Towards a one shot entanglement theory

Towards a one-s hot entanglement theory

Francesco Buscemi and Nilanjana Datta

\Beyond i.i.d. in information theory,"University of Cambridge, 9 January 2013

Part One:Introduction

Resource theory of (bipartite) entanglement

Entanglement is useful (quantum information processing) but expensive(difficult to establish and fragile to preserve)

+ study of entanglement as a resource

raw resources: bipartite quantum systems (in pure and/or mixedstates)

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) anddilution (creation of generic bipartite states from singlets) by LOCC

Asymptotic manipulation of (bipartite) quantumcorrelations

ρAB ⊗ ρAB ⊗ · · · ⊗ ρAB︸ ︷︷ ︸Min

L∈LOCC7−−−−−−→ σA′B′ ⊗ · · ·σA′B′︸ ︷︷ ︸Nout

where A-systems belong to Alice, B-systems belong to Bob, and thetransformation 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 thetarget state σA′B′ are distilled from Min copies of the initial state ρAB

Task is optimized with respect to the resources created (optimaldistillation, N = N(Min)) or those consumed (optimal dilution,M = M(Nout))

Optimal rates are computed as limMin→∞N(Min)/Min (optimaldistillation rate) and limNout→∞M(Nout)/Nout (optimal dilutionrate)

Asymptotic entanglement distillation and dilution

Entanglement distillation

ρAB ⊗ · · · ⊗ ρAB︸ ︷︷ ︸Min

L∈LOCC7−−−−−−→ 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∈LOCC7−−−−−−→ σA′B′ ⊗ · · ·σA′B′︸ ︷︷ ︸Nout

entanglement cost: E∞C (σA′B′) = limNout→∞M(Nout)/Nout

Criticisms to this approach

The asymptotic framework is operational but not practical, for two reasons:

asymptotic achievability (and often without knowing how fast thelimit is approached)

i.i.d. assumption: hardly satisfied in practical scenarios

A third remark: the asymptotic i.i.d. argument mixes information theoryand probability theory. As noticed by Han and Verdu, we’d like todistinguish what is information theory from what is probability theory.

The one-shot case

One-shot entanglement distillation:

ρABL∈LOCC7−−−−−−→ Ψ−A′B′ ⊗ · · ·Ψ−A′B′︸ ︷︷ ︸

Nmax(ρAB)

.

One-shot entanglement dilution:

Ψ−AB ⊗ · · · ⊗Ψ−AB︸ ︷︷ ︸Mmin(σA′B′ )

L∈LOCC7−−−−−−→ σA′B′ .

Correspondingly,

I one-shot distillable entanglement: E(1)D (ρAB) = Nmax(ρAB);

I one-shot entanglement cost: E(1)C (σA′B′) = Mmin(σA′B′)

Allowing for finite accuracy

Again, with an eye to practical implementations:

One-shot entanglement ε-distillation:

ρABL∈LOCC7−−−−−−→ 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∈LOCC7−−−−−−→ 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)

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 Renyi entropies and derived quantities

mixed state case: one-shot results

comparison and discussion

Part Two:The Strange Case of Pure States

Case study: pure bipartite states

|ψAB〉 L∈LOCC7−−−−−−→ |φA′B′〉True in this case (but grossly false in general):

all the properties of a pure bipartite state ψAB are determined by thelist of eigenvalues ~λψ of the reduced density matrix ψA = TrB[ψAB];

Lo and Popescu: the action of a general LOCC map on a pure statecan be also obtained by another one-way, one-round LOCC map;

Nielsen: there exists an LOCC transformation mapping ψAB intoφA′B′ if and only if ψA ≺ φA′ , i.e.,

∑ki=1 λ

↓iψ 6

∑ki=1 λ

↓iφ , for all k;

asymptotic reversibility (total ordering):E∞D (ψAB) = E∞C (ψAB) = S(ψA).

One-shot zero-error distillable entanglement: E(1)D (ψAB; 0)

Nielsen: given an initial pure state ψAB, a maximally entangled state ofrank R, i.e. R−1/2

∑Ri=1 |i〉|i〉, can be distilled if and only if

λmaxψ ≡ λ↓1ψ 6 R−1, λ↓1ψ + λ↓2ψ 6 2R−1, and so on.

+ A maximally entangled state of rank R =⌊

1λmaxψ

⌋can always be

distilled exactly, i.e.,

E(1)D (ψAB; 0) > log2

⌊1

λmaxψ

⌋.

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

λmaxψ

⌋can always be

distilled up to an ε-error, i.e.,

E(1)D (ψAB; ε) > max

ψ∈B∗ε (ψ)log2

⌊1

λmaxψ

⌋.

Getting the right smoothing

With B∗ε (ψAB) :={|ψAB〉 : ψAB

ε≈ ψAB}

:

E(1)D (ψAB; ε) > max

ψ∈B∗ε (ψ)log2

⌊1

λmaxψ

⌋︸ ︷︷ ︸

f(ψAB ,ε)

≡ Sεmin(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) Sεmin(ρ) := maxρ∈Bε(ρ) [− log2 λmax(ρ)].

Smin is the one-shot distillable entanglement

A converse also holds:

Sεmin(ψA) 6 E(1)D (ψAB; ε) 6 Sε

′min(ψA)− log2(1− 2

√ε).

[ε′ = 2

54 ε

18

]

+ min-entropy of the reduced state ≈ one-shot distillable entanglement ofa pure bipartite state.

Smin is the one-shot distillable entanglement

Beside the inequality E(1)D (ψAB; ε) � Sεmin(ψA), a converse can also be

proved:

E(1)D (ψAB; ε) � Sεmin(ψA)− log2(1− 2

√ε).

This corroborates the idea that the min-entropy of the reduced state reallyis the natural quantity measuring the one-shot distillable entanglement ofa pure bipartite state.

Smin(ψA)(α=∞)

· · · S(ψA)(α=1)

· · · Smax(ψA)(α=0)

Figure: is Smax associated with anything?

Smax is the one-shot entanglement cost

Vidal, Jonathan, and Nielsen: a pure bipartite state ψAB can beobtained by LOCC from a maximally entangled state of rank R with aminimum error of ε = 1−∑R

i=1 λ↓iψ .

As a consequence, E(1)C (ψAB; 0) = log2 rankψA = Smax(ψA).

With finite accuracy:

E(1)C (ψAB; ε) ' Sεmax(ψA),

where Sεmax(ρ) = minρ∈Bε(ρ) Smax(ρ).

Summary of the pure state case

E(1)D (ψAB; ε) ' Sεmin(ψA) 6 Sεmax(ψA) ' E

(1)C (ψAB; ε)

↓ ↘ ↙ ↓

E∞D (ψAB) = S(ψA) = E∞C (ψAB)

where “F (ρ; ε)→ G(ρ)” means limε→0 limn→∞ 1nF (ρ⊗n; ε) = G(ρ)

+ asymptotic reversibility holds for pure states

One-shot irreversibility gap for pure states

Reversibility only holds asymptotically. Define the one-shot irreversibilitygap as

∆(ψAB; ε) : = E(1)C (ψAB; ε)− E(1)

D (ψAB; ε)

' Sεmax(ψA)− Sεmin(ψA)

This quantity is related with the communication cost C of transforming aninitial pure state ψiAB into a final state ψfA′B′ (Hayden and Winter, 2003):

2C > ∆(ψfA′B′ ; 0)−∆(ψiAB; 0).

+ “Increasing irreversibility requires communication.”

Part Three:The Complicated Case of Mixed

States(an overview)

Mixed state case: asymptotic i.i.d. results

Distillable entanglement and entanglement cost are naturally quantified bydifferent functions of ρAB (Hayden, Horodecki, Terhal, 2001; Devetak,Winter, 2005):

E∞D (ρAB) E∞C (ρAB)o o

IA→Bc (ρAB)pure states↪−−−−−−→ S(ρA)

pure states←−−−−−−↩ minE∑

i piS(ψiA)

where:

IA→Bc (ρAB) = S(ρB)− S(ρAB) = −H(ρAB|B): coherentinformation

minE∑

i piS(ψiA) is done over all pure-state ensemble decompositionsρAB =

∑i piψ

iAB: entanglement of formation EF (ρAB)

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→Bc (ρAB) := −H(ρAB |B)

Relative Renyi 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→B0 (ρAB) := −H0(ρAB |B)

Technical remark: quasi-entropies

In our proofs we employed the notion of quasi-entropies (Petz, 1986)

SPα (ρ‖σ) =1

α− 1log2 Tr

[√Pρα√P σ1−α

],

defined for ρ, σ > 0, 0 6 P 6 1, and α ∈ (0,∞)/{1}.

In particular, we enjoyed working with

SP0 (ρ‖σ) = limα↘0

SPα (ρ‖σ) = − log2 Tr[√

PΠρ

√P σ

],

smoothing w.r.t. ρ or P , depending on the problem at hand.

Mixed state case: one-shot results

Keeping in ming the asymptotic i.i.d. case:

E∞D (ρAB) E∞C (ρAB)o o

IA→Bc (ρAB)pure↪−−→ S(ρA)

pure←−−↩ minE∑

i piS(ψiA)‖

minEH(ρRA|R)

Here are the one-shot analogues:

E(1)D (ρAB; ε) E

(1)C (ρAB; ε)

o oIA→B0,ε (ρAB)

pure↪−−→ Sεmin(ρA) 6 Sεmax(ρA)

pure←−−↩ minEHε0(ρRA|R)

where minEHε0(ρRA|R) is done over all cq-extensions

ρRAB =∑

i pi|i〉〈i|R ⊗ ψiAB, such that TrR[ρRAB] = ρAB

A by-product worth noticing

Since E∞C (ρAB) = limε→0 limn→∞ 1nE

(1)C (ρ⊗nAB; ε), from the previous slide:

minEHε

0(ρRA|R)limε→0

1n

limn→∞−−−−−−−−−−−→ minEH(ρRA|R)

Both well-known guests of the zoo of entanglement measures:

minEH(ρRA|R) is the entanglement of formation (Bennett et al,1996) EF (ρAB) = minE

∑i piS(ψiA)

minEH0(ρRA|R) is the logarithm of the generalized Schmidt rank(Terhal, Horodecki, 2000) Esr(ρAB) = log2 minE maxi rankψiA

By introducing a smoothed Schmidt rank as follows:

Eεsr(ρAB) := minρAB∈Bε(ρAB)

Esr(ρAB),

we have implicitly proved that

limε→0

limn→∞

1

nEεsr(ρ

⊗nAB) = lim

n→∞1

nEF (ρ⊗nAB).

Conclusions and open questions

mix- and max-entropies naturally arise also in one-shot entanglementtheory

pleasant formal analogy with the asymptotic i.i.d. case: just replaceS(ρ‖σ) by S0(ρ‖σ) (but, first, find the right expression to replace!)

sometimes, the one-shot analysis uncovers new relations betweenknown functions (e.g. the regularized entanglement of formationequals the smoothed-and-regularized log-Schmidt rank)

increasing irreversibility requires communication: what about mixedstates?

other operational paradigms: SEPP done (Fernando and Nila); whatabout LOSR?

one-shot squashed entanglement: one-shot quantum conditionalmutual information?

La Fine.