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arXiv:1511.00293 · IEEE Transactions on Information Theory

Passive States Optimize the Output of Bosonic Gaussian Quantum Channels

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

Published Mar 28, 2016• 38 citations
arXivHTMLPDF

Abstract

An ordering between the quantum states emerging from a single-mode gauge-covariant bosonic Gaussian channel is proved. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e., diagonal with decreasing eigenvalues) produces an output, which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of Mari et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others.

DOI
10.1109/tit.2016.2547426

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arXiv:1607.05262•Jul 18, 2016•7 citations
An ordering between the quantum states emerging from a single-mode gauge-covariant bosonic Gaussian channel is proved. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e., diagonal with decreasing eigenvalues) produces an output, which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of Mari et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others.
The passive states of a quantum system minimize the average energy among all the states with a given spectrum. We prove that passive states are the optimal inputs of single-jump lossy quantum channels. These channels arise from a weak interaction of the quantum system of interest with a large Markovian bath in its ground state, such that the interaction Hamiltonian couples only consecutive energy eigenstates of the system. We prove that the output generated by any input state $\ensuremath{\rho}$ majorizes the output generated by the passive input state ${\ensuremath{\rho}}_{0}$ with the same spectrum of $\ensuremath{\rho}$. Then, the output generated by $\ensuremath{\rho}$ can be obtained applying a random unitary operation to the output generated by ${\ensuremath{\rho}}_{0}$. This is an extension of De Palma et al. [IEEE Trans. Inf. Theory 62, 2895 (2016)], where the same result is proved for one-mode bosonic Gaussian channels. We also prove that for finite temperature this optimality property can fail already in a two-level system, where the best input is a coherent superposition of the two energy eigenstates.
The passive states of a quantum system minimize the average energy among all the states with a given spectrum. We prove that they are the optimal inputs of a large class of lossy quantum channels, in the sense that the output generated by a passive state is still passive and majorizes (i.e. it is less noisy than) the output generated by any other state with the same spectrum. This is an extension of De Palma et al. [arXiv:1511.00293], where the same result is proved for one-mode bosonic Gaussian channels. We also prove that for finite temperature this optimality property can fail already in a two-level system, where the best input is a coherent superposition of the two energy eigenstates.
We show the existence of a majorization ladder in bosonic Gaussian channels, that is, we prove that the channel output resulting from the nth energy eigenstate (Fock state) majorizes the channel output resulting from the (n+1)th energy eigenstate (Fock state). This reflects a remarkable link between the energy at the input of the channel and a disorder relation at its output as captured by majorization theory. This result was previously known in the special cases of a pure-loss channel and quantum-limited amplifier, and we achieve here its non-trivial generalization to any single-mode phase-covariant (or -contravariant) bosonic Gaussian channel. The key to our proof is the explicit construction of a column-stochastic matrix that relates the outputs of the channel for any two subsequent Fock states at its input. This is made possible by exploiting a recently found recurrence relation on multiphoton transition probabilities for Gaussian unitaries [Jabbour and Cerf, Phys. Rev. Res. 3, 043065 (2021)]. Possible generalizations and implications of these results are then discussed.
It is shown that phase-insensitive Gaussian bosonic channels are majorization-preserving over the set of passive states of the harmonic oscillator. This means that comparable passive states under majorization are transformed into equally comparable passive states by any phase-insensitive Gaussian bosonic channel. Our proof relies on a new preorder relation called Fock-majorization, which coincides with regular majorization for passive states but also induces another order relation in terms of mean boson number, thereby connecting the concepts of energy and disorder of a quantum state. The consequences of majorization preservation are discussed in the context of the broadcast communication capacity of Gaussian bosonic channels. Because most of our results are independent of the specific nature of the system under investigation, they could be generalized to other quantum systems and Hamiltonians, providing a new tool that may prove useful in quantum information theory and especially quantum thermodynamics.
Abstract We introduce a class of quantum channels called passive-environment bosonic channels. These channels are relevant from a quantum thermodynamical viewpoint because they correspond to the energy-preserving linear coupling of a bosonic system with a bosonic environment that is in a passive state (no energy can be extracted from it with a unitary transformation) followed by discarding the environment. The Fock-majorization relation defined in Jabbour et al (2016 New J. Phys . 18 073047) happens to be particularly useful in this context as, unlike regular majorization, it connects the disorder of a state together with its energy. Our main result here is the preservation of Fock-majorization across all passive-environment bosonic channels. This implies a similar preservation for regular majorization over the set of passive states, and also extends to the passive-environment bosonic channels whose Stinespring dilation involves an active Gaussian unitary. Beyond bosonic systems, the class of passive-environment operations as defined here naturally generalizes the set of thermal operations and is expected to provide new insights into the thermodynamics of quantum systems.
Recently de Palma et al. [IEEE Trans. Inf. Theory 63, 728 (2017)] proved---using Lagrange multiplier techniques---that under a non-zero input entropy constraint, a thermal state input minimizes the output entropy of a pure-loss bosonic channel. In this note, we present our attempt to generalize this result to all single-mode gauge-covariant Gaussian channels by using similar techniques. Unlike the case of the pure-loss channel, we cannot prove that the thermal input state is the only local extremum of the optimization problem. %It is unclear to us why the same techniques do not lead to a proof for amplifier channels. However, we do prove that, if the conjecture holds for gauge-covariant Gaussian channels, it would also hold for gauge-contravariant Gaussian channels. The truth of the latter leads to a solution of the triple trade-off and broadcast capacities of quantum-limited amplifier channels. We note that de Palma et al. [Phys. Rev. Lett. 118, 160503 (2017)] have now proven the conjecture for all single-mode gauge-covariant Gaussian channels by employing a different approach from what we outline here. Proving a multi-mode generalization of de Palma et al.'s above mentioned result---i.e., given a lower bound on the von Neumann entropy of the input to an nnn-mode lossy thermal-noise bosonic channel, an nnn-mode product thermal state input minimizes the output entropy---will establish an important special case of the conjectured Entropy Photon-number Inequality (EPnI). The EPnI, if proven true, would take on a role analogous to Shannon's EPI in proving coding theorem converses involving quantum limits of classical communication over bosonic channels.