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arXiv:quant-ph/9906099 · Journal of Optics B Quantum and Semiclassical Optics

DiscreteQ- andP-symbols for spins

Jean-Pierre Amiet, Stefan Weigert

Published Apr 1, 2000• 24 citations
arXivHTMLPDF

Abstract

Non-orthogonal bases of projectors on coherent states are introduced to expand Hermitean operators acting on the Hilbert space of a spin s . It is shown that the expectation values of a Hermitean operator  in a family of (2 s + 1) 2 spin-coherent states determine the operator unambiguously. In other words, knowing the Q -symbol of  at (2 s + 1) 2 points on the unit sphere is already sufficient in order to recover the operator. This provides a straightforward method to reconstruct the mixed state of a spin since its density matrix is explicitly parametrized in terms of expectation values. Furthermore, a discrete P -symbol emerges naturally which is related to a basis dual to the original one.

DOI
10.1088/1464-4266/2/2/309

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Non-orthogonal bases of projectors on coherent states are introduced to expand Hermitean operators acting on the Hilbert space of a spin s . It is shown that the expectation values of a Hermitean operator  in a family of (2 s + 1) 2 spin-coherent states determine the operator unambiguously. In other words, knowing the Q -symbol of  at (2 s + 1) 2 points on the unit sphere is already sufficient in order to recover the operator. This provides a straightforward method to reconstruct the mixed state of a spin since its density matrix is explicitly parametrized in terms of expectation values. Furthermore, a discrete P -symbol emerges naturally which is related to a basis dual to the original one.
To reconstruct a mixed or pure quantum state of a spin s is possible through coherent states: its density matrix is fixed by the probabilities to measure the value s along 4s(s+1) appropriately chosen directions in space. Thus, after inverting the experimental data, the statistical operator is parametrized entirely by expectation values. On this basis, a symbolic calculus for quantum spins is developed, the `expectation-value representation.' It resembles the Moyal representation for SU(2) but two important differences exist. On the one hand, the symbols take values on a discrete set of points in phase space only. On the other hand, no quasi-probabilities - that is, phase-space distributions with negative values - are encountered in this approach.
The expectation values of a Hermitian operator in (2 s +1) 2 specific coherent states of a spin are known to determine the operator unambiguously. As shown here, (almost) any other set of (2 s +1) 2 coherent state projectors also provide a basis for self-adjoint operators. This is proved by considering the determinant of the Gram matrix associated with the coherent state projectors as a Hamiltonian of a fictitious classical spin system. The result guarantees that (almost) any experimentally desirable choice of directions is appropriate for reconstructing the state of a quantum spin by means of a Stern–Gerlach apparatus.
The density matrix of a spin s is fixed uniquely if the probabilites to obtain the value s upon measuring . are known for 4 s ( s +1) appropriately chosen directions in space. These numbers are just the expectation values of the density operator in coherent spin states, and they are easily obtained in an experiment carried out with a Stern-Gerlach apparatus. Furthermore, the experimental data is non-redundant, and it can be inverted explicitly which allows for a parametrization of the statistical operator by the 4 s ( s +1) positive parameters.
In Moyal's formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e., by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P or Q symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels that give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)2{(2s+1)}^{2}(2s+1)2 points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism.
Coherent states bridge the gap between quantum and classical physics, but their overcomplete and nonorthogonal nature makes it difficult to identify the minimal discrete set needed to reconstruct quantum information. Finite spin-coherent tomography and discrete coherent-state operator bases are known, but here we address the more specific rank-resolved problem of preserving the canonical contravariant-symbol representation. We show that the canonical finite coherent-state formula reconstructs every operator in the rank-SSS sector exactly if and only if the sampling points form a spherical (2J+S)(2J+S)(2J+S)-design. We call the associated configurations spin-coherent quantum designs. We further give a fully explicit positive-weight Gauss-Legendre construction that avoids the need for an equal-weight spherical design. Together, these results establish a unified framework for reading out physical observables from a handful of measurement samples, playing for spin systems the role that the so-called von Neumann lattice plays for canonical coherent states. Finally, we derive practical protocols for estimating moments of spin operators from these constructions, with direct applications to polarimetry, magnetometry, and quantum state tomography.
State reconstruction for quantum spins is reviewed. Emphasis is on non-tomographic approaches which are based on measurements performed with a Stern-Gerlach apparatus. Two consequences of successfully implemented state reconstruction are pointed out. First, it allows one to determine experimentally the expectation value of an arbitrary operator without a device measuring it. Second, state reconstruction suggests a reformulation of Schroedinger's equation in terms of expectation values only, without explicit reference to a wave function or a density operator.