- and max-entropies naturally arise also in one-shot entanglement theory pleasant formal analogy with the asymptotic i.i.d. case: just replace S(ρ σ) by S0(ρ σ) (but, first, find the right expression to replace!) sometimes, the one-shot analysis uncovers new relations between known functions (e.g. the regularized entanglement of formation equals the smoothed-and-regularized log-Schmidt rank) increasing irreversibility requires communication: what about mixed states? other operational paradigms: SEPP done (Fernando and Nila); what about LOSR? one-shot squashed entanglement: one-shot quantum conditional mutual information? La Fine.

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