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arXiv:1607.01215

Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient

Attila Lovas, Attila Andai

Published Jul 5, 2016• 1 citation
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Abstract

The simplest building blocks for quantum computations are the qbit-qbit quantum channels. In this paper we analyse the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels over real and complex state spaces with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. Moreover an algorithm is presented to generate uniformly distributed channels with respect to the Lebesgue measure, which enables further studies. With this algorithm the distribution of trace-distance contraction coefficient (Dobrushin) is investigated numerically by Monte-Carlo simulations, which leads to some conjectures and points out the strange behaviour of the real state space.

DOI
10.48550/arxiv.1607.01215

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The simplest building blocks for quantum computations are the qbit-qbit quantum channels. In this paper we analyse the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels over real and complex state spaces with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. Moreover an algorithm is presented to generate uniformly distributed channels with respect to the Lebesgue measure, which enables further studies. With this algorithm the distribution of trace-distance contraction coefficient (Dobrushin) is investigated numerically by Monte-Carlo simulations, which leads to some conjectures and points out the strange behaviour of the real state space.
The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of the general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.
The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of the general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.
The volume of the quantum mechanical state space over n-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endowed with a monotone metric or a pullback metric is considered too, we give formulas for the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the volume of the space of qubits can be infinite too. We characterize those monotone metrics which generates infinite volume.
The contraction coefficient of the trace distance is a central tool in quantum information, quantifying how strongly a quantum channel degrades the distinguishability of states. However, being an extremal ratio, it captures only the most optimistic behaviour of the channel and is often trivial, even for very noisy channels. Moreover, a single scalar is poorly suited to describe how contraction accumulates under channel composition. In this work we introduce the \emph{contraction and expansion values}, two monotone sequences that refine the contraction and expansion coefficients in the same way singular values refine the operator norm. They arise from a min--max variational principle over subspaces of traceless Hermitian operators, admit an operational interpretation in terms of two state-discrimination games, and are shown to coincide with the Gel'fand or Bernstein numbers of the channel restricted to traceless operators. This identification places the sequences within Pietsch's theory of sss-numbers and yields, in particular, bounds under channel composition that the contraction coefficient alone cannot provide. We establish their main structural properties and compute or estimate them for single-qubit channels, ddd-dimensional amplitude damping channels, and direct-sum channels.
Several techniques of generating random quantum channels, which act on the set of d-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show that they are mathematically equivalent. We discuss under which conditions they give the uniform Lebesgue measure on the convex set of quantum operations and compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute the mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity, and the 2-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed, and their spectral properties are studied using the Bloch representation of a classical probability vector.
The volume of the quantum mechanical state space over n -dimensional real, complex and quaternionic Hilbert spaces with respect to the canonical Euclidean measure is computed, and explicit formulae are presented for the expected value of the determinant also in the general setting. The case when the state space is endowed with a monotone metric or a pull-back metric is also considered; we give formulae for the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the volume of the space of qubits can also be infinite. We characterize those monotone metrics which generate infinite volume.