. The notion of nonclassicality distance is used to show that the initial environment superposition state becomes maximally nonclassical as the constraint energy is increased. When the system is initially prepared in a coherent state, application of the quantum channel defined by

U_{\text{BS}}

results in a nonclassical state for all values of the environment energy constraint. We also discuss the following properties of the quantum channels: 1) the maximal noise that a coherent system can tolerate, beyond which the linear bosonic attenuator channel defined by

U_{\text{BS}}

cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by

U_{\text{TM}}

, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.

DOI: 10.48550/arxiv.1703.02405

Related papers

Linear bosonic quantum channels defined by superpositions

Tyler J. Volkoff

A minimal energy quantum superposition of two maximally distinguishable, isoenergetic single mode Gaussian states is used to construct the system-environment representation of a class of linear bosonic quantum channels acting on a single bosonic mode. The quantum channels are further defined by unitary dynamics of the system and environment corresponding to either a passive linear optical element U_{BS} or two-mode squeezing U_{TM}. The notion of nonclassicality distance is used to show that the initial environment superposition state becomes maximally nonclassical as the constraint energy is increased. When the system is initially prepared in a coherent state, application of the quantum channel defined by U_{BS} results in a nonclassical state for all values of the environment energy constraint. We also discuss the following properties of the quantum channels: 1) the maximal noise that a coherent system can tolerate, beyond which the linear bosonic attenuator channel defined by U_{BS} cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by U_{TM}, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.A minimal energy quantum superposition of two maximally distinguishable, isoenergetic single mode Gaussian states is used to construct the system-environment representation of a class of linear bosonic quantum channels acting on a single bosonic mode. The quantum channels are further defined by unitary dynamics of the system and environment corresponding to either a passive linear optical element U_{BS} or two-mode squeezing U_{TM}. The notion of nonclassicality distance is used to show that the initial environment superposition state becomes maximally nonclassical as the constraint energy is increased. When the system is initially prepared in a coherent state, application of the quantum channel defined by U_{BS} results in a nonclassical state for all values of the environment energy constraint. We also discuss the following properties of the quantum channels: 1) the maximal noise that a coherent system can tolerate, beyond which the linear bosonic attenuator channel defined by U_{BS} cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by U_{TM}, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.

Quantum Information and ComputationMay 1, 20180 citations

Fock majorization in bosonic quantum channels with a passive environment

Michael G. Jabbour, Nicolas J. Cerf

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.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.

arXiv:1806.06044Journal of Physics A Mathematical and TheoreticalNov 14, 20185 citations

Maximal trace distance between isoenergetic bosonic Gaussian states

T. J. Volkoff

We locate the set of pairs (ρ1, ρ2) of Gaussian states of a single mode electromagnetic field that exhibit maximal trace distance subject to the energy constraint ⟨a†a⟩ρ1=⟨a†a⟩ρ2=E. Any such pair allows to achieve the minimum possible error in the task of binary distinguishability of two single mode, isoenergetic Gaussian quantum signals. In particular, we show that the logarithm of the minimal error probability for distinguishing two maximally trace distant, isoenergetic Gaussian states scales as −E2, less than the achievable scaling of the minimal error probability for distinguishing, e.g., a pair of isoenergetic Heisenberg-Weyl coherent states with energy E or a pair of isoenergetic quadrature squeezed states with energy E. For the case of a field consisting of M > 1 modes, we locate the set of pairs of maximally trace distant isoenergetic, isocovariant Gaussian states. These results have basic applications in the theory of continuous variable quantum communications with Gaussian states of light.We locate the set of pairs (ρ1, ρ2) of Gaussian states of a single mode electromagnetic field that exhibit maximal trace distance subject to the energy constraint ⟨a†a⟩ρ1=⟨a†a⟩ρ2=E. Any such pair allows to achieve the minimum possible error in the task of binary distinguishability of two single mode, isoenergetic Gaussian quantum signals. In particular, we show that the logarithm of the minimal error probability for distinguishing two maximally trace distant, isoenergetic Gaussian states scales as −E2, less than the achievable scaling of the minimal error probability for distinguishing, e.g., a pair of isoenergetic Heisenberg-Weyl coherent states with energy E or a pair of isoenergetic quadrature squeezed states with energy E. For the case of a field consisting of M > 1 modes, we locate the set of pairs of maximally trace distant isoenergetic, isocovariant Gaussian states. These results have basic applications in the theory of continuous variable quantum communications with Gaussian states of light.

arXiv:1706.05807Journal of Mathematical PhysicsDec 1, 20172 citations

Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels

Kunal Sharma, Mark M. Wilde, Sushovit Adhikari +1 more

We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the ‘data-processing bound,’ is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the ‘ ε -degradable bound’ and the ‘ ε -close-degradable bound,’ are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the ‘data-processing bound,’ is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the ‘ ε -degradable bound’ and the ‘ ε -close-degradable bound,’ are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.

arXiv:1708.07257New Journal of PhysicsApr 30, 201859 citations

Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels

Kunal Sharma, Mark M. Wilde, Sushovit Adhikari +1 more

We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the 'data-processing bound,' is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the 'ϵ-degradable bound' and the 'ϵ-close-degradable bound,' are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the 'data-processing bound,' is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the 'ϵ-degradable bound' and the 'ϵ-close-degradable bound,' are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.

Civil War Book ReviewJun 1, 201850 citations

Gaussian Optimality of Energy-Constrained One-Shot Communication through Single-Mode Bosonic Gaussian Channels

Se-Wan Ji

We prove Gaussian optimality for the energy-constrained one-shot Holevo capacity of single-mode bosonic Gaussian channels, including phase-sensitive channels with arbitrarily squeezed thermal noise. The unresolved regime is the low energy branch, where the Gaussian optimizer modulates only one quadrature and minimum-output-entropy arguments cannot decouple the average state from the letters. For pure one-mode dilations we retain a stronger pointwise result: moment-matched Gaussianization improves the fixed average Holevo function for every input. For mixed environments, whose purification produces a 1:2 entanglement-of-formation problem, we avoid any generic 1:2 Gaussian extremality conjecture. The optimal Gaussian letter selects an effective environmental Schmidt mode and an affine two-mode EPR witness. Its null direction is exactly the modulated quadrature, while its slope equals the negative derivative of the Gaussian letter-output entropy. This produces a supporting lower bound on the dilated entanglement of formation; Gaussian maximum entropy and concavity then give a global upper bound tangent at the Gaussian optimizer. Consequently the known Gaussian formulas for attenuating, amplifying, phase-conjugating, and additive-noise fiducial channels are exact over unrestricted ensembles at every input energy. Via the passive-input fiducial decomposition and energy-constrained continuity, the result extends to every single-mode Gaussian channel, including lower-rank canonical limits. No additivity across channel uses is assumed.We prove Gaussian optimality for the energy-constrained one-shot Holevo capacity of single-mode bosonic Gaussian channels, including phase-sensitive channels with arbitrarily squeezed thermal noise. The unresolved regime is the low energy branch, where the Gaussian optimizer modulates only one quadrature and minimum-output-entropy arguments cannot decouple the average state from the letters. For pure one-mode dilations we retain a stronger pointwise result: moment-matched Gaussianization improves the fixed average Holevo function for every input. For mixed environments, whose purification produces a 1:2 entanglement-of-formation problem, we avoid any generic 1:2 Gaussian extremality conjecture. The optimal Gaussian letter selects an effective environmental Schmidt mode and an affine two-mode EPR witness. Its null direction is exactly the modulated quadrature, while its slope equals the negative derivative of the Gaussian letter-output entropy. This produces a supporting lower bound on the dilated entanglement of formation; Gaussian maximum entropy and concavity then give a global upper bound tangent at the Gaussian optimizer. Consequently the known Gaussian formulas for attenuating, amplifying, phase-conjugating, and additive-noise fiducial channels are exact over unrestricted ensembles at every input energy. Via the passive-input fiducial decomposition and energy-constrained continuity, the result extends to every single-mode Gaussian channel, including lower-rank canonical limits. No additivity across channel uses is assumed.

quant-pharXiv:2608.17239Aug 19, 20260 citations

U_{\text{BS}}

or two-mode squeezing

U_{\text{TM}}

U_{\text{BS}}

U_{\text{BS}}

cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by

U_{\text{TM}}

, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.