Maximal trace distance between isoenergetic bosonic Gaussian states

Abstract

We locate the set of pairs $(\rho_{1},\rho_{2})$ of Gaussian states of a single mode electromagnetic field that exhibit maximal trace distance subject to the energy constraint $\langle a^{\dagger}a \rangle_{\rho_{1}}=\langle a^{\dagger}a \rangle_{\rho_{2}} = E$. 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 $-E^{2}$, 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.Comment: 11 pages, 2 figure

Linear bosonic quantum channels defined by superpositions of maximally distinguishable Gaussian environments

“…In this work, we define and examine several properties of linear bosonic quantum channels that arise from the linear coupling of a quantum oscillator to a single mode environment, when the initial environment state is prepared in a superposition |Ω + of pure, normalized Gaussian states |ψ 1 , |ψ 2 that are maximally distant subject to the energy constraint a † a |ψ 1(2) = E. The relevant pairs of maximally distant, energy-constrained Gaussian states were derived in Ref. [20]. Taking H S(E) to be the Hilbert spaces of the system and environment, both isomorphic to 2 (C), the quantum channel Ξ is expressed in terms of its Stinespring form…”

“…To explore the properties of a non-Gaussian quantum environment prepared in a superposition of maximally distant isoenergetic Gaussian pure states, it is first necessary to find a pair of Gaussian pure states |ϕ 1 , |ϕ 2 that exhibit minimal fidelity subject to the energy constraint ϕ j |a † a|ϕ j = E, j = 1, 2 [20]. Using the definitions S(z) := e 1 2 (za 2 −za †2 ) for the unitary squeezing operator and D(α) := e αa † −αa for the unitary displacement operator, and defining |(α, z) := D(α)S(z)|0 , where |0 is the Fock vacuum, a pair of minimal fidelity, isoenergetic Gaussian states (unique up to U(1) phase shifts) is given by | ±r(d c (E)), 1 2 ln d c (E) , where…”

Efficient trainability of linear optical modules in quantum optical neural networks

“…This fact contrasts with the case of m qubit registers, for which the relevant unitary group is U(2 m ). Secondly, distinguishability of continuous variable Gaussian states depends on intensity [12], which is preserved by linear optical transformations and can be tuned relative to the circuit size. By contrast, the orbit of U(2 m ) is dense in the m qubit pure state space, from which it follows that the output of Haar distributed quantum circuits is independent of the input state.…”