Test of a hypothesis of realism in quantum theory using a Bayesian approach

In this paper we propose a time-independent equality and time-dependent inequality, suitable for an experimental test of the hypothesis of realism. The derivation of these relations is based on the concept of conditional probability and on Bayes' theorem in the framework of Kolmogorov's axiomatics of probability theory. The equality obtained is intrinsically different from the well-known Greenberger-Horne-Zeilinger (GHZ) equality and its variants, because violation of the proposed equality might be tested in experiments with only two microsystems in a maximally entangled Bell state $|{\mathrm{\Psi }}^{-}\rangle$, while a test of the GHZ equality requires at least three quantum systems in a special state $|{\mathrm{\Psi }}^{\mathrm{GHZ}}\rangle$. The obtained inequality differs from Bell's, Wigner's, and Leggett-Garg inequalities, because it deals with spin $s=1/2$ projections onto only two nonparallel directions at two different moments of time, while a test of the Bell and Wigner inequalities requires at least three nonparallel directions, and a test of the Leggett-Garg inequalities requires at least three distinct moments of time. Hence, the proposed inequality seems to open an additional experimental possibility to avoid the “contextuality loophole.” Violation of the proposed equality and inequality is illustrated with the behavior of a pair of anticorrelated spins in an external magnetic field and also with the oscillations of flavor-entangled pairs of neutral pseudoscalar mesons.

Figure 1

Functions $F_{1,5}(x,r,\zeta ,\lambda )$ for $B_s$ mesons. The top axis corresponds to $ct$ (in mm), the bottom axis to the time in units of lifetime $z=({\mathrm{\Gamma }}_H+{\mathrm{\Gamma }}_L)t/2=\mathrm{\Gamma }t=t/{\tau }_{B_s}$, where time $t$ is calculated in the $B_s$-meson rest frame. The plots are produced for $r=1.004$ and $\zeta ={185}^{\circ }$.

Figure 2

Functions $F_{3,4}(x,r,\zeta ,\lambda )$ for neutral $K$ mesons. The top axis corresponds to $ct$ (in mm), the bottom axis to the time in units of lifetime $z=({\mathrm{\Gamma }}_S+{\mathrm{\Gamma }}_L)t/2=\mathrm{\Gamma }t=t/{\tau }_K$, where $t$ is calculated in the $K$-meson rest frame. The plots are produced for $r=0.997$ and $\zeta =-0.{18}^{\circ }$.

Figure 3

Functions $F_{3,5}(x,r,\zeta ,\lambda )$ for neutral $D$ mesons. The top axis corresponds to $ct$ (in mm), the bottom axis to the time in units of lifetime $z=({\mathrm{\Gamma }}_H+{\mathrm{\Gamma }}_L)t/2=\mathrm{\Gamma }t=t/{\tau }_D$, where $t$ is calculated in $D$-meson rest frame. The plots are produced for $r=1.1$ and $\zeta =-{10}^{\circ }$.