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arXiv:quant-ph/0205121

Strategies for Estimating Quantum Lossy Channels

Xian-Ting Liang

Published May 20, 2002• 0 citations
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Abstract

Due to the anisotropy of quantum lossy channels one must choose optimal bases of input states for best estimating them. In this paper, we obtain that the equal probability Schrödinger cat states are optimal for estimating a single lossy channel and they are also the optimal bases of input states for estimating composite lossy channels. On the other hand, by using the symmetric logarithmic derivative (SLD) Fisher information of output states exported from the lossy channels we obtain that if we take the equal probability Schrödinger cat states as the bases of input states the maximally entangled inputs are not optimal, however if the bases of the input states are not the equal probability Schrödinger cat states the maximally entangled input states may be optimal for the estimating composite lossy channel.

DOI
10.48550/arxiv.quant-ph/0205121

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Due to the anisotropy of quantum lossy channels one must choose optimal bases of input states for best estimating them. In this paper, we obtain that the equal probability Schrödinger cat states are optimal for estimating a single lossy channel and they are also the optimal bases of input states for estimating composite lossy channels. On the other hand, by using the symmetric logarithmic derivative (SLD) Fisher information of output states exported from the lossy channels we obtain that if we take the equal probability Schrödinger cat states as the bases of input states the maximally entangled inputs are not optimal, however if the bases of the input states are not the equal probability Schrödinger cat states the maximally entangled input states may be optimal for the estimating composite lossy channel.
We investigate an optimal distance of two components in an entangled coherent state for quantum phase estimation in lossy interferometry. The optimal distance is obtained by an economical point, representing the quantum Fisher information that we can extract per input energy. Maximizing the formula of the quantum Fisher information over an input mean photon number, we show that, as the losses of the interferometer increase, it can be more beneficial to prepare an initially entangled coherent state that is less entangled. This represents that the optimal distance of the two-mode components decreases with more loss in the interferometry. Under the constraint of the input mean photon number, we obtain that the optimal entangled coherent state is more robust than a separable coherent state, even in a high photon loss rate. The optimal entangled coherent state preserves quantum advantage over the standard interferometric limit of the separable coherent state. We also show that the corresponding optimal measurement requires correlation measurement bases.
The passive states of a quantum system minimize the average energy among all the states with a given spectrum. We prove that passive states are the optimal inputs of single-jump lossy quantum channels. These channels arise from a weak interaction of the quantum system of interest with a large Markovian bath in its ground state, such that the interaction Hamiltonian couples only consecutive energy eigenstates of the system. We prove that the output generated by any input state $\ensuremath{\rho}$ majorizes the output generated by the passive input state ${\ensuremath{\rho}}_{0}$ with the same spectrum of $\ensuremath{\rho}$. Then, the output generated by $\ensuremath{\rho}$ can be obtained applying a random unitary operation to the output generated by ${\ensuremath{\rho}}_{0}$. This is an extension of De Palma et al. [IEEE Trans. Inf. Theory 62, 2895 (2016)], where the same result is proved for one-mode bosonic Gaussian channels. We also prove that for finite temperature this optimality property can fail already in a two-level system, where the best input is a coherent superposition of the two energy eigenstates.
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The problem of finding the optimal strategy for estimating a generalized amplitude damping channel ${\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ by means of the extension $\mathrm{id}\ensuremath{\bigotimes}{\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ is addressed. We first evaluate the quantum Fisher information of output states based on the symmetric logarithmic derivative and specify all pure-state inputs that maximize the quantum Fisher information. We next investigate the ${\ensuremath{\nabla}}^{e}$ autoparallelity of output state manifolds and characterize the condition for the existence of an efficient estimator. A comparison of these results concludes that, while there is no uniformly optimal input for all ppp and $\ensuremath{\eta}$, a maximally entangled input is an admissible one under a nonasymptotic setting.
The passive states of a quantum system minimize the average energy among all the states with a given spectrum. We prove that they are the optimal inputs of a large class of lossy quantum channels, in the sense that the output generated by a passive state is still passive and majorizes (i.e. it is less noisy than) the output generated by any other state with the same spectrum. This is an extension of De Palma et al. [arXiv:1511.00293], where the same result is proved for one-mode bosonic Gaussian channels. We also prove that for finite temperature this optimality property can fail already in a two-level system, where the best input is a coherent superposition of the two energy eigenstates.
We derive a general criterion to detect entangled states in multipartite systems based on the symmetric logarithmic derivative quantum Fisher information. This criterion is a direct consequence of the fact that separable states do not improve the accuracy upon estimating the one-parameter family of quantum channels. Our result is a generalization of the previously known criterion for the one-parameter unitary channel to any one-parameter quantum channel. Several variants of the proposed criterion are also given, and then the general structure is revealed behind this sort of entanglement criteria based on quantum Fisher information. We discuss several examples to illustrate our criterion. In the last part, we briefly show how the proposed criterion can be extended to a more general setting that is applicable for a certain class of open quantum systems, and we discuss how to detect entangled states even in the presence of decoherence.