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CRC Press eBooks

Quantum coding with systems with finite Hilbert space

Apostol Vourdas

Published Nov 30, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781482269079-121

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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.
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.
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.
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.
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.
Squeezing transformations in the discrete Z2j+1{\mathit{Z}}_{2\mathit{j}+1}Z2j+1​\ifmmode\times\else\texttimes\fi{}Z2j+1{\mathit{Z}}_{2\mathit{j}+1}Z2j+1​ angle--angular-momentum phase space are shown to be associated with the SL(2,Z2j+1{\mathit{Z}}_{2\mathit{j}+1}Z2j+1​) group and important special cases are explicitly constructed. ``Spherical'' bases are introduced in the direct sum of all the Hilbert spaces H2j+1{\mathit{H}}_{2\mathit{j}+1}H2j+1​ and the corresponding representations are defined. Transformations of these bases, using area preserving diffeomorphisms on a sphere, are studied and potential applications in quantum-optics models are discussed.
Phase operators and phase states are introduced in the Hilbert space H2j+1{H}_{2j+1}H2j+1​ associated with the SU(2) group. The phase operators obey the SU(2) algebra and play a dual role to the standard angular-momentum operators. A finite Weyl group plays a fundamental role in those ideas. In the SU(1,1) case the exponential of the phase operators is nonunitary, and the phase states form an overcomplete set which is used to formulate an analytic representation.