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arXiv:1512.08225 · New Journal of Physics

Majorization preservation of Gaussian bosonic channels

Michael G. Jabbour, Raúl García−Patrón, Nicolas J. Cerf

Published Jul 27, 2016• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/18/7/073047

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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.
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.
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.
We compare two sets of multimode quantum channels acting on a finite collection of harmonic oscillators: (a) the set of linear bosonic channels, whose action is described as a linear transformation at the phase space level; and (b) Gaussian dilatable channels, that admit a Stinespring dilation involving a Gaussian unitary. Our main result is that the set (a) coincides with the closure of (b) with respect to the strong operator topology. We also present an example of a channel in (a) which is not in (b), implying that taking the closure is in general necessary. This provides a complete resolution to the conjecture posed in Sabapathy and Winter (2017 Phys. Rev. A 95 062309). Our proof technique is constructive, and yields an explicit procedure to approximate a given linear bosonic channel by means of Gaussian dilations. It turns out that all linear bosonic channels can be approximated by a Gaussian dilation using an ancilla with the same number of modes as the system. We also provide an alternative dilation where the unitary is fixed in the approximating procedure. Our results apply to a wide range of physically relevant channels, including all Gaussian channels such as amplifiers, attenuators, phase conjugators, and also non-Gaussian channels such as additive noise channels and photon-added Gaussian channels. The method also provides a clear demarcation of the role of Gaussian and non-Gaussian resources in the context of linear bosonic channels. Finally, we also obtain independent proofs of classical results such as the quantum Bochner theorem, and develop some tools to deal with convergence of sequences of quantum channels on continuous variable systems that may be of independent interest.
A long-standing open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum entropy conjecture for a quantum-limited amplifier is actually sufficient to confirm this capacity conjecture, and we provide a strong argument towards this proof by exploiting a connection between quantum entanglement and majorization theory.
A long-standing open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum entropy conjecture for a quantum-limited amplifier is actually sufficient to confirm this capacity conjecture, and we provide a strong argument towards this proof by exploiting a connection between quantum entanglement and majorization theory.