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Journal of Physics A Mathematical and General

Symplectically entangled states and their applications to coding

Apostol Vourdas

Published Feb 18, 2004• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/37/9/017

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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.
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.
Quantum systems with finite Hilbert space are considered. Position and mo- \nmentum states and their relation through a Fourier transform, displacement \nin the position-momentum phase-space, and symplectic transformations are \nintroduced and their properties are studied. Symplectic Sp(2l;Zp) trans- \nformations in l-partite finite system are explicit constructed. The general \nmethod is applied to bi-partite and tri-partite systems. The effect of these \ntransformations on the correlations is discussed. Entanglement calculations \nbetween the subsystems in a bi-partite system and a tri-partite system are \npresented. The effect of measurements is also studied.
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.