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arXiv:0908.0421 · Physica Scripta

Depolarization for quantum channels with higher symmetries

A. B. Klimov, Luis Lorenzo Sánchez-Soto

Published Sep 1, 2010• 7 citations
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Abstract

The depolarization channel is usually modelled as a quantum operation that destroys all input information, replacing it by a completely chaotic state. For qubits this has a quite intuitive interpretation as a shrinking of the Bloch sphere. We propose a way to deal with depolarizing dynamics (in the Markov approximation) for systems with arbitrary symmetries.

DOI
10.1088/0031-8949/2010/t140/014009

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The depolarization channel is usually modelled as a quantum operation that destroys all input information, replacing it by a completely chaotic state. For qubits this has a quite intuitive interpretation as a shrinking of the Bloch sphere. We propose a way to deal with depolarizing dynamics (in the Markov approximation) for systems with arbitrary symmetries.
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The noisy quantum channel significantly causes the qubits to lose coherence. The depolarizing quantum channel is one of the most pessimistic quantum channels, which possibly causes the qubit coherence to be entirely lost. The success of the quantum information process depends on the strategy used to prevent the quantum system from interacting with the noisy quantum channel. Therefore, understanding depolarizing quantum operations is essential for investigating the effects of noisy channels on various quantum information processes. In this work, we have successfully implemented the depolarizing quantum channel for polarization photonic qubits in the laboratory. The depolarizing quantum channel was capable of continuously varying the degree of depolarization for all input-state polarizations.
A simple model describing depolarization channels with zero-bandwidth environment is presented and exactly solved. The environment is modeled by Lorentzian, telegraphic, and Gaussian zero-bandwidth noises. Such channels can go beyond the standard Markov dynamics and therefore can illustrate the influence of memory effects of the noisy communication channel on the transmitted information. To quantify the disturbance of quantum states the entanglement fidelity between arbitrary input and output states is investigated.
In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.
We investigate quantum channels, which after a finite number kkk of repeated applications erase all input information, i.e., channels whose kkk-th power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension ddd, the order is bounded by k≤d2−1k\leq d^2-1k≤d2−1, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e., channels with an additional input and output in every step, which after exactly kkk steps retain no information about the initial memory state. We establish an explicit representation for such channels showing that the same bound applies for the memory depth kkk in terms of the memory dimension ddd.