[Paper Review] Linear bosonic quantum channels defined by superpositions of maximally distinguishable Gaussian environments

Tyler Volkoff|arXiv (Cornell University)|Mar 7, 2017

Quantum Information and Cryptography38 references3 citations

TL;DR

This paper introduces a new class of linear bosonic quantum channels by coupling a system mode to an environment prepared in a superposition of two maximally distinguishable, isoenergetic Gaussian states. Using unitary dynamics via beam splitters (U_BS) or two-mode squeezing (U_TM), the channels generate nonclassical states even from coherent inputs, with the key result that nonclassicality persists under arbitrarily high energy constraints, and a lower bound on the trace norm contraction coefficient is derived for energy-constrained Gaussian states.

ABSTRACT

A minimal energy quantum superposition of two maximally distinguishable, isoenergetic single mode Gaussian states is used to construct the system-environment representation of a class of linear bosonic quantum channels acting on a single bosonic mode. The quantum channels are further defined by unitary dynamics of the system and environment corresponding to either a passive linear optical element $U_{ ext{BS}}$ or two-mode squeezing $U_{ ext{TM}}$. The notion of nonclassicality distance is used to show that the initial environment superposition state becomes maximally nonclassical as the constraint energy is increased. When the system is initially prepared in a coherent state, application of the quantum channel defined by $U_{ ext{BS}}$ results in a nonclassical state for all values of the environment energy constraint. We also discuss the following properties of the quantum channels: 1) the maximal noise that a coherent system can tolerate, beyond which the linear bosonic attenuator channel defined by $U_{ ext{BS}}$ cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by $U_{ ext{TM}}$, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.

Motivation & Objective

• To study linear bosonic quantum channels defined by superpositions of maximally distinguishable, energy-constrained Gaussian states in the environment.
• To analyze how such channels generate nonclassical states when the system is initially coherent.
• To derive a lower bound for the trace norm contraction coefficient on the convex hull of energy-constrained Gaussian states.
• To quantify the noise tolerance and amplification effects of these channels, particularly under phase-preserving dynamics.
• To establish a foundation for future analysis of communication capacities and entropy production in non-Gaussian linear bosonic channels.

Proposed method

• Constructs the system-environment representation of the channel using a minimal-energy superposition of two maximally trace-distant, isoenergetic single-mode Gaussian states in the environment.
• Implements unitary dynamics via a beam splitter (U_BS) or two-mode squeezing (U_TM) to define the channel evolution.
• Uses the nonclassicality distance metric to quantify the degree of nonclassicality in the output state as a function of environment energy.
• Applies the Husimi Q-function and trace norm distance to derive a lower bound on the contraction coefficient for all diameters of the convex hull of energy-constrained Gaussian states.
• Leverages covariance under a Z2 group to generalize the contraction coefficient bound to any channel with such symmetry.
• Analyzes the channel's behavior under increasing energy constraints and compares nonclassicality generation to even coherent state environments.

Experimental results

Research questions

• RQ1 Can a linear bosonic channel defined by a superposition of maximally distinguishable Gaussian environments generate nonclassical states from a coherent input, even under arbitrarily high energy constraints?
• RQ2 What is the maximal thermal noise level beyond which the channel fails to impart nonclassical correlations to a coherent input?
• RQ3 How does the noise added by the phase-preserving linear amplifier channel (U_TM) compare when the environment is initialized in the superposition state versus an even coherent state?
• RQ4 What is the lower bound for the trace norm contraction coefficient of the channel on the closed, convex hull of energy-constrained Gaussian states?
• RQ5 How does the nonclassicality distance of the output state compare between the superposition environment and an even coherent state environment at high energy?

Key findings

• The channel defined by U_BS maps a coherent state to a nonclassical state for all values of the environment energy constraint, including in the asymptotic limit of infinite energy.
• The critical thermal noise threshold beyond which nonclassicality cannot be imparted to a coherent input is derived, establishing a limit on the noise tolerance of the channel.
• The second moment noise generated by the phase-preserving linear amplifier channel (U_TM) is computed explicitly for both coherent and superposition-environment initial states.
• A general lower bound for the trace norm contraction coefficient of the Ξ channels is derived, valid for all diameters of the convex hull of energy-constrained Gaussian states.
• The nonclassicality distance of the output state under the superposition environment is shown to be larger than that under an even coherent state environment, indicating enhanced nonclassicality generation.
• The construction ensures that the environment superposition state becomes maximally nonclassical as the energy constraint increases, validating its use as a resource for non-Gaussian dynamics.