1. Introduction
Probability, as formally viewed in stochastic processes (see, e.g., [
1]), is a fundamental ingredient to understand countless natural phenomena. For instance, it is ubiquitous in the general framework of classical statistical mechanics. Probability is also a keystone in quantum mechanics, where the concept of probability amplitudes relates to the distribution of outcomes upon measurements. Nonetheless, there are fundamental distinctions between the idea of randomness [
2] in classical and quantum physics since probability can have a contrasting character in these two realms [
3]. A context where their differences is particularly manifest is in the analysis of correlations of physical quantities [
2,
4].
Although many restrictions may apply [
5], correlations can be used as an measure of the degree of determinism/randomness in a system. Indeed, certain situations might be relatively easily to pinpoint. On the one hand, if there is a well-behaved mapping between
a and
b (with
a and
b possible values for
bona fide observables
A and
B describing a problem
1) — e.g.,
and
r in a classical Keplerian orbit — the correlation is “perfect” and there should be a fully deterministic causation between them. On the other hand, if a specific value for
A determines a range of allowable values for
B as well as their frequencies of occurrence (with the same being true for
A regarding
B), this would indicate a stochastic connection between
A and
B. So, at least in principle one could infer a joint probability for
A and
B, allowing to define a proper correlation function for
A and
B,
.
But as already mentioned,
will display different features if resulting from either classical or quantum processes
2. Quantum are often more general than classical correlations [
4,
9,
10] and in some cases stronger [
11,
12,
13], for example, creating scalings (due to entanglement) in many-body systems which are of course classically absent [
14,
15]. Actually, there are distinct types of quantum correlations [
9], from the most basic — associated to the quantum formalism itself, which we simply call `quantum’ — to those presenting an increasing order of restrictness, namely, entanglement, steering and nonlocality,
Figure 1.
In brief, entanglement can be thought of as the primordial quantum correlation since it includes all possible forms of interrelations in pure bipartite states. When a composite state
, with
and
the Hilbert space of systems I and II,
cannot be written as the direct product
, for
and
, we say that
is entangled. A second, greater, interaction between systems I and II is that in which the state of one of them (e.g., I) can be driven or steered through measurements on the other (e.g., II); nevertheless the contrary does not hold true. In such inseparability context, the composite system exhibits a
steering correlation (reviews in [
9,
16,
17]). All steered systems are entangled, but not all entangled systems are steered. Finally, suppose we measure the observable
A for system II and
C for system I (with I and II assumed away apart), obtaining respectively
a and
c. In quantum mechanics, one finds that usually the joint probability
. The explanation is that the quantum world is nonlocal, a notion heavily criticized by the famous EPR (Einstein, Podolsky, Rosen) paper [
18]. An alternative would be to assume that quantum mechanics is incomplete, i.e., there exist
local hidden variables inaccessible by the theory. In Bell’s groundbreaking work [
19], it has been demonstrated that then
and some inequalities for the associated quantum correlations should be observed. However, the local hidden variables have been ruled out through concrete experiments [
20,
21,
22,
23,
24,
25,
26,
27,
28,
29] showing the violation of the Bell’s inequalities. Thus, in opposition to the EPR local realism, nonlocality is inherent to quantum mechanics and indeed its most restrictive correlation [
9,
30].
A rather significant example of Bell’s inequality for qubits (two levels systems) is the so called CHSH’s [
31]. Consider an ensemble of a composite formed by systems I and II. For all copies, I and II are put apart (conceivably without disturbing the original properties we shall measure). So, detectors 1 and 2 can infer certain observables for I (
and
, whose results are denoted as
e
) and II (
and
, of results
e
),
Figure 2. From appropriate averages we can calculate the CHSH correlation function
— explicit expressions are given in
Section 5. For arbitrary observables
X’s and
Y’s, local hidden variables would dictate
. But quantum mechanics also allows
, corresponding to a Bell’s inequality violation.
All the previous discussion eventually might give the wrong impression that to identify and classify quantum correlations is a rather straightforward task. In fact, from a technical point of view this is not usually the case [
9,
32,
33,
34]. But perhaps more surprisingly, in certain instances even a conceptual intuition about correlations can fail in the quantum context. Consider the observables
for I and
for II, with
. For any
j the allowed outcomes are the same,
and
. Also,
is a quantum correlation function of these possible observables considered in pairs
. With the exception of too specific functional forms for
, typically we could anticipate a greater (smaller) dispersion for
to be related to a weaker (stronger) interdependence among
. To be more concrete, let us presume the two conditions: (a) Regardless of
j, the corresponding
are all equally likely to describe the composite system state, i.e., in an ensemble sense we have an uniform distribution for these values so that any pair
n of contributes with a same probability
. Hence, for
and fixed
N, this assumption maximizes the normalized entropy
(note that
is well-defined even in the hypothetical limit of
). (b) There are no reinforcement dependence between the
’s and between the
’s. On the contrary, to know
(
) precludes to know with certainty the value of
(
),
. This is the physical consequence of the non-commutation,
and
, of the corresponding self-adjoint operators for the observables
and
in quantum mechanics. Therefore, given (a) and (b) we could expect a broader range of variation for
. So, for
being the CHSH (thence with
), presumably we would have the non-observance of the Bell’s inequality.
The arguments
3 in [
18] were completely based on particular bipartite
’s, the EPR states which do verify the assumption (a) (see next for detailed definitions). The conclusions in [
18] — hence, somehow indirectly the structure of these
’s — motivated the investigations in [
19] (refer also to [
36,
37]). But as aforementioned, incidentally local realism has been overturned by testing the Bell’s inequalities for arbitrary
’s.
In the present contribution we prove generally that the ’s also comply with the assumption (b). So, one could speculate the CHSH inequality to be overruled when considered for EPR states. We check this possibility by calculating only for observables (of , so essentially qubits) associated to . We find that . This result is a bit unforeseen taking into account the above ponderations. Moreover, it clearly typifies the subtleties of correlations in quantum mechanics, with the seemingly high dispersive interdependence of EPR states observables given rise to a more limited interval of values for , not exceeding the nonlocality threshold of 2.
The paper is organized as the following. We review important aspects of EPR states in
Section 2. We demonstrate the condition (b) for EPR states in
Section 3. We exemplify the results of
Section 3 for three levels systems in
Section 4. We show that EPR states do not violate Bell inequalities in the case of CHSH correlations in
Section 5. Final remarks and conclusion are drawn in
Section 6. More technical analysis are left to the Appendices.
2. Some Key Aspects in Forming and Measuring EPR States
Here we review the basic characteristics of EPR states
[
38], essentially those considered in the original work by Einstein-Podolsky-Rosen [
18]. A much more general and rigorous definition for finite dimensional systems can be found in [
39,
40] (for infinite-dimensional spaces, see [
41,
42]). Furthermore, we briefly discuss: the kind of measurements one can perform on
, the associated possible outcomes, and the consequent state reductions.
The main relevant steps in the preparation and posterior measurement of
are schematically depicted in
Figure 3. Initially (
), one has two non-interacting systems I and II, with their composite state
being simply the direct product of the individual states of I and II. Then, during the time interval
, I and II are brought to interact in such way to quantum correlate certain physical observables pertaining to I (say
C; associated to the Hermitian operator
with eigenvalues
) with some of those pertaining to II (say
A; associated to
with
), yielding the entangled state
(we assume
). Next, for
, I and II are somehow put sufficiently away so to cease any eventual mutual influence between the two systems. In other words, the type of interaction normally described by a potential
in the Schrödinger equation — coupling the systems I and II degrees of freedom — should be null. However, one must guarantee that regardless the separation procedure and the subsequent dynamics (for
), the attained entanglement (along
) between I and II are maintained. Obviously, this until an eventual measurement at any
. Hence, this last process would not alter the components of
corresponding to the eigenvectors
of I and
of II, or (
)
In Eq. (
1),
and
denote how other observables, distinctly from
C of I and
A of II, will evolve in time given that the systems no longer interact.
If by means of measurements we aim to assess only the correlated quantities
C and
A, the information given by
and
are not really relevant. Thus, one can drop these explicit dependences in Eq. (
1), just written
for
. We mention that it is a common practice in the analysis of EPR states [
38,
39]: (i) To suppose maximally entangled states, i.e., all
in
being equally probable, namely, for any
n to assume that
. (ii) To disregard eventual relative phases
by trivially reincorporating them into the states definition, or
. Thus, the existence or not of phases multiplying the eigenvectors is totally irrelevant for all our general results next.
A second fundamental feature of a EPR state is that
(with
, but prior to any measurement) can be written in the following distinct ways
where the set
(
) is composed by the eigenvectors of
(
), or
Therefore, from Eq. (
2) we have that from a measurement performed only on I at
— determining the observable
C (
D) — we should obtain complete knowledge about the value of
A (
B) for II,
Figure 3. In fact, if afterwards (
) we test system II for
A (
B), we should find exactly this previously inferred value.
The last feature defining an EPR state is to have for the observables
A and
B — associated to system II, cf. Eq. (
2) — operators which do not commute, i.e.,
. Actually, an important and involved issue related to this property is, given an EPR state, to determine all its associated pairs of non-commuting observables [
40]. We shall remark that
was a key assumption in [
18], in the attempt to show (of course misguidedly; see a summary of many valid objections to the EPR arguments in [
35] as well as in the refs. therein) that quantum mechanics is incomplete.
In the next Sec. we unveil a necessary condition for
to be written as in Eq. (
2), where also
. Then, in
Section 5 we analyze the consequences of such prerequisite for the measurement outcomes of EPR states.
3. A Necessary Condition for EPR States: The Observables Are Pairwisely Associated to Non-Commuting Operators
Hereafter we assume that for the composed systems (refer to Eq. (
1)),
. Note that
and
distinguishes I from II. On the other hand, all the specific aspects [
43] we wish determining for our individual systems (either I or II) are described by a proper separable Hilbert space
of dimensions
N, finite or infinite, spanned by a countable, i.e., discrete, basis (see Eq. (
4) below). The reasons are twofold.
First, EPR-like states have been vital to test certain fundamental predictions of QM, e.g., motivating the development of the Bell inequalities [
19,
36]. But although some theoretical proposals [
44], and even devised experimental arrangements [
45,
46,
47], are based on continuous variables (for a review, see [
48]), historical breakthroughs [
20,
21,
22,
23,
24,
25] and recent loophole-free measurements [
26,
27,
28,
29,
49] consider discrete observables. Second, the spectrum theorem for self-adjoint operators — relevant to determine suitable basis for
— holds true in very general terms [
50]. However, to establish solid grounded properties of transformations (similar to those presented in
Section 3.1) between continuous basis might demand extra technicalities, going beyond the scope of this contribution. For instance, for continuous basis associated to operators such as position and momentum, one should work with generalized eigenvectors in a rigged Hilbert space [
51].
As quantum observables we consider linear self-adjoint — or Hermitian, the usual jargon in physics — operators, whose domains are the whole
(or linear subsets of
which are dense in
if the self-adjoint operators are unbounded [
52,
53],
Appendix A). Also, for our goals it is not necessary to explicit address formal constructions of the join probability measures associated to any assessment of observables, e.g., as rigorously done in [
39]. Pragmatically, we just suppose fairly well-established definitions (refer to [
39]), consistent with actual procedures in concrete measurement realizations [
20,
21,
22,
23,
24,
25].
For
and
being the set of eigenvectors of
and
and representing orthonormal basis (ONB) of
, the basis change
are implemented through (for
N finite or infinite)
To derive the main result of the next Sec., we will rely on some features of such unitary matrix
— which are dependent on the commutation relation of
and
and discussed in the
Appendix A. We also leave to the
Appendix A considerations about appropriate self-adjoint operators
and
to represent observables for EPR states.
3.1. Correlations in the Observables of EPR States
From the above discussions and the remarks in
Section 2, for the systems I and II described by EPR states in Eq. (
2), we can concentrate on the “reduced” Hilbert space
. Moreover, for the self-adjoint operators
,
,
,
(see Eq. (
3)), we assume the features described in the
Appendix A.2 and that
and
(
and
) are defined on I (II). Thus, the set of eigenvectors
and
, respectively of
and
, are suitable ONBs for
. The same is true regarding
and
) (of
and
) with respect to
. We also recall that
.
Next, by inserting the second relation in Eq. (
4) into the second equality in Eq. (
2), we get
Comparing Eq. (
5) with Eq. (
2), we must have (recalling that
is an ONB)
Hence, from Eq. (
6) we see that up the complex conjugation of the elements of
(an operation which certainly should not change their matrix structural relations), the basis transformation
is fully akin to
. In other words, Eq. (
6) has exactly the same functional form of the mapping between
and
.
But the matrix
— hence also
— represents a change of basis associated to non-commuting operators,
and
. So, according to
Appendix A.2 it cannot be transformed into an identity, or more generally in a permutation, matrix. In the
Appendix A.3 we illustrate such universal fact considering the basis transformations of a spin-
system in arbitrary directions. So,
,
cannot commute, otherwise it would be possible to find a basis where they are simultaneously diagonal leading to
(or
), which is a contradiction.
Thus, if a state
can be expanded as in Eq. (
2), for the different basis related to the Hermitian operators
,
,
,
, and
, we conclude that necessarily
. As far as we know, this rather straightforward property of EPR states has surprisingly gone unexplored in the literature, even though it is implicit in particular contexts, like in the Bohm construction of EPR states [
54] and in the Bohr’s response [
55] to the EPR paper [
18].
4. An Explicit Example: Three Levels System
To illustrate the previous discussion, we assume such that the systems can be described in terms of three components angular moment states. The procedure here, although implemented to a particular situation, also exemplifies how to generate different expansions for a EPR state .
In this
case, the equivalent to the Pauli matrices read (with the usual direction association: 1 for
x, 2 for
y, 3 for
z)
where
for
the permutation symbol of
and
(
) for
. Further, we have for the eigenstates (
and
)
For the basis transformation
(with
), we find
whereas for
(with
and
), we get
For any
w and
, obviously
and
.
From the above one finds , , . Furthermore, for our purposes next it is useful to consider the notation meaning (notice the complex conjugation “” of ). Hence
- •
(1,2) :
- •
(2,1) :
- •
(1,3) :
- •
(3,1) :
- •
(2,3) :
- •
(3,2) :
In this way, in all situations with , results in an eigenvector of either , or , having the general form , where:
(i) , but always with ,
(ii) ,
(iii) .
Now, let us suppose the state (for
arbitrary)
then for any
we have
For
, the operators
and
for system II do not commute. Also in such case,
necessarily will be one of the eigenvalues (eventually multiplied by a phase
) of the operator
, which according to the condition (i) above does not commute with
.
Thence, for
f an one-to-one index function from
to
and
integer numbers, we find that
so that
is an EPR state (note we could simply relabeling
) and, as it should be, displays all the necessary properties regarding the non-commutation of the associated observables since
and
.
As a last example we discuss a degenerated case. We should mention that using the general expressions in [
56], it is straightforward to engender Hermitian matrices displaying degenerated eigenvalues. Thus, suppose the eigenvalues
(so
,
) of the eigenvectors
of
We set
Changing the basis from
to
, as done in Eq. (
12), and from the explicit form of
, we get (for
, thus assuming the values 1,-1,0 for
, respectively)
for
of eigenvalues
(so
,
) such that
, with
Finally,
and given the exact form of Eqs. (
15) and (
16), we find that
is an EPR state as expected.
5. EPR Observable Correlations for Qubits Do Not Violate the Bell Inequalities
As it should become clear from our previous analysis, EPR states display a rather unusual link between their observables. They are pairwisely incompatible, namely, considering Eq. (
2) we cannot simultaneously determine
A and
B for II and
C and
D for I. So, an interesting question is which are the possible values for the CHSH correlation function
[
31], obtained only from EPR states.
Assuming the EPR-like state in Eq. (
2), let
with the possible eigenvalues of
) being
(so
,
). For simplicity, we disregard eventual relative phases between the states. Nonetheless, we emphasize that introducing them through,
,
, and
, does not change the following results. Thus
The transformation from the basis
to
is given by an arbitrary unitary matrix
, whereas from
to
is given by the complex conjugated of
(cf., Eqs. (
5)–(
6)), or
. Using an appropriate parameterization for U(2) matrices (see, e.g., [
57]), we generally have
with
complex numbers such that
and
. Moreover, we suppose
so that
(see
Appendix A.3). Thus,
Therefore, for the observables
, the CHSH correlation function
[
31] reads
with (see
Figure 2)
To obtain
(as well
) and
(as well
) we consider in Eq. (
22)
given, respectively, by the first and second expansion in Eq. (
18). So, straightforwardly we find
For
and
we get
Now, using Eq. (
20) into Eq. (
24)
Finally (for
,
,
)
Analyzing all the combinations (i.e.,
,
or
,
) for
,
,
, one realizes that the only possibilities for
are
Hence, if we calculate
for any set of observables for which the state has the EPR structure, always
. However, the Bell’s inequalities violation in the CHSH construction corresponds to
(in fact,
[
58]).
The above finding is thus somewhat intriguing. The violation of the Bell inequalities [
19,
36] for a quantum system overturns the EPR claims that quantum mechanics is incomplete [
37]. Actually, many breakthrough experiments have clearly demonstrate the existence of quantum correlations violating the Bell’s inequalities [
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]. But the special correlations in EPR states seem to be of a different nature. Indeed, by probing
solely the EPR observables (here qubits) in a CHSH-like experiment, surprisingly one would not be able to discard hidden local variables.
6. Final Remarks and Conclusion
Universal quantum correlations — thus, much beyond entanglement — are notoriously more diverse and conceptually more complex than the classical counterpart. Mostly because of that, they tend to deceive our common sense expectations about their trending behavior. However, this is just a natural consequence of the involved way in which obervables are interrelated in certain specific quantum states. In the present work, we have analyzed one of these special ’s, bipartite EPR states.
For EPR states, it is prominent that all the associated observables are equally probable as the outcome results of measurements in a composite formed by systems I and II. But more than that, in the present work we have established that if can be expanded, say, in terms of eigenvectors either for the observables C (of I) and A (of II) or D (of I) and B (of II), then necessarily C is incompatible with D, and A is incompatible with B. In other words, for the corresponding self-adjoint operators it follows that and . For concrete examples, we have analyzed EPR states in the case of three-level systems
Given on the one hand the equiprobability, for any , for the observance — through experiments — of pairs of values ( or ( or etc., and on the other hand, a non-concurrence of for I and for II, then we have a considerable dispersion for the EPR observable values. Hence, for the case of qubits (), supposing the CHSH correlation function only for EPR states, one would foresee the violation of the associated Bell inequality, i.e., . But on the contrary, the we have explicitly shown that .
This finding emphasizes the intricacies of correlations in quantum mechanics. Indeed, an apparent strong dispersive interdependence of EPR states observables ends up leading to a more limited range of values for the important CHSH correlation in qubits.
Finally, a natural question relates to a possible extension of the present results to higher-dimensional systems, i.e., for
. For such a type of analysis, it is first necessary to make a generalization of the Bell’s inequalities, in particular, of the CHSH correlation. In fact, some interesting proposals have already been discussed in the literature [
59,
60,
61]. Second, irrespective of the exact analytic form of a generalized
, the computation idea should be the same as that in
Section 5. In this way, it would require unitary matrices
of order
. Given that a
U(
N) has
free parameters, general analytic calculations tend to be laborious. Presently, we are studying some concrete cases, and hopefully the results will appear in due course.