[Paper Review] Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels

Giacomo De Palma, Dario Trevisan+1 more|arXiv (Cornell University)|May 1, 2017

Quantum Information and Cryptography2 references3 citations

TL;DR

This paper establishes that thermal Gaussian states minimize the Wehrl entropy among all quantum states with a given von Neumann entropy in the multimode regime, and that they also minimize the output von Neumann entropy of multimode quantum Gaussian channels among input states diagonal in a product basis. The results are derived using additivity of constrained minimum output entropy for quantum-classical and classical-quantum channels, extending prior one-mode results and providing strong evidence for the long-standing minimum output entropy conjecture in quantum information theory.

ABSTRACT

We prove in the multimode scenario a fundamental relation between the Wehrl and the von Neumann entropy, stating that the minimum Wehrl entropy among all the quantum states with a given von Neumann entropy is achieved by thermal Gaussian states. We also prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers and phase-contravariant channels among all the input states diagonal in some product basis and with a given entropy. This result constitutes a major step towards the proof of the same property for generic input states, which is still an open conjecture. This conjecture is necessary to determine the maximum communication rates for the triple trade-off coding and broadcast communication with the Gaussian quantum-limited attenuator. Finally, we prove that the tensor product of n identical geometric input probability distributions minimizes the output Shannon entropy of the n-mode thinning among all the input probability distributions with a given entropy.

Motivation & Objective

• To extend the fundamental relation between Wehrl and von Neumann entropies from the one-mode to the multimode quantum regime.
• To prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers, and phase-contravariant channels among diagonal-in-product-basis inputs with fixed entropy.
• To provide a major step toward proving the general minimum output entropy conjecture for quantum Gaussian channels with arbitrary input states.
• To establish the multimode extension of the constrained minimum output entropy property for the thinning map, showing geometric distributions minimize output Shannon entropy.

Proposed method

• Proved additivity of the constrained minimum output entropy (CMOE) for quantum-classical and classical-quantum channels using induction and convexity arguments.
• Applied the additivity result to prove that the minimum Wehrl entropy for a given von Neumann entropy is achieved by thermal Gaussian states in the multimode scenario.
• Used the additivity framework to show that n-mode thermal Gaussian states minimize the output von Neumann entropy of n-mode tensor-product quantum Gaussian channels when inputs are diagonal in a product basis.
• Extended the CMOE property to the n-mode thinning map by leveraging the additivity result and the known one-mode result on geometric distributions.
• Employed convexity and monotonicity analysis of entropy functions, particularly involving the entropy function g(λ) and its inverse, to prove the required inequalities.
• Utilized auxiliary lemmas on the convexity and monotonicity of composite functions involving the entropy function g and its inverse.

Experimental results

Research questions

• RQ1 Does the minimum Wehrl entropy for a given von Neumann entropy remain achieved by thermal Gaussian states in the multimode regime, as in the one-mode case?
• RQ2 Can the constrained minimum output entropy conjecture for quantum Gaussian channels be proven for input states diagonal in a product basis in the multimode setting?
• RQ3 Is the output entropy of the n-mode thinning map minimized by the tensor product of identical geometric distributions among all input distributions with a given entropy?
• RQ4 Does the additivity of the constrained minimum output entropy hold for quantum-classical and classical-quantum channels in the multimode setting?
• RQ5 Can the multimode extension of the minimum output entropy property for the thinning map be rigorously established using the additivity framework?

Key findings

• The minimum Wehrl entropy among all quantum states with a given von Neumann entropy is achieved by thermal Gaussian states in the multimode scenario, confirming a conjecture previously proven only in the one-mode case.
• For multimode quantum Gaussian channels (attenuators, amplifiers, phase-contravariant), thermal Gaussian input states minimize the output von Neumann entropy among all input states diagonal in a product basis with a fixed entropy.
• The result provides strong evidence for the general minimum output entropy conjecture, which remains open for arbitrary input states, and is crucial for determining maximum communication rates in triple trade-off and broadcast quantum communication.
• The tensor product of n identical geometric input probability distributions minimizes the output Shannon entropy of the n-mode thinning map among all input distributions with a given entropy, extending a known one-mode result.
• The additivity of the constrained minimum output entropy was rigorously proven for quantum-classical and classical-quantum channels, enabling the extension of one-mode results to the multimode regime.
• The proof relies on novel convexity and monotonicity properties of entropy functions, particularly involving the inverse of the entropy function g(λ), which were established through detailed analysis of logarithmic and rational functions.