Source-linked AI summary
One-shot rates for entanglement manipulation under non-entangling maps
Fernando G. S. L. Brandao, Nilanjana Datta
TL;DR
Mixed-state entanglement manipulation can be irreversible, motivating the study of single-copy rates under operations that generate negligible entanglement. The paper relates these rates to smoothed entanglement measures and, in the many-copy limit, partially recovers reversibility, with catalysis required for this recovery.
Problem
Mixed-state entanglement manipulation can be irreversible, motivating study of alternative operation classes and their conversion rates.
Source: §I · p. 1Method
The paper derives one-shot manipulation rates under non-entangling and approximately non-entangling maps and relates them to smooth entanglement measures.
Source: §I · p. 1, §III · p. 3Results
The optimal one-shot distillation and dilution rates are characterized by smoothed min- and max-relative entropies of entanglement and logarithmic robustness measures.
Source: §I · p. 1, §III · p. 3Takeaways & Limitations
The formulas give operational interpretations to these measures and partially recover asymptotic reversibility through catalytic manipulation.
Source: §I · p. 1, §III · p. 3The asymptotic reversibility recovery here requires catalytic maximally entangled states to keep operations negligibly entangling at the single-copy level.
Source: §III · p. 3
Abstract
from arXiv · showhide
We obtain expressions for the optimal rates of one- shot entanglement manipulation under operations which generate a negligible amount of entanglement. As the optimal rates for entanglement distillation and dilution in this paradigm, we obtain the max- and min-relative entropies of entanglement, the two logarithmic robustnesses of entanglement, and smoothed versions thereof. This gives a new operational meaning to these entanglement measures. Moreover, by considering the limit of many identical copies of the shared entangled state, we partially recover the recently found reversibility of entanglement manipu- lation under the class of operations which asymptotically do not generate entanglement.
I. INTRODUCTION
Entanglement is treated as a resource for converting shared states, but mixed-state manipulation can be irreversible. This paper studies single-copy conversion under non-entangling operations and connects its rates to entanglement measures and asymptotic reversibility.
I. INTRODUCTION
- Bound entangled mixed states can require nonzero asymptotic pure-state entanglement to create while yielding no pure-state entanglement for distillation.This contrasts with the reversibility established for bipartite pure states.Source: §I · p. 1
- Alternative manipulation scenarios were explored to simplify entanglement theory and potentially provide insight into LOCC manipulations.Source: §I · p. 1
- Under operations that generate negligible entanglement asymptotically, multipartite mixed-state manipulation is reversible and specified by regularized relative entropy of entanglement.This framework has also been applied to LOCC entanglement cost, separability criteria, and reversible transformations.Source: §I · p. 1
- The paper identifies single-copy cost and distillation rates under non-entangling maps with logarithmic robustnesses and min-relative entropy of entanglement.These identifications give operational interpretations to the measures.Source: §I · p. 1
- Taking asymptotic limits of the finite-copy formulas partially recovers reversibility under asymptotically non-entangling maps, using catalytic entanglement manipulations.The paper notes that this recovery uses catalysis, unlike the cited earlier results.Source: §I · p. 1
II. NOTATION AND DEFINITIONS
This section sets notation for finite-dimensional operators and states, defines separability and entanglement measures, and specifies the operation classes and one-shot cost and distillation quantities used later.
II. NOTATION AND DEFINITIONS
- B(H) denotes linear operators, B+(H) positive operators, and D(H) positive unit-trace states on finite-dimensional Hilbert spaces.Source: §II · p. 2
- A multipartite state is separable when it is a probabilistic mixture of product states across the parties.Source: §II · p. 2
- The max- and min-relative entropies of entanglement are introduced, with max-relative entropy identified as logarithmic global robustness and a distinct logarithmic robustness also defined.The paper also defines smoothed versions and regularized smooth min- and max-relative entropies.Source: §II · p. 2
- A non-entangling map preserves separability, while a δ-non-entangling map bounds the output global robustness on separable inputs by δ.Source: §II · p. 3
- The one-shot framework defines entanglement cost, catalytic cost, and distillable entanglement for general operation classes, then specializes to SEPP and δ-SEPP maps.Protocols convert multipartite states to or from maximally entangled states shared by the first two parties.Source: §II · p. 3
III. MAIN RESULTS
The theorems give operational meanings to smooth entanglement measures through optimal one-shot manipulation rates under non-entangling maps. In the many-copy limit, the results partially recover reversibility, with catalytic maximally entangled states required to keep entanglement generation negligible already at the single-copy level.
III. MAIN RESULTS
- The first theorem identifies smoothed min-relative entropy of entanglement with single-shot distillable entanglement under non-entangling maps.Source: §III · p. 3
- The second theorem relates smooth logarithmic robustness of entanglement to one-shot entanglement cost under non-entangling maps.Source: §III · p. 3
- The third theorem relates logarithmic global robustness, also called max-relative entropy of entanglement, to one-shot catalytic cost under δ-non-entangling maps.Source: §III · p. 3
- Distillable entanglement and catalytic cost coincide under asymptotically non-entangling maps, but catalysis is needed to ensure negligible entanglement generation at the single-copy level.The earlier result cited in the paper does not require catalysis because negligible generation is required only over many copies.Source: §III · p. 3
- The regularized smooth min-relative entropy of entanglement equals the regularized relative entropy of entanglement.Source: §III · p. 4
IV. PROOF OF THEOREM 1
The proof constructs a separability-preserving map that distills a maximally entangled state at a rate given by the floored smooth min-relative entropy of entanglement. An optimal measurement element sets the map’s output dimension while ensuring its fidelity and separability-preserving property.
IV. PROOF OF THEOREM 1
- The proof’s upper-bound argument transfers a bounded measurement operator through Λ† and uses that Λ maps separable states to separable states.The adjoint Λ† preserves the operator bounds needed for the min-relative-entropy comparison.Source: §IV · p. 4
- ⌊Eε min(ρ)⌋ is an achievable one-shot distillation rate for ρ.The construction chooses an optimal POVM element and sets M = 2⌊Eε min(ρ)⌋, yielding fidelity at least 1−ε while the map remains SEPP.Source: §IV · p. 4
- The construction uses an operator 0 ≤ A ≤ I to define a CPTP map ΛA for the distillation protocol.The map’s action is used to convert the input state into an isotropic output whose overlap with ΨM determines its fidelity.Source: §IV · p. 4
- For every separable input, the output isotropic state is separable when its overlap with ΨM is at most 1/M.The proof uses this criterion to establish that the constructed map is SEPP.Source: §IV · p. 4
V. PROOF OF THEOREM 2
The proof bounds one-shot entanglement dilution under non-entangling maps using smoothed logarithmic robustness. Its achievability construction maps a maximally entangled state to a nearby target state, while the converse invokes monotonicity under SEPP maps.
V. PROOF OF THEOREM 2
- For separable inputs, the constructed map remains non-entangling because its output has zero robustness.The proof uses convexity of robustness and R(π)=0 for separable π to establish this property.Source: §V · p. 4
- Λ(ΨM) = ρε at rate log M = log(1 + ⌈R(ρε)⌉) ≤ LRε(ρ) + 1, where ρε attains the minimum defining smooth logarithmic robustness over Bε(ρ).Source: §V · p. 4
- The converse starts from a SEPP map producing a state within fidelity 1−ε of ρ and applies monotonicity of log robustness under SEPP maps.This supplies the proof’s lower-bound argument for dilution cost.Source: §V · p. 4
VI. PROOF OF THEOREM 3
The proof establishes dilution bounds for approximately non-entangling maps by using an auxiliary maximally entangled state ΨK. The protocol returns ΨK unchanged, using it to ensure the map is δ-SEPP while deriving an achievable rate bound.
VI. PROOF OF THEOREM 3
- A smoothing lemma supplies a state ρε with F(ρε, ρ) ≥ 1−ε and controls Emax(ρε ⊗ ΨK) using Eε max(ρ ⊗ ΨK).The stated control includes an additive −log(1−ε) term.Source: §VI · p. 5
- The proof uses LOCC monotonicity of global robustness and RG(ΨK) = K − 1 when deriving its bound.The choice of K is then used to complete the inequality.Source: §VI · pp. 5–6
- R = log M is achievable with a δ-SEPP map using an auxiliary ΨK that is retrieved unchanged.The stated bound is R ≤ Eε max(ρ ⊗ ΨK) − log K − log(1−ε) + 1.Source: §VI · p. 6
- The auxiliary ΨK ensures the protocol’s operation is δ-SEPP for any given δ > 0.The proof bounds the global robustness of the output on separable inputs using a separable state and convexity.Source: §VI · pp. 5–6
- The converse relates Eε max(ρ ⊗ ΨK) to log M + log K + log(1 + δ).The proof uses monotonicity under the map and the δ-SEPP condition to obtain this bound.Source: §VI · p. 6
ENTROPY OF ENTANGLEMENT
The section proves an asymptotic entanglement result using a generalized Quantum Stein’s Lemma for separable states. Its proof combines direct and strong-converse measurement bounds and then takes limits in n and ε.
ENTROPY OF ENTANGLEMENT
- The proof invokes a generalized Quantum Stein’s Lemma, specialized to the set of separable states, as its main ingredient.The cited generalization is attributed to Refs. [16],.Source: ENTROPY OF ENTANGLEMENT · p. 6
- The lemma provides both a direct part and a strong converse for sequences of POVMs against separable states.The direct and strong-converse statements are used as the two sides of the proof.Source: ENTROPY OF ENTANGLEMENT · p. 6
- Earlier work established the relation used in the proof in Refs. [16],,.The section applies those established results in proving Theorem 4.Source: ENTROPY OF ENTANGLEMENT · p. 6
- The proof compares Emax and Emin bounds for ρ⊗n, divides by n, and takes n → ∞ followed by ε → 0.It uses Emax(ρ) ≥ Emin(ρ) and an optimal POVM sequence from the direct part of the lemma.Source: ENTROPY OF ENTANGLEMENT · p. 6