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Physica Scripta

Superpositions of generalized Schrödinger cat states exposed to the amplitude damping noise

Filippo Giraldi

Published May 19, 2025• 1 citation
Publisher

Abstract

Abstract Superpositions of even and odd Schrödinger cat states, exposed to the amplitude damping noise, exhibit regularities in the time evolution that allow to encode a logical qubit. In the present scenario, we evaluate how the above-mentioned encoding procedure is affected by perturbations of the canonical coherent states that form the initial superposition. The above-mentioned regularities are maintained, over determined times, by perturbations of the canonical coherent states given by Wright generalized coherent states, but do not hold for perturbations given by Mittag-Leffler generalized coherent states. Thus, the above-mentioned encoding procedure can be performed with mutually orthogonal Wright generalized Schrödinger cat states exposed to the amplitude damping noise. The encoding procedure holds for special values of the label over time intervals that are determined by the distribution of the negative real zeros of the Wright function.

DOI
10.1088/1402-4896/adda9a

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Abstract Superpositions of even and odd Schrödinger cat states, exposed to the amplitude damping noise, exhibit regularities in the time evolution that allow to encode a logical qubit. In the present scenario, we evaluate how the above-mentioned encoding procedure is affected by perturbations of the canonical coherent states that form the initial superposition. The above-mentioned regularities are maintained, over determined times, by perturbations of the canonical coherent states given by Wright generalized coherent states, but do not hold for perturbations given by Mittag-Leffler generalized coherent states. Thus, the above-mentioned encoding procedure can be performed with mutually orthogonal Wright generalized Schrödinger cat states exposed to the amplitude damping noise. The encoding procedure holds for special values of the label over time intervals that are determined by the distribution of the negative real zeros of the Wright function.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
The Wigner distribution function for the time-dependent quadratic Hamiltonian system in the coherent Schrodinger cat state is investigated. The type of state we consider is a superposition of two coherent states, which are by an angle of π out of phase with each other. The exact Wigner distribution function of the system is evaluated under a particular choice of phase, δc, q. Our development is employed for two special cases, namely, the Caldirola–Kanai oscillator and the frequency stable damped harmonic oscillator. On the basis of the diverse values of the Wigner distribution function that were plotted, we analyze the nonclassical behavior of the systems.
We analyze a generalized quantum error-correction code against photon loss where a logical qubit is encoded into a subspace of a single oscillator mode that is spanned by distinct multicomponent cat states (coherent-state superpositions). We present a systematic code construction that includes the extension of an existing one-photon-loss code to higher numbers of losses. When subject to a photon loss (amplitude damping) channel, the encoded qubits are shown to exhibit a cyclic behavior where the code and error spaces each correspond to certain multiples of losses, half of which can be corrected. As another generalization we also discuss how to protect logical qudits against photon losses, and as an application we consider a one-way quantum communication scheme in which the encoded qubits are periodically recovered while the coherent-state amplitudes are restored as well at regular intervals.