Geometry on the manifold of Gaussian quantum channels

Abstract

In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamiołkowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaussian entanglement breaking and incompatibility breaking channels. Finally, we show that, when given the purities of the Choi-Jamiołkowski state of the channel, one can determine whether or not such channel is incompatibility breaking. arXiv:1908.07285v1 [quant-ph]

Nonequilibrium readiness and precision of Gaussian quantum thermometers

“…The MCP theorem states that each Riemannian metric for discrete-variable systems is characterized by a correspondence with a set of functions that, by satisfying very restrictive conditions, can lead unambiguously to the quantum Fisher information metric [51,52]. The restriction to Gaussian states, however, greatly simplifies the problem as the first and second moments of the system are the only elements to be considered for its complete description [53,54].…”

Non-equilibrium readiness and accuracy of Gaussian Quantum Thermometers

“…The MCP theorem states that each Riemannian metric for discrete variable systems is characterized by a correspondence with a set of functions that, by satisfying very restrictive conditions, can lead unambiguously to the quantum Fisher information metric [48,49]. The restriction to Gaussian states, however, greatly simplifies the problem as the first and the second moments of the system are the only elements to be considered for its complete description [50,51].…”

Geometry of symmetric and noninvertible Pauli channels

“…Additionally, for the Pauli channels, the volume of non-Markovian dynamical maps was found for the convex combinations of Markovian semigroups [10] and their simple generalization [11]. For continuous variable systems, the geometry of the Gaussian channels has been analyzed for the Bures-Fisher [12] and Hilbert-Schmidt [13] metrics.…”