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SPIN TOMOGRAPHY AND STAR-PRODUCT KERNEL FOR QUBITS AND QUTRITS

Vladimir I. Man’ko

Published Aug 5, 2013• 15 citations
OpenAlex

Abstract

Using the irreducible tensor-operator technique, we establish the relation between different forms of spin tomograms. Quantizer and dequantizer operators are presented in simple explicit forms and are specified for the low-spin states. The kernel of the star-product is evaluated for qubits and qutrits and its connection with a generic formula is found.

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Using the irreducible tensor-operator technique, we establish the relation between different forms of spin tomograms. Quantizer and dequantizer operators are presented in simple explicit forms and are specified for the low-spin states. The kernel of the star-product is evaluated for qubits and qutrits and its connection with a generic formula is found.
We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.
The spin-state tomogram is presented as the modulus squared of the matrix element of the SU (2)-group irreducible representation. For the spin system, we established explicitly a realization of the set of operators defining for spin tomograms the star-product formalism in terms of irreducible tensors. On the set of spin tomograms, the delta-function and ‘standard’ and Moyal-like kernels of the tomogram star-product are calculated explicitly in terms of Clebsch–Gordan and Racah coefficients and matrix elements of the SU (2) irreducible representation. In the limit of infinite spin, the spin tomogram is shown to become the tomogram of the harmonic-oscillator state.
The scheme of photon-number tomography is discussed in the framework of star- product quantization. The connection of dual quantization scheme and observables is reviewed. The quantizer and dequantizer operators and kernels of star-product of tomograms in photon-number tomography scheme and its dual one are presented in explicit form. The fidelity and state purity are discussed in photon-number tomographic scheme, and the expressions for fidelity and purity are obtained in the form of integral of the product of two photon-number tomograms with integral kernel which is presented in explicit form. The properties of quantumness are discussed in terms of inequalities on state photon-number tomograms.
Mutually unbiased bases (MUBs) are considered within the framework of a generic star-product scheme. We rederive that a full set of MUBs is adequate for a spin tomography, i.e. knowledge of all probabilities to find a system in each MUB-state is enough for a state reconstruction. Extending the ideas of the tomographic-probability representation and the star-product scheme to MUB tomography, dequantizer and quantizer operators for MUB symbols of spin states and operators are introduced, ordinary and dual star-product kernels are found. Since MUB projectors are to obey specific rules of the star-product scheme, we reveal the Lie algebraic structure of MUB projectors and derive new relations on triple- and four-products of MUB projectors. An example of qubits is considered in detail. MUB tomography by means of the Stern–Gerlach apparatus is discussed.