Measuring Observable Quantum Contextuality

Abstract

Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide.

A Proofs of Statements

In this appendix, we describe how the analytic results of the main text were obtained for each of the expressions (4), (10), (12), and (16) of the main text.

1.1 A.1 EPR-Bell: Contextuality-by-Default

Following the computations of Dzhafarov and Kujala [10, TextS3], or the more general formulation in Ref. [16], it can be shown that the observable distributions with probabilities given by the matrices

(18)

are compatible with the connections

(19)

if and only if

$$\begin{aligned} \begin{array}{l} s_{u}\!\left( \left\langle \mathbf {A}_{1,1}\mathbf {B}_{1,1}\right\rangle \!,\left\langle \mathbf {A}_{1,2}\mathbf {B}_{1,2}\right\rangle \!,\left\langle \mathbf {A}_{2,1}\mathbf {B}_{2,1}\right\rangle \!,\left\langle \mathbf {A}_{2,2}\mathbf {B}_{2,2}\right\rangle \right) \\ \le 6-s_{1-u}\!\left( \left\langle \mathbf {A}_{1,1}\mathbf {A}_{1,2}\right\rangle \!,\left\langle \mathbf {A}_{2,1}\mathbf {A}_{2,2}\right\rangle \!,\left\langle \mathbf {B}_{1,1}\mathbf {B}_{2,1}\right\rangle \!,\left\langle \mathbf {B}_{1,2}\mathbf {B}_{2,2}\right\rangle \right) , \end{array} \end{aligned}$$

(20)

where u stands for 0 or 1, and

$$\begin{aligned} s_{0}(x_{1},\dots ,x_{n})&=\max \{\pm x_{1}\pm \dots \pm x_{n}:\text {even }\# \text { of } -'s \},\\ s_{1}(x_{1},\dots ,x_{n})&=\max \{\pm x_{1}\pm \dots \pm x_{n}:\text {odd }\# \text { of } -'s \}. \end{aligned}$$

Here we use the parameterization by the 12 expectation variables defined as (with $i,j\in \{ 1,2 \}$)

$$\begin{aligned} \left\langle \mathbf {A}_{i,j}\mathbf {B}_{i,j}\right\rangle =\left( 4p_{ij}-1\right) -\left( 2a_{i}-1\right) -\left( 2b_{j}-1\right) , \end{aligned}$$

(21)

$$\begin{aligned} \left\langle \mathbf {A}_{i,1}\mathbf {A}_{i,2}\right\rangle =1-4\alpha _{i}=1-2\Pr \left[ \mathbf {A}_{i,1}\ne \mathbf {A}_{i,2}\right] , \end{aligned}$$

(22)

$$\begin{aligned} \left\langle \mathbf {B}_{1,j}\mathbf {B}_{2,j}\right\rangle =1-4\beta _{j}=1-2\Pr \left[ \mathbf {B}_{1,j}\ne \mathbf {B}_{2,j}\right] , \end{aligned}$$

(23)

$$\begin{aligned} \left\langle \mathbf {A}_{i}\right\rangle =2a_{i}-1,\quad \left\langle \mathbf {B}_{j}\right\rangle =2b_{j}-1. \end{aligned}$$

(24)

Writing the inequality (20) in terms of these expectations rather than in terms of probabilities is the most economic way of presenting the 128 non-trivial inequalities of the system, as the marginal probabilities $a_{1},a_{2},b_{1},b_{2}$ (or expectations $\left\langle \mathbf {A}_{1}\right\rangle ,\left\langle \mathbf {A}_{2}\right\rangle ,\left\langle \mathbf {B}_{1}\right\rangle ,\left\langle \mathbf {B}_{2}\right\rangle$) vanish in this form. However, it should be noted that in addition to these 128 inequalities, the form of the observed distributions and connections itself imposes further 28 trivial constraints on the 12 expectation variables of the system: the probabilities within each $2\times 2$ matrix in (18) and (19) should be nonnegative and sum to one. 16 of these trivial constraints pertain to the observed distributions and 12 to the connections. In terms of the expectations, these trivial constraints correspond to

$$\begin{aligned} -1+|\left\langle \mathbf {A}\right\rangle +\left\langle \mathbf {B}\right\rangle |\le \left\langle \mathbf {A}\mathbf {B}\right\rangle \le 1-|\left\langle \mathbf {A}\right\rangle -\left\langle \mathbf {B}\right\rangle |, \end{aligned}$$

(25)

for given marginals for each pair $(\mathbf {A},\mathbf {B})$ of random variables in (21)–(23). This expands to four inequalities for each of the observed distributions and to three inequalities for each of the connections (the two upper bounds in (25) coincide when $\left\langle \mathbf {A}\right\rangle =\left\langle \mathbf {B}\right\rangle$). Although these trivial constraints can usually be assumed implicitly, it is important to keep them explicitly in the system for the next step.

Adding the equation

$$\begin{aligned} \varDelta =&\Pr \left[ \mathbf {A}_{1,1}\ne \mathbf {A}_{1,2}\right] +\Pr \left[ \mathbf {A}_{2,1}\ne \mathbf {A}_{2,2}\right] +\Pr \left[ \mathbf {B}_{1,1}\ne \mathbf {B}_{2,1}\right] +\Pr \left[ \mathbf {B}_{1,2}\ne \mathbf {B}_{2,2}\right] \\ =&~2-\frac{1}{2}\left( \left\langle \mathbf {A}_{1,1}\mathbf {A}_{1,2}\right\rangle +\left\langle \mathbf {A}_{2,1}\mathbf {A}_{2,2}\right\rangle +\left. \left\langle \mathbf {B}_{1,1}\mathbf {B}_{2,1}\right\rangle +\left\langle \mathbf {B}_{1,2}\mathbf {B}_{2,2}\right\rangle \right) \right. \end{aligned}$$

to the system and then eliminating the connection correlations $\left\langle \mathbf {A}_{1,1}\mathbf {A}_{1,2}\right\rangle$, $\left\langle \mathbf {A}_{2,1}\mathbf {A}_{2,2}\right\rangle$, $\left\langle \mathbf {B}_{1,1}\mathbf {B}_{2,1}\right\rangle$, $\left\langle \mathbf {B}_{1,2}\mathbf {B}_{2,2}\right\rangle$ from the system using the Fourier–Motzkin elimination algorithm, we obtain the system

$$\begin{aligned}&-1+\frac{1}{2}S_{CHSH}\le \varDelta \le 4-\left[ -1+\frac{1}{2}S_{CHSH}\right] ,\end{aligned}$$

(26)

$$\begin{aligned}&0\le \varDelta \le 4-\left( \left| \left\langle \mathbf {A}_{1}\right\rangle \right| +\left| \left\langle \mathbf {A}_{2}\right\rangle \right| +\left| \left\langle \mathbf {B}_{1}\right\rangle \right| +\left| \left\langle \mathbf {B}_{2}\right\rangle \right| \right) , \end{aligned}$$

(27)

where we denote

$$\begin{aligned} \begin{array}{l} S_{CHSH}=s_{1}\big (\left\langle \mathbf {A}_{1,1}\mathbf {B}_{1,1}\right\rangle ,\left\langle \mathbf {A}_{1,2}\mathbf {B}_{1,2}\right\rangle ,\left\langle \mathbf {A}_{2,1}\mathbf {B}_{2,1}\right\rangle ,\left\langle \mathbf {A}_{2,2}\mathbf {B}_{2,2}\right\rangle \big )\end{array} \end{aligned}$$

(28)

as in the main text. This means that $\varDelta$ is compatible with the given observed probabilities if and only if the above inequalities are satisfied. Since the set of possible values of $\varDelta$ constrained by (26) and (27) is known to be nonempty, it follows that the minimum value of $\varDelta$ is always given by

$$\begin{aligned} \varDelta _{\min }=\max \left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} . \end{aligned}$$

1.2 A.2 EPR-Bell: Negative Probabilities

The analogous result for the negative probabilities approach is that the observable distributions (18) are obtained as the marginals of some negative probability joint of $\mathbf {A}_{1}=\mathbf {A}_{1,1}=\mathbf {A}_{1,2},$ $\mathbf {A}_{2}=\mathbf {A}_{2,1}=\mathbf {A}_{2,2}$, $\mathbf {B}_{1}=\mathbf {B}_{1,1}=\mathbf {B}_{2,1},$ and $\mathbf {B}_{2}=\mathbf {B}_{1,2}=\mathbf {B}_{2,2}$ given by

$$\begin{aligned}&\Pr \left[ \mathbf {A}_{1}=a'_{1},\mathbf {\,A}_{2}=a'_{2},\,\mathbf {B}_{1}=b'_{1},\,\mathbf {B}_{2}=b'_{2}\right] \,=p^{+}(a'_{1},a'_{2},b'_{1},b'_{2})-p^{-}(a'_{1},a'_{2},b'_{1},b'_{2}), \end{aligned}$$

$a'_{1},a'_{2},b'_{1},b'_{2}\in \{1,-1\}$, for some nonnegative functions $p^{+}$ and $p^{-}$ having a total probability mass value

$$\begin{aligned} M=\sum _{a'_{1},a'_{2},b'_{1},b'_{2}}p^{+}(a'_{1},a'_{2},b'_{1},b'_{2})+p^{-}(a'_{1},a'_{2},b'_{1},b'_{2}) \end{aligned}$$

if and only if $M\ge 1+\varGamma _{\min }$, where

$$\begin{aligned} \varGamma _{\min }=\max \left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} . \end{aligned}$$

The computations are similar to those of the CbD approach, but there are two general differences. First, in the CbD approach, the convex range of the possible observed and connection expectations (21)–(24) over the convex polytope of all possible joints is obtained by looking at these expectations at the $2^{8}$ vertices defining the polytope of all joints and then applying a computer algorithm to find the set of inequalities delineating the extreme values of the expectations at these vertices. However, in the negative probabilities approach, the joint is represented by the $2^{4}$ differences of the positive and negative components of the distribution and so, although these $2\cdot 2^{4}$ components are nonnegative as in the CbD approach, they do not need to sum to one. Hence, the joint is represented by a convex cone rather than a bounded polytope. Still, a convex cone is a special case of a general polytope and can be handled by the same algorithms that we have used in the CbD approach.

Second, we do not need to apply the Fourier–Motzkin elimination algorithm here as we have defined M directly by the representation of the joint so there are no extra variables we would need to eliminate. This difference, however, is not really a difference between the two approaches, as we could have done the same in the CbD approach as well: we could have defined ${\varDelta }$ directly based on the joint of all eight variables without explicitly defining the connection correlations (22)–(23), and then we would have obtained the result directly from the half-space representation, as we do in the negative probabilities approach.

1.3 A.3 Leggett–Garg: Contextuality-by-Default

The results for Leggett–Garg $\mathbf {Q}_{1},\mathbf {Q}_{2},\mathbf {Q}_{3}$ can be obtained in the same way as for the EPR-Bell systems. In the CbD approach, the observed correlations $\left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{2,1}\right\rangle$, $\left\langle \mathbf {Q}_{1,3}\mathbf {Q}_{3,1}\right\rangle$, $\left\langle \mathbf {Q}_{2,3}\mathbf {Q}_{3,2}\right\rangle$, $\left\langle \mathbf {Q}_{1}\right\rangle =\left\langle \mathbf {Q}_{1,2}\right\rangle =\left\langle \mathbf {Q}_{1,3}\right\rangle$, $\left\langle \mathbf {Q}_{2}\right\rangle =\left\langle \mathbf {Q}_{2,1}\right\rangle =\left\langle \mathbf {Q}_{3,2}\right\rangle$, $\left\langle \mathbf {Q}_{3}\right\rangle =\left\langle \mathbf {Q}_{3,1}\right\rangle =\left\langle \mathbf {Q}_{3,2}\right\rangle$ are consistent with the connection correlations $\left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{1,3}\right\rangle$, $\left\langle \mathbf {Q}_{2,1}\mathbf {Q}_{2,3}\right\rangle$, $\left\langle \mathbf {Q}_{3,1}\mathbf {Q}_{3,2}\right\rangle$ if and only if these connection correlations are realizable with the given marginals (i.e., each correlation $\left\langle \mathbf {AB}\right\rangle$ has to satisfy $-1+|\left\langle \mathbf {A}\right\rangle +\left\langle \mathbf {B}\right\rangle |\le \left\langle \mathbf {A}\mathbf {B}\right\rangle \le 1-|\left\langle \mathbf {A}\right\rangle -\left\langle \mathbf {B}\right\rangle |$ as discussed in the EPR-Bell case above) and satisfy

$$\begin{aligned} \begin{array}{l} s_{u}\left( \left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{2,1}\right\rangle ,\left\langle \mathbf {Q}_{1,3}\mathbf {Q}_{3,1}\right\rangle ,\left\langle \mathbf {Q}_{2,3}\mathbf {Q}_{3,2}\right\rangle \right) \\ +s_{1-u}\left( \left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{1,3}\right\rangle ,\left\langle \mathbf {Q}_{2,1}\mathbf {Q}_{2,3}\right\rangle ,\left\langle \mathbf {Q}_{3,1}\mathbf {Q}_{3,2}\right\rangle \right) \le 4, \end{array} \end{aligned}$$

(29)

These two inequalities (with u standing for 0 or 1) expand to 32 linear inequalities and there are 21 trivial constraints.

Denoting

$$\begin{aligned} \varDelta&=\Pr \left[ \mathbf {Q}_{1,2}\!\ne \!\mathbf {Q}_{1,3}\right] +\Pr \left[ \mathbf {Q}_{2,1}\!\ne \!\mathbf {Q}_{2,3}\right] +\Pr \left[ \mathbf {Q}_{3,1}\!\ne \!\mathbf {Q}_{3,2}\right] \\&=\frac{3}{2}-\frac{1}{2}\left( \left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{1,3}\right\rangle +\left\langle \mathbf {Q}_{2,1}\mathbf {Q}_{2,3}\right\rangle +\left\langle \mathbf {Q}_{3,1}\mathbf {Q}_{3,2}\right\rangle \right) \end{aligned}$$

and eliminating the connection correlations from the system using the Fourier–Motzkin algorithm, we obtain the system

$$\begin{aligned} -\frac{1}{2}+\frac{1}{2}S_{LG}\le \varDelta&\le 3-\left[ -\frac{1}{2}+\frac{1}{2}S_{LG}^{0}\right] ,\end{aligned}$$

(30)

$$\begin{aligned} 0\le \varDelta&\le 3-\left| \left\langle \mathbf {Q}_{1}\right\rangle \right| -\left| \left\langle \mathbf {Q}_{2}\right\rangle \right| -\left| \left\langle \mathbf {Q}_{3}\right\rangle \right| , \end{aligned}$$

(31)

where we denote

$$\begin{aligned} S_{LG}&=s_{1}\left( \left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{2,1}\right\rangle ,\left\langle \mathbf {Q}_{1,3}\mathbf {Q}_{3,1}\right\rangle ,\left\langle \mathbf {Q}_{2,3}\mathbf {Q}_{3,2}\right\rangle \right) ,\\ S_{LG}^{0}&=s_{0}\left( \left\langle \mathbf {Q}_{1,2}\mathbf {Q}_{2,1}\right\rangle ,\left\langle \mathbf {Q}_{1,3}\mathbf {Q}_{3,1}\right\rangle ,\left\langle \mathbf {Q}_{2,3}\mathbf {Q}_{3,2}\right\rangle \right) . \end{aligned}$$

That is, $\varDelta$ is consistent with the observed probabilities if and only if the above inequalities are satisfied. It follows that the minimum value of $\varDelta$ is given by

$$\begin{aligned} \varDelta _{\min }=\max \left\{ 0,\frac{1}{2}S_{LG}-\frac{1}{2}\right\} . \end{aligned}$$

1.4 A.4 Leggett–Garg: Negative Probabilities

With the same additional comments as in the negative probability calculations for the EPR-Bell case, our calculations show that the observable probabilities $\Pr \left[ \mathbf {Q}_{12}=q_{1},\mathbf {Q}_{21}=q_{2}\right]$, $\Pr \left[ \mathbf {Q}_{13}=q_{1},\mathbf {Q}_{31}=q_{3}\right]$, $\Pr \left[ \mathbf {Q}_{23}=q_{2},\mathbf {Q}_{32}=q_{3}\right]$, $q_{1},q_{2},q_{3}\in \{0,1\},$ can be obtained as the marginals of a negative probability jpd of $\mathbf {Q}_{1}=\mathbf {Q}_{1,2}=\mathbf {Q}_{1,3}$, $\mathbf {Q}_{2}=\mathbf {Q}_{2,1}=\mathbf {Q}_{3,2}$, and $\mathbf {Q}_{3}=\mathbf {Q}_{3,1}=\mathbf {Q}_{3,2}$ with the total probability mass of M if and only if $M\ge 1+\varGamma _{\min }$, where

$$\begin{aligned} \varGamma _{\min }=\max \left\{ 0,\frac{1}{2}S_{LG}-\frac{1}{2}\right\} . \end{aligned}$$