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Quantum Information and Computation

On unital qubit channels

Man-Duen Choi, Chi-Kwong Li

Published May 1, 2023• 2 citations
Publisher

Abstract

A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients $p_1, \dots, p_m$ as long as $2(p_1, \dots, p_m)$ is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.

DOI
10.26421/qic23.7-8-2

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•Jan 1, 2013•5 citations
A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_mp1​,…,pm​ as long as 2(p1,…,pm)2(p_1, \dots, p_m)2(p1​,…,pm​) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.
A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_mp1​,…,pm​ as long as 2(p1,…,pm)2(p_1, \dots, p_m)2(p1​,…,pm​) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.
We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.
We investigate the set of quantum channels acting on a single qubit. We provide an alternative, compact generalization of the Fujiwara–Algoet conditions for complete positivity to non-unital qubit channels, which we then use to characterize the possible geometric forms of the pure output of the channel. We provide universal sets of quantum channels for all unital qubit channels as well as for all extremal (not necessarily unital) qubit channels, in the sense that all qubit channels in these sets can be obtained by concatenation of channels in the corresponding universal set. We also show that our universal sets are essentially minimal.
The Bloch representation of an n-qubit channel provides a way to represent quantum channels as certain affine transformations on [special characters omitted]. In higher dimensions (n > 1), the correspondence between quantum channels and their Bloch representations is not well-understood. Partly motivated by the ability to simplify the calculation of information theoretic quantities of a qubit channel using the Bloch representation, in this thesis we investigate the correspondence between a channel and its Bloch representation with an emphasis on unital n-qubit channels, in which case the Bloch representation is linear. The thesis is divided into three main sections. First we focus our attention on qubit channels. For certain sets of quantum channels, we establish the surprising existence of a special isomorphism into the set of classical channels. We classify the sets of qubit channels with this property and show that information theoretic quantities are preserved by such classical representations. In a natural progression, we prove some well-known facts about SO(3), the proofs of which are either nonexistent or difficult to find in the literature. Some of this work is based on [12, 13]. In the next section, we consider the multi-qubit channels and show that every finite group can be realized as a subgroup of the quantum channels; this approach allows for the construction of a quantum representation for the free affine monoid over any finite group and gives a classical representation for it. We extend some fundamental results from [26, 28] to the multi-qubit case, including that the set of diagonal Bloch matrices is equal to the free affine monoid over the involution group [special characters omitted]. Some of this work appeared in [10]. Lastly, we study the extreme points for the set of n-qubit channels. There are two types of extreme points: invertible and non-invertible; invertible channels are non-singular maps for which the inverse is also a channel. We briefly study the non-invertible extreme points and then parameterize and analyze the invertiblen-qubit Bloch matrices, which form a compact connected Lie group. We calculate the Lie algebra and give an explicit generating set for the invertible Bloch matrices and a maximal torus.