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A Fully Quantum Asymptotic Equipartition Property

Marco Tomamichel, Roger Colbeck, Renato Renner

arXiv:0811.1221v3quant-ph

TL;DR

The paper addresses how to extend the classical asymptotic equipartition property when both outcomes and side information are quantum. It proves a fully quantum version using quantum Rényi-like conditional entropies, with finite-n convergence bounds independent of the side-information dimension. The von Neumann entropy appears as the asymptotic special case.

  • Problem

    The classical AEP does not directly extend to settings where quantum side information may be entangled with the outcome, because quantum conditional probability distributions cannot be defined.

  • Method

    The paper develops quantum Rényi-like conditional entropies and inequalities that yield finite-n bounds for bipartite quantum states and their i.i.d. tensor powers.

  • Results

    The paper proves a fully quantum AEP whose finite-n convergence bound is independent of the Hilbert-space dimension of the side information, with von Neumann entropy as the asymptotic limit.

  • Takeaways & Limitations

    The result provides an AEP for quantum-information problems involving single-shot min- and max-entropies, including settings where the side-information dimension is difficult to bound.

Abstract

from arXiv · show

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, we prove a fully quantum generalization of this property, where both the output of the experiment and side information are quantum. We give an explicit bound on the convergence, which is independent of the dimensionality of the side information. This naturally leads to a family of Renyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.

I. INTRODUCTION

The paper proves a fully quantum asymptotic equipartition property for experiments whose outcomes and side information may both be quantum. It first reformulates the classical AEP to guide this generalization.

  • I. INTRODUCTION: The fully quantum AEP addresses experiments where both outcomes and side information require quantum descriptions.Quantum side information may be entangled with the outcome, so a quantum analogue of conditional probability distributions is unavailable.
  • I. INTRODUCTION: The proof begins by rewriting the classical AEP in a form that can be generalized to the fully quantum setting.The authors sketch the classical proof and follow similar lines for the quantum result.

A. Classical AEP

The classical AEP characterizes long i.i.d. sequences by a typical set of about 2^nH(X) outcomes, each with probability near 2^-nH(X), and underlies source compression at H(X) bits per symbol. It can also be expressed through smooth min- and max-entropies and generalized to conditional entropies.

  • A. Classical AEP: The AEP is central to classical information theory and has been generalized to conditional entropies.The classical statement connects typical sequences with entropy-based compression and randomness extraction.
  • A. Classical AEP: For large n, an i.i.d. sequence is almost certainly among approximately 2^nH(X) typical events, each occurring with probability close to 2^-nH(X).This formulation follows from the weak law of large numbers.
  • A. Classical AEP: The AEP implies that source compression needs approximately H(X) bits per sequence element when non-typical events are ignored.A small probability of failure is tolerated, while the typical set is retained.
  • A. Classical AEP: The AEP is equivalently formulated using smooth min- and max-entropies, which discard respectively the most probable or least probable events.The smooth min-entropy ignores events up to total probability ε, while the smooth max-entropy is closely related to H0.
  • A. Classical AEP: Rényi entropies interpolate between H∞, H0, and the Shannon entropy as α approaches infinity, zero, and one, respectively.For α > 1 they are close to smooth min-entropy, while for α < 1 they are close to smooth max-entropy.
  • A. Classical AEP: For i.i.d. sequences, the apparent divergence in the Rényi-to-Shannon error term can be controlled by choosing α dependent on n.The proof uses additivity Hα(X^n) = nHα(X), with α = 1 + 1/√n for a lower bound and Hmin(X) ≤ H(X) for the upper bound.

B. Fully Quantum AEP

The paper extends the AEP to bipartite quantum states with potentially quantum side information, proving finite-n bounds whose convergence is independent of the side-information dimension. The proof uses quantum Rényi-like quantities and technical inequalities, while typical-subspace approaches generally retain dimension dependence.

  • B. Fully Quantum AEP: The fully quantum AEP allows both the outcome system A and side-information system B to be quantum, including possibly entangled systems.Earlier generalizations allowed quantum outcomes with classical side information; the paper treats quantum B explicitly.
  • B. Fully Quantum AEP: The result serves as the quantum-information analogue of the classical AEP and recovers the classical AEP as a special case.It connects single-shot min- and max-entropy formulations with asymptotic i.i.d. results expressed using von Neumann entropies.
  • B. Fully Quantum AEP: Theorem 1 establishes the fully quantum AEP for finite-dimensional bipartite states and i.i.d. tensor powers.The statement uses a bipartite state ρAB on HA ⊗ HB and its n-fold tensor power.
  • B. Fully Quantum AEP: The non-asymptotic theorem gives a finite-n lower bound on Hε_min(A^n|B^n) whose deviation depends on conditional entropies but not on the dimension of B.This dimension independence is important when an adversary controls a quantum system whose dimension is difficult or impossible to bound.
  • B. Fully Quantum AEP: Typical-subspace proofs generally depend on the dimensions of both HA and HB and are therefore qualitatively weaker for high-dimensional systems.The paper also notes that an earlier result explicitly depends on dim(HA).
  • B. Fully Quantum AEP: The proof uses quantum generalizations of Rényi entropies together with inequalities that extend classical bounds to the quantum domain.These ingredients lead to generalized Rényi entropies and the von Neumann entropy as the asymptotic quantity.

II. QUANTUM R´ENYI ENTROPIES

This section defines quantum conditional min-, max-, and α-entropies, along with smoothing and their principal structural properties. The α-entropies generalize Rényi entropies, recover von Neumann entropy at α = 1, and satisfy monotonicity, additivity, representation invariance, data processing, and duality properties within stated ranges.

  • Conditional min- and max-entropies: Quantum conditional min-entropy is defined relative to a state σB, with finiteness determined by whether σB covers the support of ρB.If supp{σB} does not contain supp{ρB}, the quantity is −∞.
  • Conditional min- and max-entropies: The conditional max-entropy is defined through a purification of ρAB and is dual to the conditional min-entropy.The purification uses an auxiliary system C.
  • Smoothing: Smoothed min-entropies optimize the corresponding entropy over states in an ε-ball defined using a fidelity-based distance measure.The chosen distance is invariant under purifications and relates directly to trace distance for pure states.
  • α-entropies: The α-entropy of A conditioned on B is defined relative to σB; for trivial B it recovers classical Rényi entropy, while continuous extension at α = 1 gives von Neumann entropy.The limits α → 0 and α → ∞ define H0 and H∞, respectively.
  • α-entropies: Quantum conditional min- and max-entropies are not special cases of the α-entropies, unlike their classical counterparts.This is an explicit distinction between the quantum conditional entropy families.
  • Properties: The α-entropies decrease monotonically with α, are additive for i.i.d. states, and are invariant under local isometric changes of representation.They also satisfy data processing for α ∈ [0, 2].
  • Properties: For pure tripartite states, α-entropies obey a duality relation whose α → 1 limit yields the von Neumann entropy duality.The limiting relation is H(A|B)ρ = −H(A|C)ρ.

III. LOWER BOUND ON SMOOTH MIN-ENTROPY

This section establishes inequalities connecting smooth conditional min-entropy with quantum α-entropies. The resulting bound is a quantum generalization of the corresponding classical relation and is proved for α ∈ (1, 2].

  • Main inequality: The main tool is a family of inequalities relating smooth conditional min-entropy to quantum α-entropies.These inequalities provide the bridge needed for the fully quantum AEP proof.
  • Main inequality: For ε > 0 and α ∈ (1, 2], Theorem 7 gives a lower bound on smooth conditional min-entropy in terms of α-entropy.The theorem assumes a bipartite state ρAB and reference state σB.
  • Proof strategy: The proof handles divergent α-entropy separately and otherwise reduces to the case supp{ρB} ⊆ supp{σB}.An isometry is used so that σB has full support without changing the relevant entropies.
  • Proof strategy: The proof constructs X = ρAB − λ1A ⊗ σB and projects onto its positive eigenspace to control the entropy inequality.The positive-eigenspace projector produces the operator used in the subsequent bounds.
  • Proof strategy: Operator convexity of t^α for α ∈ (1, 2] enables the application of a functional inequality and completes the proof.The map is applied through a trace-preserving completely positive map associated with the positive-eigenspace projection.

IV. LOWER BOUND ON α-ENTROPIES

This section bounds α-entropies near α = 1 in terms of the conditional von Neumann entropy and a convergence parameter. Combined with the preceding min-entropy inequality, these estimates quantify convergence in the fully quantum AEP.

  • Convergence control: The convergence bound depends on the smoothing parameter ε and Υ(A|B), which describes how rapidly α-entropies approach von Neumann entropy.This parameter controls the deviation associated with choosing α near 1.
  • Relation to min-entropy: For σB = ρB, the dual relation and H3/2(A|B)ρ|ρ ≥ Hmin(A|B)ρ connect α-entropies with conditional min- and max-entropies.The resulting bound can be expressed using H1/2 and H3/2.
  • Near-α = 1 bound: For α sufficiently close to 1, Lemma 8 provides an inequality controlled by η = Υ(A|B)ρ|σ.The stated range is 1 < α < 1 + log 3/(4 log η), subject to the displayed condition.
  • Proof strategy: The proof approximates Hα for small β = α − 1 by expanding t^β around β = 0 and bounding the remainder.The remainder is controlled using Taylor expansion and bounds involving cosh(β ln t).
  • Proof strategy: The argument applies concavity and monotonicity properties of sβ together with spectral bounds on X + 1 to derive the final estimate.The proof uses the condition β ≤ 1/2 and the range constraints on the eigenvalues.

V. QUANTUM AEP

The paper derives finite-n bounds for smooth min- and max-entropies that converge to conditional von Neumann entropy, yielding a generalized quantum AEP. The finite-size deviation depends on conditional entropies but is independent of the side-information dimension.

  • Finite-size bound: Theorem 9 provides a finite-n bound for conditional min-entropy, from which the asymptotic min-entropy relation follows.The bound uses the α-entropy, additivity, and an optimized parameter μ.
  • Finite-size bound: For sufficiently large n, the optimized parameter μ* can be reached despite finite-n restrictions imposed by the condition on α.For fixed n, the restriction can make δ(ε, η) a function of n.
  • Asymptotic result: The generalized asymptotic equipartition property follows as a corollary of Theorem 9.The proof takes the n →∞ limit of the finite-size theorem and relates smooth min-entropy to von Neumann entropy.
  • Asymptotic result: The corresponding max-entropy relation follows by substituting the duals of smooth min-entropy and von Neumann entropy into the min-entropy relation.

APPENDIX A: TECHNICAL RESULTS

Appendix A develops functionals built from operators and proves their invariance under isometries and monotonicity under trace-preserving completely positive maps. These properties support the paper’s entropy arguments.

  • Relation to relative entropy: The appendix notes that the paper’s results can also be expressed using relative entropies instead of conditional entropies.
  • Functional construction: For continuous f with f(0)=0, the functional Sf(A, B) is defined using positive operators A and B and a fully entangled state.The construction uses an unnormalized fully entangled state and transposes relative to a chosen basis.
  • Invariance: Sf(A, B) is invariant under isometric representations of A and B.The proof uses preservation of eigenvalues and scalar products under the isometry.
  • Monotonicity: For operator-convex f, Sf(A, B) is monotone under a trace-preserving completely positive map E.The inequality is Sf(A, B) ≥ Sf(E(A), E(B)).
  • Monotonicity: The monotonicity proof reduces a trace-preserving completely positive map to an isometry followed by a partial trace.The partial-trace step is established using operator Jensen’s inequality, with an invertibility assumption handled by continuity.

APPENDIX B: PROOFS OF CLAIMS IN SECTION II

Appendix B proves properties of the min-, max-, and α-entropies by expressing them through the Appendix A functionals. The proofs use isometric invariance, data processing, duality, and convexity or concavity.

  • Entropy properties: The appendix establishes properties of the min-, max-, and α-entropies introduced in Section II.
  • Entropy relations: The min-entropy is bounded by conditional von Neumann entropy, and duality transfers this relation to max-entropy.The proof uses optimized operators and operator monotonicity of the logarithm.
  • Entropy properties: α-entropy representation follows by applying the Appendix A functional construction to gα(t)=t^α and h(t)=−t log t.
  • Entropy properties: Isometric invariance of the α-entropies follows from Lemma 13 applied to the corresponding functionals.
  • Entropy properties: Data processing for the entropies follows from Lemma 14 applied to I ⊗ E and the relevant functionals.The proof uses operator concavity or convexity of the functions defining the entropies.

APPENDIX C: ESTIMATE OF Hε

Appendix C proves an explicit smooth min-entropy estimate by constructing a nearby state whose conditional min-entropy is controlled by λ. The construction uses the positive part of ρAB−λ1A⊗σB and a contraction-based fidelity bound.

  • Explicit estimate: Lemma 15 constructs an explicit smooth min-entropy estimate for ρAB relative to σB and λ>0.
  • Construction: The positive operator ∆ is defined as the positive part of ρAB−λ1A⊗σB.
  • Construction: The constructed state ˜ρAB satisfies ˜ρAB≤λ1A⊗σB and therefore Hmin(A|B)˜ρ|σ≥−log λ.
  • Fidelity bound: A purification of ˜ρAB is obtained by applying the operator G to a purification of ρAB, allowing Uhlmann’s theorem to bound the fidelity.
  • Fidelity bound: The construction reaches an ε-ball around ρAB because 2tr(∆)=ε, while G is controlled using its contraction property.The proof uses ||G||≤1 and operator monotonicity of the square root.
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