Operator-sum representation for bosonic Gaussian channels

Abstract

Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. The fact that the two-mode metaplectic operators acting as unitary purification of these channels do not, in their canonical form, mix the position and momentum variables is exploited to present a procedure which applies uniformly to all families in the Holevo classification. In this procedure the Kraus operators of every quantum-limited Gaussian channel can be simply read off from the matrix elements of a corresponding metaplectic operator. Kraus operators are employed to bring out, in the Fock basis, the manner in which the antilinear, unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel, denoted D(κ) in the Holevo classification. The matrix transposition channels D(κ), D(κ −1 ) turn out to be a dual pair in the sense that their Kraus operators are related by the adjoint operation. The amplifier channel with amplification factor κ and the beamsplitter channel with attenuation factor κ −1 turn out to be mutually dual in the same sense. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasiprobabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. The semigroup property of the amplifier and attenuator families leads in both cases to a Zeno-like effect arising as a consequence of interrupted evolution. In the cases of entanglement-breaking channels a description in terms of rank one Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that every noisy Gaussian channel can be realised as product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.

Bounding the quantum limits of precision for phase estimation with loss and thermal noise

“…One possible Kraus operator description is found by decomposing the lossy thermal-noise channel into a pure loss channel of transmissivity τ = η/G followed by a quantum-limited amplifier with gain G = 1 + (1 − η)n B [26]. This provides a Kraus representation for the lossy thermal-…”

“…k , where A l are the Kraus operators of the pure loss channel, while B k (G) are the Kraus operators of the quantum limited amplifier [26] given by…”

“…First of all we decompose the lossy thermal-noise channel into a pure loss channel followed by quantum-limited amplifier [26], as depicted in Fig. 3.…”

Strong Converse for the Classical Capacity of Optical Quantum Communication Channels

“…A bosonic Gaussian quantum channel is said to be "quantum-limited' if the inequality above (involving det X and det Y ) is saturated [22], [13], [14], [15]. For instance, phase-insensitive Gaussian channels are quantum-limited when their environment is initially in a vacuum state.…”

“…Phase-insensitive Gaussian channels are invariant with respect to phase-space rotations [10], [11], [12], [13], and they are considered to be one of the most practically relevant models to describe free space or optical fiber transmission, or transmission of classical messages through dielectric media, etc. In fact, phase-insensitive Gaussian channels constitute a broad class of noisy bosonic channels, encompassing all of the following: thermal noise channels (in which the signal photon states are mixed with a thermal state), additive noise channels (in which the input states are randomly displaced in phase space), and noisy amplifier channels [10], [14], [15], [16].…”