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Journal of Physics A Mathematical and Theoretical

Dissonance as a resource for probing the qubit depolarizing channel

Michael R. Frey, Theodore J. Yoder

Published Aug 31, 2012• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1751-8113/45/38/385301

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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.
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.
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.
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.
A fundamental problem in quantum information is to explore what kind of quantum correlations is responsible for successful completion of a quantum information procedure. Here we study the roles of entanglement, discord, and dissonance needed for optimal quantum state discrimination when the latter is assisted with an auxiliary system. In such process, we present a more general joint unitary transformation than the existing results. The quantum entanglement between a principal qubit and an ancilla is found to be completely unnecessary, as it can be set to zero in the arbitrary case by adjusting the parameters in the general unitary without affecting the success probability. This result also shows that it is quantum dissonance that plays as a key role in assisted optimal state discrimination and not quantum entanglement. A necessary criterion for the necessity of quantum dissonance based on the linear entropy is also presented. PACS numbers: 03.65.Ta, 03.67.Mn, 42.50.Dv.
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.