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Quantum Channels and Representation Theory

Ritter, W G

Published Feb 23, 2005• 10 citations
OpenAlex

Abstract

In the study of d-dimensional quantum channels $(d \\geq 2)$, an assumption which is not very restrictive, 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 Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.

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In the study of d-dimensional quantum channels (dgeq2)(d \\geq 2)(dgeq2), an assumption which is not very restrictive, 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 nnn, but for most other representations this is not the case. Since the Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.
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.
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.
By the Majorana representation, for any d>1d > 1d>1 there is a one-one correspondence between a quantum state of dimension ddd and d−1d-1d−1 qubits represented as d−1d-1d−1 points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing d−1d-1d−1 points on the Bloch sphere and the corresponding d−1d-1d−1 qubits representing a ddd-dimensional quantum state. Additionally, we demonstrate how the inner product of two ddd-dimensional quantum states can be expressed as a permanent of a matrix related to their (d−1)(d-1)(d−1)-qubit state representations. Extension of the result to mixed states is also considered.
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of operators belonging to the enveloping Lie algebra that satisfy an approximate symmetry and have the properties required by physics.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
Interest in the Werner-Holevo channel ${\mathrm{\ensuremath{\Lambda}}}_{1}(\ensuremath{\rho})=\frac{1}{2}[\mathrm{tr}(\ensuremath{\rho})I\ensuremath{-}{\ensuremath{\rho}}^{T}]$ has been mainly due to its abstract mathematical properties. We show that in three dimensions and with a slight modification, this channel can be realized as the rotation of qutrit states in random directions by random angles. Our modification takes the form ${\mathrm{\ensuremath{\Lambda}}}_{x}(\ensuremath{\rho})=(1\ensuremath{-}x)\ensuremath{\rho}+x{\mathrm{\ensuremath{\Lambda}}}_{1}(\ensuremath{\rho})$. Therefore, and in view of the potential use of qutrits in quantum processing tasks and their realization in many different platforms, the modified Werner-Holevo channel can be used as a very simple and realistic noise model, in the same way that the depolarizing channel is for qubits. We will make a detailed study of this channel and derive its various properties. In particular, we will use the recently proposed flag extension and other techniques to derive analytical expressions and bounds for the different capacities of this channel. The role of symmetry is revealed in these derivations. We also rigorously prove that the channel ${\mathrm{\ensuremath{\Lambda}}}_{x}$ is antidegradable and hence has zero quantum capacity, in the region $\frac{4}{7}\ensuremath{\le}x\ensuremath{\le}1.$