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arXiv:1408.3100

A Bayesian Probability Calculus for Density Matrices

Manfred K. Warmuth, Dima Kuzmin

Published Aug 9, 2014• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1408.3100

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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.
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 can be seen as a special case when the density matrix is restricted to be diagonal. We develop a probability calculus based on these more general distributions that includes definitions of joints, conditionals and formulas that relate these, including analogs of the Theorem of Total Probability and various Bayes rules for the calculation of posterior density matrices. The resulting calculus parallels the familiar “conventional” probability calculus and always retains the latter as a special case when all matrices are diagonal. We motivate both the conventional and the generalized Bayes rule with a minimum relative entropy principle, where the Kullbach-Leibler version gives the conventional Bayes rule and Umegaki’s quantum relative entropy the new Bayes rule for density matrices. Whereas the conventional Bayesian methods maintain uncertainty about which model has the highest data likelihood, the generalization maintains uncertainty about which unit direction has the largest variance. Surprisingly the bounds also generalize: as in the conventional setting we upper bound the negative log likelihood of the data by the negative log likelihood of the MAP estimator.
The classical Bayes rule computes the posterior model probability from the prior probability and the data likelihood. We generalize this rule to the case when the prior is a density matrix (symmetric positive definite and trace one) and the data likelihood a covariance matrix. The classical Bayes rule is retained as the special case when the matrices are diagonal. In the classical setting, the calculation of the probability of the data is an expected likelihood, where the expectation is over the prior distribution. In the generalized setting, this is replaced by an expected variance calculation where the variance is computed along the eigenvectors of the prior density matrix and the expectation is over the eigenvalues of the density matrix (which form a probability vector). The variances along any direction is determined by the covariance matrix. Curiously enough this expected variance calculation is a quantum measurement where the co-variance matrix specifies the instrument and the prior density matrix the mixture state of the particle. We motivate both the classical and the generalized Bayes rule with a minimum relative entropy principle, where the Kullbach-Leibler version gives the classical Bayes rule and Umegaki’s quantum relative entropy the new Bayes rule for density matrices. 1
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.
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.
We consider the problem of gambling on a quantum experiment and enforce rational behavior by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalized to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalized Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalization (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.
Density matrices defined with respect to a finite basis set are considered as elements in a vector space. A basis set is introduced in the space whose elements are Hermitian matrices. By a suitable transformation of the basis, one component only will contribute to the trace, so the space of unit trace matrices is a translated linear subspace. The positivity constraint is then examined in terms of distance from a multiple of the unit matrix, which is in the interior of the set of density matrices. The boundary of this set is shown to lie between two concentric hyperspheres. The set whose elements are NNN-representable 1 matrices has similar properties.