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arXiv:quant-ph/9606028

Correct interpretation of trace normalized density matrices as ensembles

Paul M. Sheldon

Published Jun 25, 1996• 0 citations
arXivHTMLPDF

Abstract

A density operator, $ρ= {P}_{α} |α> = 0$, ${P}_{α}$ and ${P}_{β}$ cannot be interpreted as probabilities of finding $|α>$ and $|β>$ respectively. We show that a density matrix comprised of two (${P}_{α}$ and ${P}_{β}$ nonzero) non-orthogonal projectors have unique spectral decomposition into diagonal form with orthogonal projectors. Only then, according to axioms of Von Neumann and Fock, can we have probability interpretation of that density matrix, only then can the diagonal elements be interpreted as probabilities of an ensemble. Those probabilities on the diagonal are not ${P}_{α}$ and ${P}_β$. Further, only in the case of orthogonal projectors can we have the degenerate situation in which multiple ensembles are permitted.

DOI
10.48550/arxiv.quant-ph/9606028

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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.
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In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.
Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions are a special case where the density matrix is restricted to be diagonal. Density matrices are mixtures of dyads, where a dyad has the form uu' for any any unit column vector u. These unit vectors are the elementary events of the generalized probability space. Perhaps the simplest case to see that something unusual is going on is the case of uniform density matrix, i.e. 1/n times identity. This matrix assigns probability 1/n to every unit vector, but of course there are infinitely many of them. The new normalization rule thus says that sum of probabilities over any orthonormal basis of directions is one. We develop a probability calculus based on these more general distributions that includes definitions of joints, conditionals and formulas that relate these, i.e. analogs of the theorem of total probability, various Bayes rules for the calculation of posterior density matrices, etc. The resulting calculus parallels the familiar 'classical' probability calculus and always retains the latter as a special case when all matrices are diagonal. Whereas the classical Bayesian methods maintain uncertainty about which model is 'best', the generalization maintains uncertainty about which unit direction has the largest variance. Surprisingly the bounds also generalize: as in the classical setting we bound the negative log likelihood of the data by the negative log likelihood of the MAP estimator.
A division of the 2-matrix Γ, for a system of N identical fermions, into 2 parts is proposed in such a way that Γ has the same 1-matrix γ as does Γ and Γ has a vanishing 1-matrix (and therefore vanishing trace). This is accomplished by evaluating the natural spin orbital (NSO) occupation numbers Γ of Γ from (where the ni are NSO occupation numbers of γ); and letting γ0 have the same NSO's as does γ. The physical interpretation of Γ should be easier with this decomposition, since Γ is completely determined by γ and has a Hartree-Fock-like form, while Γ corrects the pair density of Γ without disturbing its 1-density.