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arXiv:1608.06221 · Quantum Measurements and Quantum Metrology

Quantum channel-estimation with particle loss: GHZ versus W states

Julien Mathieu Elias Fraïsse, Daniel Braun

Published Jan 30, 2016• 6 citations
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Abstract

Abstract We consider quantum channel-estimation for depolarizing channels and phase-flip channels extended by ancilla qubits and fed with a GHZ orWstate. After application of the channel one or several qubits can be lost, and we calculate the impact of the loss on the quantum Fisher information that determines the smallest uncertainty with which the parameters of these channels can be estimated.

DOI
10.1515/qmetro-2016-0009

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Abstract We consider quantum channel-estimation for depolarizing channels and phase-flip channels extended by ancilla qubits and fed with a GHZ orWstate. After application of the channel one or several qubits can be lost, and we calculate the impact of the loss on the quantum Fisher information that determines the smallest uncertainty with which the parameters of these channels can be estimated.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the kkk-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most kkk, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗k^*k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.
In this work we probe the impact of channel estimation on the performance of quantum LDPC codes. Our channel estimation is based on an optimal estimate of the relevant decoherence parameter via its quantum Fisher information. Using state-of-the art quantum LDPC codes designed for the quantum depolarization channel, and utilizing various quantum probes with different entanglement properties, we show how the performance of such codes can deteriorate by an order of magnitude when optimal channel identification is fed into a belief propagation decoding algorithm. Our work highlights the importance in quantum communications of a viable channel identification campaign prior to decoding, and highlights the tradeoff between entanglement consumption and quantum LDPC code performance.
We consider estimating the parameter associated with the qubit depolarizing channel when the available initial states that might be employed are mixed. We use quantum Fisher information as a measure of the accuracy of estimation to compare protocols which use collections of qubits in product states to one in which the qubits are in a correlated state. We show that, for certain parameter values and initial states, the correlated-state protocol can yield a greater accuracy per channel invocation than the product-state protocols. We show that, for some parameters and initial states, using more than two qubits and channel invocations is advantageous. These results stand in contrast to the known optimal case that uses pure initial states and a single channel invocation on a pair of entangled qubits.
We consider protocols for estimating the parameter which characterizes a single qubit depolarizing channel, with the goal of attaining the most accurate estimate per channel use. The accuracy of any quantum estimation protocol will be quantified via the quantum Fisher information (QFI) since the Cramer-Rao bound implies thata larger QFI yields a smaller lower bound on the possible variance in any estimate of the parameter. Within this framework, the choice of input state prior to channel invocation affects estimation accuracy. The known optimal estimation scheme uses pairs of maximally entangled pure input states, giving a gain over any protocol using unentangled input states. Entangling more than two qubits initially in apure state provides no further gain in the QFI per channel use. We ask if,when the available input states are not pure,these gains persist and if correlating more than two qubits is advantageous. We present a protocol using input states correlated over any number of qubits and compare this to an independent channel useprotocol using uncorrelated mixed states. We show that the correlated state protocol yields gains in the QFI per channel use for certain physically reasonable parameter ranges. We show that, unlike the pure state case, using more than two correlated qubits can be advantageous and we show that, as the initial qubit states become highly mixed, adding additional correlated qubits can provide substantial gains in estimation accuracy. We show that for two qubits, such gainsare attained even when the state prior to channel invocationare separable. Thedepolarizing channelmaps a qubit as
The quantum Fisher information is derived for the task of identifying the qudit depolarizing channel with any dimension. The identification scheme in which a pure state channel input is entangled with an ancilla system is shown to dominate other schemes for all qudit depolarizing channels of any dimension and any depolarizing rate. This extends known results for the qubit depolarizing channel. This dominance, though, is not robust; if the ancilla system itself undergoes any degree of depolarization, no matter how small, entanglement with the ancilla is no longer necessarily optimal. The quantum score operator for the qudit depolarizing channel has a special "quasi-classical" form, readily yielding these various results.