- The document discusses one-shot entanglement theory, which considers manipulating entanglement without assuming many identical copies or an i.i.d. setting. This generalizes the traditional asymptotic i.i.d. entanglement theory. - For pure states, the one-shot distillable entanglement is characterized by the smoothed min-entropy of the reduced state. The one-shot entanglement cost is characterized by the smoothed max-entropy. - For mixed states, the one-shot results involve smoothed min- and max-entropies as well as smoothed relative entropies. These generalize the corresponding quantities in the asymptotic i.i.d. setting, providing a one-to-one correspondence between the two frameworks.

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. Part One:
Introduction

3. Resource theory of (bipartite) entanglement
Entanglement is useful (quantum information processing) but expensive
(difficult to establish and fragile to preserve)

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. 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. 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. 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. 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. 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. 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. Part Two:
The Strange Case of Pure States

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. 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. 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. Finite accuracy: E
(1)
D (ψAB; ε)
Consider the set of pure states B∗
ε (ψAB) := | ¯ψAB : ¯ψAB
ε
≈ ψAB

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. 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. 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. 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. 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. 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. asymptotic reversibility holds for pure states

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. “Increasing irreversibility requires communication.”

25. Part Three:
The Complicated Case of Mixed
States
(an overview)

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. 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. 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. 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. 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. 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.