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arXiv:1008.1557 · Journal of Physics A Mathematical and Theoretical

Probing the qudit depolarizing channel

Michael R. Frey, David C. Collins, K. Gerlach

Published Apr 21, 2011• 27 citations
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Abstract

For the quantum depolarizing channel with any finite dimension, we compare three schemes for channel identification: unentangled probes, probes maximally entangled with an external ancilla, and maximally entangled probe pairs. This comparison includes cases where the ancilla is itself depolarizing and where the probe is circulated back through the channel before measurement. Compared on the basis of (quantum Fisher) information gained per channel use, we find broadly that entanglement with an ancilla dominates the other two schemes, but only if entanglement is cheap relative to the cost per channel use and only if the external ancilla is well shielded from depolarization. We arrive at these results by a relatively simple analytical means. A separate, more complicated analysis for partially entangled probes shows for the qudit depolarizing channel that any amount of probe entanglement is advantageous and that the greatest advantage comes with maximal entanglement.

DOI
10.1088/1751-8113/44/20/205306

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For the quantum depolarizing channel with any finite dimension, we compare three schemes for channel identification: unentangled probes, probes maximally entangled with an external ancilla, and maximally entangled probe pairs. This comparison includes cases where the ancilla is itself depolarizing and where the probe is circulated back through the channel before measurement. Compared on the basis of (quantum Fisher) information gained per channel use, we find broadly that entanglement with an ancilla dominates the other two schemes, but only if entanglement is cheap relative to the cost per channel use and only if the external ancilla is well shielded from depolarization. We arrive at these results by a relatively simple analytical means. A separate, more complicated analysis for partially entangled probes shows for the qudit depolarizing channel that any amount of probe entanglement is advantageous and that the greatest advantage comes with maximal entanglement.
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.
We consider the problem of identifying the degree of correlation θ in the depolarization rates of an array of qubits, such as might be found in a holding register in a quantum information processing architecture. We treat this as a problem in quantum channel identification and use the quantum Fisher information about θ in the channel output to compare different candidate probe states. We consider channel probes in general pure N-qubit cat states and find that partial entanglement is always suboptimal; the best cat probe state is either separable or maximally entangled, and we lay out for any given depolarizing rate p and correlation θ, the (p, θ)-regions for which separable and maximally entangled probes each yield the most information. In the case of very large qubit arrays, we find that the quantum Fisher information obtained about θ using any pure cat probe state is asymptotically independent of the state and identically equal to the information available about a classical coin's bias from a single coin toss. Finally, we turn our attention to arrays of N qudits (with common dimension D) that are depolarizing with some correlation θ. We consider both separable and maximally entangled cat probe states and find and compare their quantum Fisher informations in both the large N and large D regimes.
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.
Among the intriguing features of quantum theory, the problem of distinguishing quantum channels is of fundamental interest. In this paper, we focus on the single-shot discrimination of two noisy quantum channels using two distinct classes of probes: single-system (product) probes and entangled probes. Our aim is to identify optimal probing state for specific discrimination tasks and to analyze the necessity and role of entanglement in enhancing channel distinguishability. We show that maximally entangled probes are optimal for discriminating two qubit depolarizing channels, with any nonzero entanglement providing an advantage over single-system probes. In contrast, for dephasing channels in arbitrary dimensions, we prove that single-system probe can be optimal and that entanglement offers no improvement, even when the dephasing unitary is generalized. For qubit amplitude-damping channels, we identify distinct noise-dependent regimes in which either single-system probe outperforms maximally entangled probes and vice-versa. Moreover, we demonstrate that non-maximally entangled probes can act as the optimum probe if the noise parameters restricted to certain values in this task. We also present examples of noisy unitary channels for which discrimination is possible using non-maximally entangled probe, while both single-systems and maximally entangled probes fail. We introduce another class of noisy unitary channels for which perfect discrimination is achievable with a single system, while maximally entangled probes are insufficient. Finally, we show that two erasure channels can be optimally discriminated using any pure single-system probe, with no advantage gained from entanglement.
Quantum mutual information defined in terms of von Neumann entropy captures and quantifies all correlations, quantum and classical, between the two parts of a bipartite quantum system. Within this framework entanglement is the most distinguished type of quantum correlation, and a rich body of theory and experiment establishes that entanglement is a potent and fungible resource for quantum information processing broadly. Bipartite systems can exhibit quantum correlations beyond entanglement. Such non-classical states are called discordant in general and strictly discordant (or dissonant) when the quantum state is separable. We show that strict discord can increase the amount of information available from probing a quantum channel. We focus in this study on the qubit depolarizing channel, using quantum Fisher information to measure the information available about the channel depolarizing probability. We consider channel probes prepared, along with an ancilla, in a separable two-qubit Bell-diagonal state. We prove for Bell-diagonal probes of the qubit depolarizing channel that, in the absence of entanglement and controlling for marginal purity and degree of classical correlation, any increase in strict discord between the probe and ancilla yields an accompanying increase in available statistical information about the channel depolarizing probability.
Quantum channel identification, a standard problem in quantum metrology, is the task of estimating parameter(s) of a quantum channel. We investigate dissonance (quantum discord in the absence of entanglement) as an aid to quantum channel identification and find evidence for dissonance as a resource for quantum information processing. We consider the specific case of dissonant Bell-diagonal probes of the qubit depolarizing channel, using quantum Fisher information as a measure of statistical information extracted by the probe. In this setting dissonant quantum probes yield more statistical information about the depolarizing probability than do corresponding probes without dissonance and greater dissonance yields greater information. This effect only operates consistently when we control for classical correlation between the probe and its ancilla and the joint and marginal purities of the ancilla and probe.