1. Introduction
The standard way to show that quantum mechanics allows correlations impossible in classical (more precisely: local hidden-variable) theories is to point to violations of some Bell inequality. The classic example is the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality [
1] by correlations between the outcomes of certain measurements on pairs of photons in a maximally entangled state. An alternative approach is to show that quantum mechanics allows correlations that seem to clash with basic logic. The work by Hardy [
2,
3] and Unruh [
4] that we examine in this paper provides intriguing examples of this approach (the most famous example, undoubtedly, is due to Greenberger, Horne and Zeilinger [
5]). In this approach, at least in principle, one combination of measurement outcomes suffices to rule out a local hidden-variable theory for the relevant quantum correlations, whereas in the more familiar approach, we need to consider the statistics of many outcomes.
Hardy [
2,
3] constructed a family of non-maximally entangled two-particle states and concomitant measurement settings such that the measurement outcomes satisfy two conditionals, but not a third, which would seem to be a direct consequence of the first two. Schematically,
This is what is known as Hardy’s paradox.
Inspired by Hardy, Unruh [
4] constructed a family of states and settings such that the outcomes satisfy three conditionals, but not a fourth, which would seem to follow directly from the first three on the basis of the transitivity of the ‘if … then’ relation. Schematically,
Such broken ‘if … then …’ arrows are allowed in quantum mechanics for the same reason that violations of Bell inequalities are. Local hidden-variable theories simultaneously assign truth values to propositions
A,
B,
C and
D above. Quantum mechanics does not. To assign truth values to all four propositions, one would simultaneously have to measure observables represented by non-commuting operators. These Hardy–Unruh chains of conditionals—as we call the sets of conditionals in Equations (
1) and (
2)—thus illustrate quantum contextuality: which observables do and do not obtain definite values depends on what measurements we decide to perform.
In this paper, we use the framework inspired by Bub [
6] and Pitowsky [
7] and developed by Janas, Cuffaro and Janssen [
8] to construct and analyze these Hardy–Unruh chains. In
Section 2, we review the elements we need from our book. In
Section 3 and
Section 4, we construct the states and measurement settings giving rise to the broken arrows in Equations (
1) and (
2). In
Section 5, we examine the relation between these broken arrows and violations of the relevant Bell inequality, which, as we will see, is a special case of the CHSH inequality. On the basis of this analysis, we conclude, in
Section 6, that broken arrows and violations of Bell inequalities are different but ultimately equivalent ways of bringing out quantum contextuality.
1 2. Preliminaries
In our book,
Understanding Quantum Raffles [
8], inspired by Bub’s
Bananaworld [
6], we used the imagery of peeling and tasting fictitious bananas mimicking the measurement of spin components of (half-)integer spin particles. We modified Bub’s banana-peeling scheme to tighten the analogy between our bananas and particles with spin. In this paper, as in most of our book, we focus on bananas mimicking the behavior of spin-
particles.
2Imagine picking a pair of such bananas, connected at the stem, from a particular species of banana tree, breaking them apart and giving one to Alice and one to Bob. Alice and Bob then choose a peeling direction, i.e., a direction in which they are required to hold their banana while peeling it. When finished peeling, they take a bite to determine whether their banana tastes yummy or nasty. It is a key feature of our banana imagery that, when the bananas are still on the banana tree, they do not possess a specific taste nor any properties predetermining their taste upon being peeled and tasted. They somehow only acquire their taste upon being peeled and tasted. Yummy and nasty are the only possible values for taste for this species of banana.
Readers put off by our
Bananaworld imagery can replace (i) bananas with spin-
particles; (ii) species of banana trees with states in which we prepare pairs of such particles (though we will also talk about pairs of bananas in particular quantum states); (iii) peeling directions (or peelings for short) with orientations of Du Bois (or Stern–Gerlach) magnets; (iv) the actual peeling with sending particles through a Du Bois magnet; (v) tasting with having a particle hit a screen with a photo-emulsion behind the magnet; and (vi) yummy and nasty with spin up and spin down, respectively. Our fictitious bananas, however, are not just a gimmick. They also underscore that the correlations examined in our book and in this paper can be realized in many different physical systems.
3Suppose Alice peels
a and Bob peels
b. The correlations between the tastes they find, which persist no matter how far apart they are, can be represented by a
correlation array (see
Figure 1). In analogy with the values
and
for spin up and spin down (where
ℏ is Planck’s constant divided by
), we assign the numerical values
and
to the tastes yummy and nasty in some appropriate units. Unless we need these values to calculate expectation values, we simply use + for yummy and − for nasty. The four entries in the correlation array give the probabilities of the four possible outcomes for this combination of peelings.
For now, we restrict our attention to species of banana trees (but this does not include the species giving rise to Hardy–Unruh chains) on which bananas grow in pairs such that the correlations between their tastes have two special properties:
No matter what peelings Alice and Bob use, the probability of them finding yummy or nasty is always .
If Alice and Bob use the same peeling, they always find opposite tastes.
Property 1 means that the entries in both rows and both columns of the correlation array in
Figure 1 add up to
. In that case, as shown in
Figure 2, the correlation array can be fully characterized by the parameter
, with
if the peelings
a and
b are the same (property 2).
Figure 1.
Correlation array for Alice peeling a and Bob peeling b.
Figure 1.
Correlation array for Alice peeling a and Bob peeling b.
Figure 2.
Parametrization of correlation array in
Figure 1 given property 1.
Figure 2.
Parametrization of correlation array in
Figure 1 given property 1.
We can simulate these correlations for any value of
with the kind of raffle introduced in our book ([
8], Section 2.5) as a model for local hidden-variable theories. In this case, the raffle consists of a basket with a mix of the two types of tickets shown in
Figure 3, with the tastes of both bananas for both peelings printed on them. We draw tickets from this basket, tear them in half along the perforation indicated by the dashed line, and randomly give one half to Alice and one half to Bob. That the values for
a and
b on the two sides of the ticket are opposite takes care of property 2. That we randomly decide which half goes to Alice and which half to Bob takes care of property 1.
A raffle that exclusively has tickets of type (i) will give a perfect anti-correlation between Alice’s result for
a and Bob’s result for
b. In that case, the entries on the diagonal in
Figure 1 and
Figure 2 are 0, while the off-diagonal ones are
. Thus, for tickets of type (i),
. A raffle that exclusively has tickets of type (ii) will give a perfect correlation. In that case, the off-diagonal entries in
Figure 1 and
Figure 2 are 0 and those on the diagonal are
. Thus, for tickets of type (i),
. To simulate the correlation in
Figure 2 for arbitrary values of
, we need a raffle with a fraction
of type-(i) tickets and a fraction
of type-(ii) tickets.
It turns out that, for all values between
and
,
is the (
Pearson)
correlation coefficient of the variables
and
, the taste Alice finds when peeling
a and the taste Bob finds when peeling
b.
4 The correlation coefficient of two stochastic variables
X and
Y is defined as the covariance,
, divided by the standard deviations,
and
, the square roots of the variances,
and
. What simplifies matters in the case of the variables
and
is that they are
balanced, i.e., their two possible values are each other’s opposite and these two values are equiprobable ([
8], p. 68). This means that their expectation values,
and
, vanish and that the correlation coefficient is given by
Inspection of the correlation arrays in
Figure 1 and
Figure 2 tells us that
that
and that, similarly,
. Substituting these results into Equation (
3), we see that the correlation coefficient is indeed equal to the parameter characterizing the correlation in
Figure 2:
As noted above, unless
, we need a mix of tickets to simulate the correlation array in
Figure 2 with one of our raffles. With our quantum bananas, as we will show below, we can produce this correlation array for arbitrary values
with pairs of bananas in the familiar fully entangled singlet state,
but with different choices for the peeling directions a and b.
Using the bases
and
of eigenvectors of the operators representing the observables ‘taste when peeled in the
a-direction’ and ‘taste when peeled in the
b-direction’ for the one-banana Hilbert space to construct bases for the two-banana Hilbert space, we can write the singlet state as:
where
, etc., is shorthand for the tensor product
, etc.
Figure 4.
Eigenvectors for ‘taste when peeled in the a-direction’ and ‘taste when peeled in the b-direction’ in the one-banana Hilbert space, where is the angle between the peeling directions a and b.
Figure 4.
Eigenvectors for ‘taste when peeled in the a-direction’ and ‘taste when peeled in the b-direction’ in the one-banana Hilbert space, where is the angle between the peeling directions a and b.
The relation between the
a-basis and the
b-basis is illustrated in
Figure 4. The angle
between these pairs of eigenvectors is equal to half the angle
between the peeling directions
a and
b. The transformation from the
b-basis to the
a-basis is given by:
its inverse by:
To find the probabilities of the various combinations of tastes when Alice peels
a and Bob peels
b, we use these transformation equations to write the singlet state in the
-basis:
The Born rule tells us that the probabilities of finding the various combinations of tastes for this combination of peelings are given by the squares of the coefficients of the corresponding terms of the singlet state in this basis. Recalling that
, we thus arrive at the correlation array in
Figure 5.
Using this correlation array to calculate the correlation coefficient (see Equation (
3)), we find:
We saw earlier (see Equation (
6)) that
is equal to the parameter
characterizing the correlation array in
Figure 2. With the appropriate choice of peeling directions, we can thus obtain this correlation array for any value
with the appropriate measurements on the same quantum state, whereas we needed a mix of tickets to obtain this correlation array with one of our raffles.
In
Understanding Quantum Raffles [
8], we used the tools introduced above to analyze the correlations found in an experimental setup due to Mermin [
10] in which Alice and Bob peel and taste bananas in the singlet state choosing between three different peeling directions,
a,
b and
c. The correlations between the tastes found by Alice and Bob in this Mermin setup can be represented by a
correlation array with cells of the form shown in
Figure 2 with
, etc. (see
Figure 5 and Equation (
11)).
Because of the symmetry of the singlet state, the cells of the correlation array on one side of the diagonal (, and ) are the same as those on the other side (, and ). In the cells on the diagonal, we have a perfect anti-correlation (if , and ). A correlation array for this Mermin setup can thus be characterized by the correlation coefficients for half of its off-diagonal cells, , and , with all three taking on values between and .
Inspired by Pitowsky [
7], we used these coefficients as coordinates of a point in a
cube, the
non-signaling polytope (
) for the Mermin setup (see
Figure 7). The part of
allowed by quantum mechanics is called the
quantum convex set (
); the part allowed by local hidden-variable theories the
local polytope (
).
5We derive the inequalities defining
and
in this case. As our model for a local hidden-variable theory, we use a raffle with a mix of the four types of tickets shown in
Figure 6.
Figure 6.
Tickets for a raffle meant to simulate the correlation array for the Mermin setup.
Figure 6.
Tickets for a raffle meant to simulate the correlation array for the Mermin setup.
The values of the correlation coefficients for raffles with only one type of ticket can be read directly off that ticket. For example, if the values for
a and
b on opposite sides of the ticket are the same,
; if they are opposite,
.
Table 1 collects the values of
,
and
for ticket types (i)–(iv).
The correlations produced by raffles with just one of these four ticket-types are represented by the vertices that are labeled (i) through (iv) in the non-signaling cube in
Figure 7. The
local polytope (
) for the Mermin setup is the tetrahedron formed by these four vertices.
Figure 7.
The non-signaling polytope (), the quantum convex set () and the local polytope () for the Mermin setup.
Figure 7.
The non-signaling polytope (), the quantum convex set () and the local polytope () for the Mermin setup.
The Bell inequality for the Mermin setup corresponds to one of the four facets of the tetrahedron, the one with the vertices (ii), (iii) and (iv). The pair of inequalities associated with this facet, which can be read off
Table 1, is:
This is the direct analogue of the CHSH inequality, the Bell inequality for a setup involving four rather than three different peelings, with Alice peeling
a or
b and Bob peeling
or
(cf. Equation (
48) below and Chapter 5 in our book [
8]). To fully characterize the local polytope for the Mermin setup, we need three more pairs of inequalities such as the ones in Equation (
12), corresponding to the other three facets of the tetrahedron in
Figure 7.
To find the
quantum convex set (
) for the Mermin setup, we consider the
matrix formed by the correlation coefficients characterizing the nine cells of its correlation array. Using that
, etc. (where
and
are unit vectors in the peeling directions
a and
b), we can write this
correlation matrix as:
This is (minus) a
Gram matrix, which has the property that its determinant cannot be negative:
. This gives us the constraint we are looking for:
This non-linear inequality defines the elliptope representing the quantum convex set (
) for the Mermin setup in
Figure 7. Taking a slice of this figure by setting one of the
’s to zero, we obtain the Vitruvian-man-like cartoon in
Bananaworld for
,
and
in an arbitrary setup ([
6], p. 107, Figure 5.2).
We now have all the ingredients we need from
Understanding Quantum Raffles [
8] to analyze the correlations found with Hardy and Hardy–Unruh states.
5. Geometrical Representation of the Correlations Found with Hardy–Unruh States
What can we say about the local polytope
and the quantum convex set
for the Hardy–Unruh setup (cf.
Figure 7)?
To answer this question, we start by comparing the correlation array in
Figure 11 for the tastes of pairs of bananas, peeled
a or
b, in the state
in Equations (
38)–(
43) (the Hardy–Unruh setup) to the correlation array in
Figure 12 for the tastes of pairs of bananas, one peeled
or
, the other peeled
or
, in the state
in Equation (
7) (the CHSH setup).
The correlation array for the CHSH setup consists of four cells of the form shown in
Figure 5 and can be fully characterized by four correlation coefficients (see Equation (
11)):
The local polytope for this setup is given by the CHSH inequality and three similar pairs of inequalities ([
8], pp. 160–161, Equations (5.4)–(5.7)):
These inequalities can be found in the same way as the pair in Equation (
12) for the Mermin setup ([
8], pp. 157–159: Figure 5.1 shows the raffle tickets for the CHSH setup, Table 5.1 lists the
values for these tickets).
The quantum convex set for the CHSH setup is given by a non-linear inequality, first obtained by Landau [
18], that follows from the straightforward generalization of the elliptope inequality in Equation (
14) if Alice and Bob have four rather than three different peelings to choose from (see [
8], p. 166, Equation (5.30), with
a,
b,
and
relabeled
,
,
and
):
To use these inequalities for the Hardy–Unruh setup we need to modify the setup somewhat. The problem is that Equations (
48) and (
49) are derived for balanced variables, i.e., their two possible values are each other’s opposite and equiprobable (see
Section 2). This guarantees that their expectation values vanish, which greatly simplifies the expressions for standard deviations and correlation coefficients (see Equations (
3) and (
5)). While the variables measured by Alice and Bob in the Hardy–Unruh setup have opposite values, their expectation values do not vanish, as these two values are not equiprobable.
We therefore introduce new variables that
are balanced but have the same covariances as the original ones. The correlations between these new balanced variables for a modified Hardy–Unruh setup can be simulated by a CHSH setup with appropriately chosen peeling directions.
9 Moreover, the modification preserves an important property of the correlation array for the Hardy–Unruh setup in
Figure 11: the
and
cells are identical. Hence, we only need three
parameters to characterize the correlation array for the CHSH setup with which we can simulate the correlations found in the modified Hardy–Unruh setup. This means that the local polytope and the quantum convex set for the modified Hardy–Unruh setup—like those for the Mermin setup (see
Figure 7)—can be pictured in three dimensions.
We introduce the new balanced variables for the modified Hardy–Unruh setup in two steps. The three panels in
Figure 13 illustrate the process for the
cell. First, we imagine that Alice and Bob, still choosing between peelings
a and
b, record the
opposite of the taste of their bananas. The correlation array for this experiment is obtained by switching the two entries on the diagonal and the two entries on the skew diagonal in each cell of the correlation array in
Figure 11 (see panel (ii) in
Figure 13 for the
cell). This obviously flips the signs of the expectation values but does not affect the covariances. As we saw in Equation (
4), in each cell, the covariance is equal to
times the sum of the two entries on the diagonal minus
times the sum of the two entries on the skew diagonal. As these sums stay the same, so do the covariances.
Next, we imagine that Alice and Bob, still choosing between peelings
a and
b, record the taste of their bananas in even runs and the
opposite of the taste in odd runs. We obtain the correlation array for this experiment by taking, for all 16 entries, the straight average of the entries in the correlation arrays for the even and the odd runs (see panel (iii) in
Figure 13 for the
cell). The four covariances are the same in all runs so the covariances for this combined correlation array will still be the same as for the original correlation array in
Figure 11. However, by having Alice and Bob alternate between recording the taste and recording minus the taste of their bananas, we ensure that the variables they measure are balanced.
Panel (iii) in
Figure 13 shows this for the
cell, but it is true for all four cells of the combined correlation array. Both entries on the diagonal are the average of the two entries on the diagonal in the original correlation array, and both entries on the skew diagonal are the average of the two entries on the skew diagonal in the original correlation array. Hence, in each cell, the sum of the two entries in each row and in each column gives
times the sum of all four probabilities in that cell. The entries in each row and in each column of each cell therefore sum to
, which means that the variables measured by Alice and Bob when they alternate between recording the taste and minus the taste of their bananas are indeed balanced.
In each cell of the correlation array for the balanced Hardy–Unruh setup, as we call it, the two entries on the diagonal and the two entries on the skew diagonal can be set equal to
times the square of, respectively, the sine and the cosine of some angle. Since two of its four cells are identical, the correlation array for the balanced Hardy–Unruh setup can thus be fully characterized by three angles. Identifying these angles with half the angles
,
and
between the peeling directions
,
,
and
, we can cast this correlation array in the form of the correlation array for the CHSH setup in
Figure 12.
The standard deviations for the variables in this correlation array are all
, so the three correlation coefficients characterizing it are given by
where we used that the covariances for this CHSH setup are the same as those for the original Hardy–Unruh setup.
Figure 14 and
Figure 15 show the local polytope
and the quantum convex set
for the subclass of correlations found in the CHSH setup if two of its four correlation coefficients are identical.
10 We obtain the inequalities defining
and
in this case by setting
in Equations (
48) and (
49). We created
Figure 14 and
Figure 15 by feeding the resulting inequalities into Mathematica. Note the similarity of these figures to
Figure 7 for the Mermin setup. In both cases,
is an inflated version of
. This ‘inflation’ corresponds to the
pushout operation in Le et al. ([
13], pp. 10–11) and was first found by Masanes [
21].
The values of the correlation coefficients in Equation (
50) parameterize the curve shown in
Figure 14 and
Figure 16 representing the correlations found between the values of the balanced variables measured on the state
for
in our balanced Hardy–Unruh setup.
We can compute the covariances on the right-hand side of Equation (
50) for these correlation coefficients with the help of the correlation array in
Figure 11 (cf. Equation (
4)):
Multiplying these expressions by 4 and feeding them into Mathematica, we found the curve in
Figure 14 and
Figure 16.
11 These figures clearly show that the correlations found with the state
are outside the local polytope. As one readily verifies, using Equations (
50) and (
51), they violate the third pair of CHSH-type inequalities in Equation (
48):
The second term on the right-hand side makes the left-hand side smaller than
. Comparison with Equation (
45) shows that this term is equal to 4 times the probability
of the outcome responsible for the broken arrow found with the state
. As the following argument will show, this is no coincidence.
Let
A and
B represent the tastes found by Alice and Bob for some combination of peelings. Let
represent the probabilities of the four possible combinations of tastes. Solving four linear equations for these four probabilities, we can express them in terms of the expectation values and the covariance of
A and
B.
12 Normalization gives us the first of these four equations:
the expectation values of
A and
B give us two more:
and the covariance of
A and
B gives us the last one:
Multiplying Equations (
54) and (
55) by 2 and Equation (
56) by 4 and solving the resulting equations for the four probabilities, we find:
If the expectation values vanish, the probabilities are equal to
plus or minus the covariance. Setting the covariance equal to
, we recover the entries in the correlation array in
Figure 2. At this point, however, we are interested in the case that the expectation values do
not vanish.
Consider the probabilities that are 0 in the
,
and
cells of the correlation array in
Figure 11 and the non-vanishing probability in the
cell that is responsible for the broken arrow in the Hardy–Unruh chain. This gives us the following four equations:
If the last three are subtracted from the first, the expectation values all cancel and we are left with:
Multiplying both sides by 4 and regrouping terms, we can rewrite this as:
Using Equation (
50) to replace 4 times the covariances by the corresponding correlation coefficients and using Equation (
45) for
, we recover Equation (
52). This shows, to reiterate, that the violation of the corresponding CHSH-type inequality is given by the probability of the outcome responsible for the broken arrow in the Hardy–Unruh chain. The maximum value of this probability is the same as the maximum value of the probability
of the outcome responsible for the broken arrow in the Hardy chain (see Equation (
27)).
6. Conclusions
Our examination of Hardy–Unruh chains has left us with a trifecta of deflating insights. The third is that we cannot claim great originality for the first two. Yet even those for whom they are hardly new will agree, we hope, that our use of the framework of
Understanding Quantum Raffles [
8] helped put these insights in sharper relief. In this short concluding section, we summarize how our analysis in terms of raffle tickets, correlation arrays and their geometrical representation led us to these insights. Our treatment of Hardy–Unruh chains also connects the literature on Hardy’s paradox with the (much more extensive) literature on correlation polytopes (see note 14) and will hopefully contribute to making the latter more widely accessible.
The first insight is that the states giving rise to the broken arrow in Hardy’s chain of conditionals in Equations (
1) and (
15) are no different from those giving rise to the broken arrow in Unruh’s chain of conditionals in Equations (
2) and (
28). All these states are part of one large family of non-maximally entangled states (how large can be gleaned from our construction of a generic member in Equations (
29)–(
36), which simplifies the constructions given by Hardy [
2] and Unruh [
4]). We exhibited these family ties by constructing correlation arrays for correlations leading to both kinds of broken arrows, the one in
Figure 9 for the Hardy states
in Equations (
16) and (
20)–(
22), and the one in
Figure 11 for the Hardy–Unruh states
in Equations (
38)–(
43). We showed how the defining properties of Hardy and Hardy–Unruh chains of conditionals can be read off these correlation arrays. We then showed that these two correlation arrays differ only in how they are labeled (the peelings
a and
b, the tastes + and −, and the angle
parametrizing the states). Although we only did this for members of a specific branch of the Hardy–Unruh family, the same could be done for any family member.
The second insight is that a broken arrow in a Hardy–Unruh chain is equivalent to the violation of some Bell inequality. We showed this (again: for a special branch of the Hardy–Unruh family) by constructing a geometrical representation of the correlation array for the Hardy–Unruh setup (see
Figure 14,
Figure 15 and
Figure 16). What complicated this task is that the two possible values of the variables measured in the Hardy–Unruh setup are not equiprobable. We took care of this problem by slightly modifying the Hardy–Unruh setup. We could then use a special case of the CHSH inequality (and similar inequalities associated with other facets of the local polytope) to characterize the class of correlations in this modified Hardy–Unruh setup allowed by a local hidden-variable theory (i.e., the class of correlations in this setup that can be simulated with one of our raffles). We showed (see Equation (
60)) that the term that expresses the violation of one of these CHSH-type inequalities is exactly the same as the expression for the probability of the very outcome that is responsible for the broken arrow in the corresponding Hardy–Unruh chain. This result may not come as a surprise to many readers, but it was still worth proving.
We agree with Mermin ([
17], pp. 883–884) that one should not exaggerate the difference between using one single outcome or the statistics of many outcomes as evidence that a correlation is not to be had in a local hidden-variable theory. If, for instance, we want to simulate the correlation array for a Hardy–Unruh setup in
Figure 9 or
Figure 11 with one of our raffles, the problem is
not to obtain a non-zero probability for one particular outcome, but to obtain it
while at the same time obtaining zero probabilities for several other outcomes.
13 In other words, rather than focusing on individual entries, we need to consider a correlation array as a whole.
14Despite being taken down a notch, Hardy–Unruh chains remain valuable. Whereas we usually consider violations of Bell inequalities by correlations found in measurements on maximally entangled states, Hardy–Unruh chains forcefully demonstrate that the slightest amount of entanglement already makes it impossible to simultaneously assign definite values to variables represented by non-commuting operators.