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Physical Review A

Coding with finite quantum systems

Apostol Vourdas

Published Apr 5, 2002• 15 citations
Publisher

Abstract

Coding using quantum states in an angular-momentum $(2j+1)$-dimensional Hilbert space H is considered in this paper. A concatenated code is studied in two steps. In the first step the space ${H}^{N}$ is considered and the code is its subspace ${H}_{A}$ spanned by the direct products of N angular-momentum states with the same m. In the second step the space ${H}_{A}^{M}$ is considered and the code is its subspace ${H}_{B}$ spanned by the direct products of M angle states with the same m. It is shown that the code introduces redundancy with respect to any transformation.

DOI
10.1103/physreva.65.042321

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Coding using quantum states in an angular-momentum (2j+1)(2j+1)(2j+1)-dimensional Hilbert space H is considered in this paper. A concatenated code is studied in two steps. In the first step the space HN{H}^{N}HN is considered and the code is its subspace HA{H}_{A}HA​ spanned by the direct products of N angular-momentum states with the same m. In the second step the space HAM{H}_{A}^{M}HAM​ is considered and the code is its subspace HB{H}_{B}HB​ spanned by the direct products of M angle states with the same m. It is shown that the code introduces redundancy with respect to any transformation.
Angle states and operators are defined in a (2 j + 1)-dimensional angular momentum Hilbert space H through Fourier transform. Displacement operators in the corresponding quantum phase space which in this case is a toroidal lattice, are generators of SU (2 j + 1) transformations in H. In this context, a concatenated code is studied. In the first step the code is the space H A spanned by the direct products of N angular momentum states with the same m. In the second step the code is the space H B spanned by the direct products of M angle states of the space H A . with the same m.
An angular momentum (2 j + 1)-dimensional Hilbert space H is considered. Symplectic transformations S on the tensor product of N of these spaces H ⊗ ⋅⋅⋅ ⊗ H are studied for the case when the 2 j + 1 is a power of a prime (Galois case). The corresponding operators are calculated numerically. The formalism is applied to quantum coding. A simple repetition code based on the space H A spanned by the direct products of N angular momentum states with the same m , has distance 1. It is shown that a code based on the symplectically transformed space SH A has (in general) larger distance. An example with three qutrits is discussed in detail.
Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.
Quantum detection in an N-dimensional Hilbert space H involves quantum states and corresponding measure ment operators which span an r-dimensional subspace U of H, with r ≤ N. Quantum detection could be restricted to this subspace, but the operations in U are still redundant, since the kets have N components. By applying the singular-value decomposition to the state matrix, it is possible to perform a compression from the subspace U onto a "compressed" space U̅, where the redundancy is removed and kets are represented by r components. The detection can be perfectly reformulated in the "compressed" space, without loss of information, with a greatly reduced complexity. The compression is particularly attractive when r ≪ N, as shown with an example of application to quantum optical communications.
Unitary transformations in an angular momentum Hilbert space H (2 j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU (2 j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H (2 j + 1) in terms of smaller qudits in Hilbert spaces H (2 j i + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.
Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum jjj with a spin 1∕21∕21∕2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.