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Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
Andreas Winter
TL;DR
Quantum entropy continuity bounds are either trivial in unrestricted infinite dimensions or not yet tight enough in finite dimensions. The paper develops sharper finite-dimensional conditional-entropy and relative-entropy bounds, extends them to entanglement measures using a quantum coupling, and establishes energy-constrained infinite-dimensional analogues. Its conclusions include almost-tight conditional-entropy bounds and energy-constrained continuity in which Gibbs entropy roughly replaces Hilbert-space dimension.
Problem
Explicit uniform continuity bounds are needed for quantum entropies, while unrestricted infinite-dimensional Fannes-type bounds are trivial because entropy is discontinuous there.
Method
The paper combines entropy concavity and classical coupling arguments, extends the principle to relative entropy distances, and introduces a quantum coupling for regularized entanglement of formation.
Results
The paper obtains an almost tight conditional-entropy continuity bound and continuity bounds for infinite-dimensional entropy under bounded expected energy.
Takeaways & Limitations
For harmonic-oscillator systems, Gibbs entropy at the given energy roughly plays the role of Hilbert-space dimension in Fannes-type bounds.
Abstract
from arXiv · showhide
We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, $E_R$, and its regularization $E_R^\infty$, as well as of the entanglement of formation, $E_F$. Using a novel "quantum coupling" of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $E_C=E_F^\infty$. Second, analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.
I. INTRODUCTION
The paper seeks explicit, dimension-sensitive continuity bounds for quantum entropies, extending finite-dimensional Fannes-type reasoning to conditional entropy and infinite-dimensional settings.
- Finite-dimensional entropy continuity becomes useful only through explicit bounds, with Fannes identified as the sharpest form presented.
- The classical proof couples distributions so that the mismatch probability equals one-half their trace-norm distance, then applies entropy monotonicity and Fano’s inequality.
- Dephasing in an eigenbasis reduces the quantum continuity problem to the classical one while preserving one state and contracting trace distance.
- The targeted bounds depend only on trace distance and a global system parameter, addressing conditional entropy, relative entropy distances, and related applications.
- Infinite-dimensional systems make unrestricted Fannes-type bounds trivial, motivating quantitative continuity statements under additional restrictions.
II. CONDITIONAL ENTROPY
The paper derives an almost tight conditional-entropy continuity bound from entropy concavity, applies it to entanglement measures, and introduces a quantum coupling for regularized entanglement of formation.
- Conditional entropy: The proof uses entropy concavity, equivalent to strong subadditivity, together with two concavity inequalities.
- Conditional entropy: The stronger conditional-entropy bound depends on the smaller subsystem dimension rather than both dimensions, including when one Hilbert space is infinite.
- Conditional entropy: The bound is asymptotically matched for large d and small ǫ by a maximally entangled-state example.
- Applications: The tighter conditional-entropy inequality yields tighter continuity bounds for quantum channel capacities and entanglement of formation.
- Applications: The paper proves asymptotic continuity for entanglement cost, using a new quantum-coupling idea for regularized entanglement of formation.
- Quantum coupling: The quantum coupling constructs purifications and an intermediate state with both fidelities at least 1−ǫ, and remains valid in separable infinite dimensions.
III. RELATIVE ENTROPY DISTANCES
The paper extends its continuity-bound method from conditional entropy to relative entropy distances from closed convex sets, then applies the result to entanglement measures. The resulting bounds cover relative entropy of entanglement and its regularization, with dimension dependence on the smaller subsystem and essentially optimal linear terms.
- General relative entropy distances: The method yields asymptotic continuity bounds for relative entropy distance from any closed, convex, bounded set containing a full-rank operator.The proof replaces conditional-entropy concavity with joint convexity of relative entropy.
- General relative entropy distances: Finiteness requires the set to contain at least one full-rank state, while boundedness ensures the relative entropy distance is bounded from below.
- Entanglement applications: For separable states, the distance becomes the relative entropy of entanglement E_R, and the bounds extend to its regularization E_R^∞.
- Entanglement applications: The entanglement bounds depend on the smaller subsystem dimension, so they also apply when the other Hilbert space is infinite dimensional.
- Optimality and scope: The coefficient of the linear trace-distance term is essentially best possible, and the proof strategy cannot improve it further.
- Optimality and scope: Removing convexity remains open; product-state and Gibbs-state families illustrate cases where continuity may hold but discontinuous behavior also occurs.
IV. BOUNDED ENERGY
In infinite-dimensional systems, entropy continuity is recovered for bounded-energy states under regularity assumptions on the Hamiltonian. The resulting bounds replace Hilbert-space dimension with Gibbs entropy evaluated at an energy cutoff, and extend to conditional entropy.
- Motivation and assumptions: Without an energy bound, Fannes-type bounds become trivial because infinite-dimensional entropy is discontinuous; bounded-energy restrictions restore continuity.The paper assumes a Hamiltonian with discrete spectrum, lower-bounded energy, and finite partition function Z(β) for every β > 0.
- Energy-constrained bounds: The entropy and conditional-entropy bounds depend on state energy and trace distance rather than Hilbert-space dimension.The Gibbs entropy at cutoff energies E/ε or E/δ plays the role of the logarithm of dimension in finite-dimensional inequalities.
- Gibbs entropy: The Gibbs entropy as a function of energy is strictly increasing and strictly concave under the Gibbs Hypothesis.This monotonicity and concavity support the maximum-entropy comparisons used in the continuity proofs.
- Proof strategy: The proofs truncate states at an energy cutoff, relate the truncated and original entropies using entropy inequalities, and then apply finite-dimensional Fannes or Alicki-Fannes bounds.The construction uses energy-cutoff projectors and a pinching map while preserving the relevant energy bound.
- Harmonic oscillators: For multiple quantum harmonic oscillators, the derived bounds are asymptotically tight apart from additive offset terms.The paper also notes that the oscillator entropy function g is concave and monotonically increasing with the excitation number.
V. CONCLUSIONS
The paper develops near-tight finite-dimensional continuity bounds and extends the same principles to energy-constrained infinite-dimensional systems. It also identifies open questions about optimality and further tightening.
- The conditional von Neumann entropy bound is improved to an almost tight form using entropy inequalities, specifically concavity.
- The optimal formula depending only on Hilbert-space dimension and trace distance remains an open problem.
- The paper leaves open the optimal form of the fidelity appearing in Proposition 5.
- A quantum-state analogue of coupling random variables is considered as a possible route to alternative or tighter conditional-entropy bounds.
- Energy-constrained continuity bounds are obtained for entropy and conditional entropy on infinite-dimensional systems, and are asymptotically tight for harmonic oscillators despite lacking a universally optimal form.