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On Variational Expressions for Quantum Relative Entropies
Mario Berta, Omar Fawzi, Marco Tomamichel
TL;DR
Measured relative entropy asks how much classical relative entropy can be extracted from quantum states by measurement, a quantity that can differ from Umegaki’s quantum relative entropy. This paper develops variational characterizations and extends the comparison to Rényi relative entropies, showing strict differences for noncommuting states and violations of data processing below order 1/2.
Problem
It was open whether measured relative entropy defined by general POVMs equals the projective version, and how the measured and quantum relative entropies compare for noncommuting states.
Source: §I · pp. 1–2, A. Umegaki’s Relative Entropy · p. 7Method
The paper derives variational characterizations of measured relative entropies and uses them to compare measurement-based quantities with quantum relative entropies.
Source: §VI · p. 17Results
Measured and sandwiched Rényi relative entropies agree only for commuting states, except at orders 1/2 and ∞, and the paper gives analytical data-processing counterexamples below order 1/2.
Source: §VI · p. 17Takeaways & Limitations
Measured Rényi relative entropies can be achieved by rank-1 projective measurements, and the paper uses its variational characterizations to analyze additivity properties of relative entropies of recovery.
Source: §VI · p. 17
Abstract
from arXiv · showhide
Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow general positive operator valued measures. Second, we extend the result to Renyi relative entropies and show that for non-commuting states the sandwiched Renyi relative entropy is strictly larger than the measured Renyi relative entropy for $\alpha \in (\frac12, \infty)$, and strictly smaller for $\alpha \in [0,\frac12)$. The latter statement provides counterexamples for the data-processing inequality of the sandwiched Renyi relative entropy for $\alpha < \frac12$. Our main tool is a new variational expression for the measured Renyi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.
I. MEASURED RELATIVE ENTROPY
Measured relative entropy maximizes the classical Kullback–Leibler divergence over measurement statistics from two quantum states. The general POVM and projective definitions coincide, and the maximum is attained by rank-1 projective measurements.
I. MEASURED RELATIVE ENTROPY
- The measured relative entropy is the maximum Kullback–Leibler divergence over statistics attainable from the two states.The general definition optimizes over finite-outcome POVMs.Source: §I · pp. 1–2
- The general measured relative entropy over POVMs equals the projectively measured relative entropy.This resolves whether the two definitions are equal.Source: §I · p. 3
- The maximum can be achieved by rank-1 projective measurements, so optimizing over general POVMs does not increase it.The quantity can be evaluated using the concave optimization problem in Lemma 1.Source: §I · pp. 2–3
II. MEASURED R´ENYI RELATIVE ENTROPY
This section extends measured relative entropy to Rényi divergences and characterizes the resulting measured quantities through variational expressions. General POVMs do not improve on rank-1 projective measurements.
II. MEASURED R´ENYI RELATIVE ENTROPY
- The section extends the previous section’s results to Rényi divergence, defined for α ∈ (1, ∞), and defines measured Rényi relative entropy as before; as α → 1, the divergence converges to Kullback-Leibler divergence.Source: §II · p. 4
- For α < 1, Rényi divergence can remain finite without absolute continuity, but diverges for orthogonal distributions.The section also records that the measured quantity is finite under its stated conditions when the states are not orthogonal.Source: §II · p. 4
- Variational expressions characterize the projectively measured Rényi quantities, with extrema attained for positive definite states.The expressions generalize Alberti’s fidelity theorem to general α in (0,1) ∪ (1,∞).Source: §II · pp. 4–5
- The measured Rényi relative entropy can be achieved by rank-1 projective measurements rather than general POVMs.The proof compares POVM statistics with the variational expression in Lemma 3.Source: §II · pp. 5–6
III. ACHIEVABILITY OF RELATIVE ENTROPY
Measured relative entropy is bounded above by Umegaki’s quantum relative entropy, with equality exactly when the states commute. Thus, for noncommuting states, measurement does not preserve all relative-entropy correlations.
A. Umegaki’s Relative Entropy
- Measured relative entropy is at most Umegaki’s relative entropy, with equality if and only if the states commute.The comparison is established for positive definite ρ and σ.Source: A. Umegaki’s Relative Entropy · p. 7
- The proof compares variational expressions for the two quantities and derives the inequality from the Golden–Thompson inequality.The equality condition is used to establish that equality forces the states to commute.Source: A. Umegaki’s Relative Entropy · pp. 7–8
- For noncommuting states, some quantum correlations measured by relative entropy do not survive measurement.The passage relates this observation to locking classical correlations in quantum states.Source: A. Umegaki’s Relative Entropy · p. 8
B. Sandwiched R´enyi Relative Entropy
Sandwiched Rényi relative entropy is compared with its measured counterpart, which is motivated by its operational significance and data-processing properties. The comparison uses the Araki–Lieb–Thirring inequality and has equality conditions tied to commutation.
B. Sandwiched R´enyi Relative Entropy
- Sandwiched Rényi relative entropies are noted as operationally significant for strong converse exponents in asymmetric binary quantum hypothesis testing.For α ≥ 1/2, they are described as the smallest additive quantum generalization of Rényi divergence satisfying data processing.Source: B. Sandwiched R´enyi Relative Entropy · p. 8
- The section states an inequality between measured and sandwiched Rényi relative entropy, with equality if and only if the states commute.The theorem assumes positive definite ρ and σ.Source: B. Sandwiched R´enyi Relative Entropy · p. 9
- The comparison is proved using the Araki–Lieb–Thirring inequality, whose direction depends on the exponent range.Its equality condition is commutation, except at r = 1.Source: B. Sandwiched R´enyi Relative Entropy · p. 9
IV. VIOLATION OF DATA-PROCESSING FOR α < 1
For noncommuting quantum states, the sandwiched Rényi relative entropy violates data processing when α < 1/2. A rank-1 measurement can attain the measured quantity and exhibit the violation.
IV. VIOLATION OF DATA-PROCESSING FOR α < 1
- For noncommuting positive ρ and σ, sandwiched Rényi relative entropy violates data processing for α < 1/2.The paper reports that a rank-1 measurement achieves the measured quantity and thus violates the inequality.Source: §IV · p. 10
- The comparison used to establish the claim is strict for noncommuting states, with equality when the states commute.The proof identifies noncommutation as the strict case and commutation as the equality case.Source: §IV · p. 10
A. Some Optimization Problems in Quantum Information
The section asks when properties of a relative entropy carry over to quantities formed by optimizing it over a set of states. It uses variational characterizations and convex optimization to derive a sufficient route to superadditivity.
A. Some Optimization Problems in Quantum Information
- The framework considers measures formed by optimizing a relative entropy over a convex, compact set of states.The relative entropy may be Umegaki, measured, or a Rényi variant.Source: A. Some Optimization Problems in… · p. 10
- Examples of the optimized state sets include separable, positive-partial-transpose, non-distillable, quantum Markov, locally recoverable, and k-extendible states.Conditional Rényi entropies are another example, using product states with a fixed marginal.Source: A. Some Optimization Problems in… · p. 10
- A minimax step and strong duality can turn the optimization into a maximization, often yielding a convex or semidefinite optimization problem.For measured relative entropies, the objective in the minimization is linear in the optimized state.Source: A. Some Optimization Problems in… · p. 11
- Superadditivity follows when the objective is itself superadditive and the optimized state sets are closed under tensor products.The argument applies these two conditions to product choices and then maximizes over the variables.Source: A. Some Optimization Problems in… · p. 11
B. Relative Entropy of Recovery
The section develops a variational characterization of measured relative entropy of recovery and uses it to prove superadditivity. It also distinguishes that result from unresolved additivity questions for other recovery quantities.
B. Relative Entropy of Recovery
- A variational characterization of measured relative entropy of recovery is obtained from its measured-entropy expression and strong duality for semidefinite programming.The derivation uses a Choi–Jamiołkowski representation and an exchange of optimization order under minimax conditions.Source: B. Relative Entropy of Recovery · pp. 12–13
- Measured relative entropy of recovery is superadditive on tensor-product inputs under the stated full-marginal support conditions.The proposition compares the quantity on a product pair with the sum of its values on each pair.Source: B. Relative Entropy of Recovery · pp. 13–14
- Additivity of relative entropy of recovery was of interest because it could strengthen a lower bound on quantum conditional mutual information.The section notes that superadditivity would permit a stronger bound; the cited motivation includes recovery and data-processing bounds [6] [16] [23] [27] [44] [45] [54].Source: B. Relative Entropy of Recovery · pp. 12, 14
- The authors do not resolve whether relative entropy of recovery is additive using the variational expression they derive.The compared dual form has a non-normalized optimization operator and a relative-entropy term that can be negative.Source: B. Relative Entropy of Recovery · p. 14
- The argument for the flipped relative entropy of recovery yields additivity under the stated support conditions.The product identity is stated for inputs whose A marginals have full support.Source: B. Relative Entropy of Recovery · p. 14
C. R´enyi Relative Entropy of Recovery
The section extends recovery quantities to Rényi variants and establishes superadditivity for measured Rényi relative entropy of recovery. It also reports fidelity-related endpoint identities and preliminary evidence concerning additivity of the sandwiched recovery quantity.
C. R´enyi Relative Entropy of Recovery
- The measured Rényi relative entropy of recovery is introduced for α ∈ (1, ∞) as a generalization of measured relative entropy of recovery.The section also introduces the corresponding Rényi relative entropy of recovery.Source: C. R´enyi Relative Entropy of Recovery · pp. 15–16
- Measured Rényi relative entropy of recovery is superadditive for all Rényi parameters.The section states this result and gives a proposition for the stated input states and support conditions.Source: C. R´enyi Relative Entropy of Recovery · p. 16
- At α = 1/2, the sandwiched Rényi relative entropy equals the negative logarithm of fidelity, and measured fidelity equals fidelity.The text presents this as the special case connecting the measured and sandwiched quantities.Source: C. R´enyi Relative Entropy of Recovery · p. 16
- Preliminary numerics indicate that sandwiched Rényi relative entropy of recovery may not be additive; weak numerical evidence is reported for violations at α = 2/3.The authors distinguish this evidence from a proof.Source: C. R´enyi Relative Entropy of Recovery · p. 16
- A cited recent preprint reports numerical examples showing that relative entropy of recovery is non-additive [15].The paper contrasts that result with its own weak numerical evidence for the Rényi recovery quantity.Source: C. R´enyi Relative Entropy of Recovery · p. 16
VI. CONCLUSION
The paper gives variational characterizations of measured relative entropies and uses them to establish results on projective measurements, Rényi entropies, and recovery-entropy additivity. It also identifies open extensions involving support conditions and continuous POVMs.
VI. CONCLUSION
- The variational characterizations show that measured relative entropies can be achieved by rank-1 projective measurements rather than general POVMs.Source: §VI · p. 17
- Measured Rényi relative entropy equals its sandwiched counterpart exactly for commuting states, except at parameters 1/2 and infinity, where they always agree.Source: §VI · p. 17
- For Rényi parameters below 1/2, the paper gives analytical counterexamples to data processing for sandwiched Rényi relative entropy.Source: §VI · p. 17
- The authors apply their variational characterizations to analyze additivity properties of various relative entropies of recovery.Source: §VI · p. 17
- The paper identifies weakening support conditions for equality results and extending measured relative entropy to continuous POVMs as open directions.It also suggests applying the variational characterizations to operational entropic quantities.Source: §VI · p. 17
Appendix A: Proofs for Results in Section V
The appendix proves recovery-entropy results through convex optimization and duality. For tensor-product inputs, feasible product operators establish the bound needed for additivity.
Appendix A: Proofs for Results in Section V
- The Lemma 10 proof formulates relative entropy of recovery as a convex optimization problem and adds a trace-one constraint to complete the proof.Source: Appendix A · pp. 17–18
- For the argument-flipped recovery quantity, the proof forms a Lagrangian and uses the measured-relative-entropy variational expression in the dual optimization.Source: Appendix A · p. 18
- For tensor-product inputs, product-feasible operators give the reverse bound to (A18), establishing additivity of the argument-flipped recovery quantity.Source: Appendix A · p. 19