Based on a recent proof of free choices in linking equations to the experiments they describe, I clarify relations among some purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and operator-valued measures), thereby allowing applications of these entities to the modeling of a wider variety of physical situations. Conditional probabilities associated with projection-valued measures are expressed by introducing conditional density operators, identical in some but not all cases to the usual reduced density operators. By lifting density operators to the extended Hilbert space featured in Neumark's theorem, I show an obstacle to extending conditional density operators to arbitrary positive operator-valued measures (POVMs); however, tensor products of POVMs are compatible with conditional density operators. By way of application, conditional density operators together with the free choice of probe particles allow the so-called postulate of state reductions to be replaced by a theorem. A second application demonstrates an equivalence between one form of quantum key distribution and another, allowing a formulation of individual eavesdropping attacks against transmitted-state BB84 to work also for entangled-state BB84.Based on a recent proof of free choices in linking equations to the experiments they describe, I clarify relations among some purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and operator-valued measures), thereby allowing applications of these entities to the modeling of a wider variety of physical situations. Conditional probabilities associated with projection-valued measures are expressed by introducing conditional density operators, identical in some but not all cases to the usual reduced density operators. By lifting density operators to the extended Hilbert space featured in Neumark's theorem, I show an obstacle to extending conditional density operators to arbitrary positive operator-valued measures (POVMs); however, tensor products of POVMs are compatible with conditional density operators. By way of application, conditional density operators together with the free choice of probe particles allow the so-called postulate of state reductions to be replaced by a theorem. A second application demonstrates an equivalence between one form of quantum key distribution and another, allowing a formulation of individual eavesdropping attacks against transmitted-state BB84 to work also for entangled-state BB84.

Jun 8, 20056 citations

Conditional probabilities in quantum mechanics

Gianni Cassinelli, Nino Zanghı̀

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BFeb 1, 198424 citations

The quantum mechanical probability density and probability current density operators in the Pauli theory

E. L. Rumyantsev, P. E. Kunavin

We present systematic construction of probability and probability current densities operators for one-band single particle Pauli equations starting from the operators in Dirac electron model within Second Quantized Approach. These operators are of importance as in probability interpretation of experimental data, so in establishing of boundary conditions. It is shown that derived operators differ significally from their convential Schrodinger-type counterparts. The generalization of continuity equation for probability density under external perturbations and physical meaning of additional source terms is discussed. The presented approach can be useful in analysis of carriers dynamics described within generic multicomponent $\vec{k} \cdot \vec{p}$ Hamiltonians in Envelope Function Approximation (EFA).We present systematic construction of probability and probability current densities operators for one-band single particle Pauli equations starting from the operators in Dirac electron model within Second Quantized Approach. These operators are of importance as in probability interpretation of experimental data, so in establishing of boundary conditions. It is shown that derived operators differ significally from their convential Schrodinger-type counterparts. The generalization of continuity equation for probability density under external perturbations and physical meaning of additional source terms is discussed. The presented approach can be useful in analysis of carriers dynamics described within generic multicomponent $\vec{k} \cdot \vec{p}$ Hamiltonians in Envelope Function Approximation (EFA).

arXiv:1708.04193Aug 14, 20171 citation

Density Operators and Other Tools of Quantum Statistics

W. Ebeling, Thorsten Pöschel

Lecture notes in physicsJan 1, 20191 citation

Probabilistic interpretation of the reduction criterion for entanglement

Zhengmin Zhang, Shunlong Luo

Inspired by the idea of conditional probabilities, we introduce a variant of conditional density operators. But unlike the conditional probabilities which are bounded by 1, the conditional density operators may have eigenvalues exceeding 1 for entangled states. This has the consequence that although any bivariate classical probability distribution has a natural separable decomposition in terms of conditional probabilities, we do not have a quantum analogue of this separable decomposition in general. The ``nonclassical'' eigenvalues of conditional density operators are indications of entanglement. The resulting separability criterion turns out to be equivalent to the reduction criterion introduced by Horodecki [Phys. Rev. A 59, 4206 (1999)] and Cerf et al. [Phys. Rev. A 60, 898 (1999)]. This supplies an intuitive probabilistic interpretation for the reduction criterion. The conditional density operators are also used to define a form of quantum conditional entropy which provides an alternative mechanism to reveal quantum discord.Inspired by the idea of conditional probabilities, we introduce a variant of conditional density operators. But unlike the conditional probabilities which are bounded by 1, the conditional density operators may have eigenvalues exceeding 1 for entangled states. This has the consequence that although any bivariate classical probability distribution has a natural separable decomposition in terms of conditional probabilities, we do not have a quantum analogue of this separable decomposition in general. The ``nonclassical'' eigenvalues of conditional density operators are indications of entanglement. The resulting separability criterion turns out to be equivalent to the reduction criterion introduced by Horodecki [Phys. Rev. A 59, 4206 (1999)] and Cerf et al. [Phys. Rev. A 60, 898 (1999)]. This supplies an intuitive probabilistic interpretation for the reduction criterion. The conditional density operators are also used to define a form of quantum conditional entropy which provides an alternative mechanism to reveal quantum discord.

Physical Review AMar 9, 20078 citations

Probability and Operators in Quantum Mechanics

Robert D. Richtmyer

Springer eBooksJan 1, 19780 citations

\vec{k} \cdot \vec{p}

Hamiltonians in Envelope Function Approximation (EFA).