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arXiv:1708.04256 · Physical Review A

Divergence-free approach for obtaining decompositions of quantum-optical processes

Krishna Kumar Sabapathy, J. Solomon Ivan, Raúl García−Patrón, R. Simon

Published Feb 27, 2018• 3 citations
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Abstract

Operator-sum representations of quantum channels can be obtained by applying the channel to one subsystem of a maximally entangled state and deploying the channel-state isomorphism. However, for continuous-variable systems, such schemes contain natural divergences since the maximally entangled state is ill defined. We introduce a method that avoids such divergences by utilizing finitely entangled (squeezed) states and then taking the limit of arbitrary large squeezing. Using this method, we derive an operator-sum representation for all single-mode bosonic Gaussian channels where a unique feature is that both quantum-limited and noisy channels are treated on an equal footing. This technique facilitates a proof that the rank-1 Kraus decomposition for Gaussian channels at its respective entanglement-breaking thresholds, obtained in the overcomplete coherent-state basis, is unique. The methods could have applications to simulation of continuous-variable channels.

DOI
10.1103/physreva.97.022339

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Operator-sum representations of quantum channels can be obtained by applying the channel to one subsystem of a maximally entangled state and deploying the channel-state isomorphism. However, for continuous-variable systems, such schemes contain natural divergences since the maximally entangled state is ill defined. We introduce a method that avoids such divergences by utilizing finitely entangled (squeezed) states and then taking the limit of arbitrary large squeezing. Using this method, we derive an operator-sum representation for all single-mode bosonic Gaussian channels where a unique feature is that both quantum-limited and noisy channels are treated on an equal footing. This technique facilitates a proof that the rank-1 Kraus decomposition for Gaussian channels at its respective entanglement-breaking thresholds, obtained in the overcomplete coherent-state basis, is unique. The methods could have applications to simulation of continuous-variable channels.
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 $\mathcal{D}(\ensuremath{\kappa})$ in the Holevo classification. The matrix transposition channels $\mathcal{D}(\ensuremath{\kappa})$, $\mathcal{D}({\ensuremath{\kappa}}^{\ensuremath{-}1})$ 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 $\ensuremath{\kappa}$ and the beam-splitter channel with attenuation factor ${\ensuremath{\kappa}}^{\ensuremath{-}1}$ 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 1 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 almost every noisy Gaussian channel can be realized as a product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for these noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.
In the first part of the paper we give a representation for entanglement-breaking channels in separable Hilbert space that generalizes the “Kraus decomposition with rank-one operators” and use it to describe complementary channels. We also note that coherent information for antidegradable channel is always nonpositive. In the second part, we give necessary and sufficient condition for entanglement breaking for the general quantum Gaussian channel. Application of this condition to one-mode channels provides several new cases where the additivity conjecture holds in the strongest form.
We present a framework for studying linear bosonic non-Gaussian channels. Our emphasis is on a class of channels that we call photon-added Gaussian channels and these are experimentally viable with current quantum-optical technologies. A strong motivation for considering these channels is the fact that it is compulsory to go beyond the Gaussian regime for numerous tasks in continuous-variable quantum information processing including universal quantum computation. These channels are obtained by extending Gaussian channels with photon addition applied to the ancilla ports giving rise to a one-parameter family of non-Gaussian channels indexed by photon number n≥1n \geq 1n≥1. We derive its operator-sum representation which is indispensable in the present context. We observe that these channels are Fock-preserving (or coherence non-generating in the Fock basis). Further, noisy Gaussian channels can be expressed as a convex mixture of these non-Gaussian channels. We then report examples of activation of nonclassicality, using this method of photon-addition, at outputs of channels that would otherwise output only classical states. We also derive many structure theorems and information theoretic properties of these channels.
We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.
Hybrid quantum systems combining discrete-variable (DV) and continuous-variable (CV) subsystems arise across a broad range of physical platforms and enable quantum information processing beyond purely qubit-based designs. Their classical simulation, however, is challenging since computational costs of both CV and DV systems exponentially increase. In this paper, we introduce a simulation framework for CV--DV hybrid systems based on linear combination of Gaussian (LCoG) decomposition. We decompose the hybrid density matrix into blocks in a DV basis and represent each block as an LCoG expansion. Within the representation, Gaussian operations on the bosonic subsystem independently act on each Gaussian function. We derive analytic formulas describing how complex Gaussian functions transform under controlled Gaussian operations. We further extend the framework to controlled Gaussian channels with state-dependent Gaussian noise. The method avoids a Fock-space truncation for the CV subsystem, and the cost of updating each Gaussian term scales polynomially with the number of modes. The framework provides a general computational tool for analyzing hybrid quantum algorithms, quantum simulation protocols, and non-Gaussian state generation schemes. As an example, we simulate Gottesman--Kitaev--Preskill (GKP) state generation based on a cavity-QED system.