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arXiv:quant-ph/0207033 · Journal of Physics A Mathematical and General

Interference and entanglement: an intrinsic approach

Vladimir I Man$apos$ko, Giuseppe Marmo, E. C. G. Sudarshan, F. Zaccaria

Published Aug 7, 2002• 45 citations
arXivHTMLPDF

Abstract

An addition rule of impure density operators, which provides a pure state density operator, is formulated. Quantum interference including a visibility property is discussed in the context of the density operator formalism. A measure of entanglement is then introduced as the norm of the matrix equal to the difference between a bipartite density matrix and the tensor product of partial traces. Entanglement for arbitrary quantum observables for multipartite systems is discussed. Star-product kernels are used to map the formulation of the addition rule of density operators onto the addition rule of symbols of the operators. Entanglement and nonlocalization of the pure state projector and allied operators are discussed. Tomographic and Weyl symbols (tomograms and Wigner functions) are considered as examples. The squeezed states and some spin states (two qubits) are studied to illustrate the formalism.

DOI
10.1088/0305-4470/35/33/311

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An addition rule of impure density operators, which provides a pure state density operator, is formulated. Quantum interference including a visibility property is discussed in the context of the density operator formalism. A measure of entanglement is then introduced as the norm of the matrix equal to the difference between a bipartite density matrix and the tensor product of partial traces. Entanglement for arbitrary quantum observables for multipartite systems is discussed. Star-product kernels are used to map the formulation of the addition rule of density operators onto the addition rule of symbols of the operators. Entanglement and nonlocalization of the pure state projector and allied operators are discussed. Tomographic and Weyl symbols (tomograms and Wigner functions) are considered as examples. The squeezed states and some spin states (two qubits) are studied to illustrate the formalism.
The problem of entanglement produced by an arbitrary operator is formulated and a related measure of entanglement production is introduced. This measure of entanglement production satisfies all properties natural for such a characteristic. A particular case is the entanglement produced by a density operator or a density matrix. The suggested measure is valid for operations over pure states as well as over mixed states, for equilibrium as well as nonequilibrium processes. Systems of arbitrary nature can be treated, described by field operators, spin operators, or any other kind of operator that is realized by constructing generalized density matrices. The interplay between entanglement production and phase transitions in statistical systems is analyzed using the examples of Bose-Einstein condensation, the superconducting transition, and magnetic transitions. The relation between the measure of entanglement production and order indices is analyzed.
The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ 2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures . The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures . The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H 1 ⊗ H 2 , but are not of the simple product form ψ 1 ⊗ ψ 2 . The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.
Quantum state interferography has recently been proposed as an interference-based technique for the measurement of the general qubit and pure qudit states [Phys. Rev. Lett. 125, 123601 (2020)]. This interferometric method exploits the relationship between the off-diagonal elements of the density matrix and the expectation values of non-Hermitian spin ladder operators. The extension of this method to a composite system is however nontrivial, as it involves the measurement of both single-photon and nonlocal multiphoton interferences. In this work, we present the theoretical analysis of the two-photon quantum state interferography of an arbitrary quantum state. We show that complete state information could in principle be obtained with two measurement settings and by recording six interferograms, using a single-photon sensitive imaging device. We then experimentally demonstrate two-photon nonlocal quantum state interferography of polarization-entangled photon pairs generated with type-I spontaneous parametric down conversion. Quantum state interferography of all four Bell states is performed. An upper bound on the separability of the bipartite states is established in terms of the visibility, phase average intensity, and phase shift of the interference patterns related to the antidiagonal elements of the matrix. Each of the four Bell states can be distinguished by the violation of these bounds uniquely.
Abstract This paper presents a review of star-product formalism. This formalism provides a description for quantum states and observables by means of the functions called’ symbols of operators’. Those functions are obtained via bijective maps of the operators acting in Hilbert space. Examples of the Wigner-Weyl symbols (Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified for the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators required for construction of bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relationship between structure constants of associative star-product of operator symbols and quantizer-dequantizer operators is reviewed.
In quantum information and communication two fundamental nonclassical features that one exploits in harnessing the true power of composite quantum systems are the interference and quantum correlations. We show how a concept akin to interference can be connected to entanglement and quantum correlation. In particular, we prove that the difference in the squared visibility for the density operator before and after a complete measurement, averaged over all unitary evolutions, is related to the quantum correlation measure based on the measurement disturbance. For pure and mixed bipartite states the unitary average of the squared visibility is related to the entanglement measure. This may provide a method to detect entanglement and quantum correlation with quantum interference setups. Furthermore, we prove that for a fixed purity of the subsystem state, there is a complementarity relation between the linear entanglement of formation and the measurement disturbance. This brings out a quantitative difference between two kinds of quantum correlations and suggests that the origins of these two can be different.