1. Review: Quantum Language (=Measurement Theory (=MT))
1.1. Introduction
Recently (cf. refs. [1] - [10] , also see (B0) - (B3) later), we proposed quantum language, which was not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the linguistic turn of dualistic idealism. And further we believe that quantum language should be regarded as the foundations of quantum information science. Quantum language is formulated as follows.
(A)
Note that this theory (A) is not physics but a kind of language based on the quantum mechanical world view. That is, we think that the location of quantum language in the history of world-descriptions is as follows.
And in Figure 1, we think that the following four are equivalent (refs. [1] [8] ):
(B0) to propose quantum language (cf. ⑩ in Figure 1, ref. [1] [8] ).
(B1) to clarify the Copenhagen interpretation of quantum mechanics (cf. ⑦ in Figure 1, refs. [2] [7] [11] ), that is, the linguistic Copenhagen interpretation is the true figure of so-called Copenhagen interpretation.
(B2) to clarify the final goal of the dualistic idealism (cf. ⑧ in Figure 1, refs. [3] [9] ).
(B3) to reconstruct statistics in the dualistic idealism (cf. ⑨ in Figure 1, refs. [4] [5] [6] [12] ).
In Bohr-Einstein debates (refs. [13] [14] ), Einstein’s standing-point (that is, “the moon is there whether one looks at it or not” (i.e., physics holds without observers)) is on the side of the realistic world view in Figure 1. On the other hand, we think that Bohr’s standing point (that is, “to be is to be perceived” (i.e., there is no science without measurements)) is on the side of the linguistic world view in Figure 1 (though N. Bohr might believe that the Copenhagen interpretation (proposed by his school) belongs to physics).
In this paper, contrary to Bell’s spirit (which inherits Einstein’s spirit), we try to discuss Bell’s inequality (refs. [15] [16] [17] [18] ) in quantum language (i.e., quantum theory with the linguistic Copenhagen interpretation). And we clarify that whether or not Bell’s inequality holds does not depend on whether classical systems or quantum systems (in Section 3), but depend on whether a combined measurement exists or not (in Section 2). And further we assert that our argument (based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell’s philosophical argument (based on Einstein’s spirit).
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Figure 1. The history of the world-descriptions.
1.2. Quantum Language (=Measurement Theory); Mathematical Preparations
Now we shall explain the measurement theory (A).
Consider an operator algebra
(i.e., an operator algebra composed of all bounded linear operators on a Hilbert space H with the norm
), and consider the pair
, called a basic structure. Here,
is a C*-algebra, and
(
) is a particular C*-algebra (called a W*-algebra) such that
is the weak closure of
in
.
The measurement theory (=quantum language) is classified as follows.
(C)
That is, when
, the C*-algebra composed of all compact operators on a Hilbert space H, the (C1) is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when
is commutative (that is, when
is characterized by
, the C*-algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space
(cf. [19] [20] )), the (C2) is called classical measurement theory (or, classical system theory).
Also, note (cf. [19] ) that, when
,
1)
,
,
(i.e., pre-dual space).
Also, when
,
2)
“the space of all signed measures on
”,
,
, where
is some measure on
(cf. [19] ). Also, the
is usually denoted by
.
Let
be a C*-algebra, and let
be the dual Banach space of
. That is,
= {r | r is a continuous linear functional on
}, and the norm
is defined by
. Define the mixed state
such that
and
for all
such that
. And define the mixed state space
such that
A mixed state
is called a pure state if it satisfies that
for some
and
implies
. Put
which is called a state space. It is well known (cf. [19] ) that
(i.e., the Dirac notation) |
, and
is a point measure at
, where
. The latter implies that
can be also identified with
(called a spectrum space or simply spectrum) such as
(1)
For instance, in the above 2) we must clarify the meaning of the “value” of
for
and
. An element
is said to be essentially continuous at
, if there uniquely exists a complex number
such that
• if
converges to
in the sense of weak* topology of
, that is,
then
converges to
.
And the value of
is defined by the
.
According to the noted idea (cf. [21] ), an observable
in
is defined as follows:
1) [s-field] X is a set,
(
, the power set of X) is a s-field of X, that is, “
”, “
”, “
”.
2) [Countable additivity] F is a mapping from
to
satisfying: a) for every
,
is a non-negative element in
such that
, b)
and
, where 0 and I is the 0-element and the identity in
respectively. (c): for any countable decomposition
of
(i.e.,
,
,
(
)), it holds that
in the sense of weak* topology in
.
Remark 1. Quantum language has two formulations (i.e., the C*-algebraic formulation and the W*-algebraic formulation). In this paper, we devote ourselves to the W*-algebraic formulation, which may, from the mathematical point of view, be superiority to the C*-algebraic formulation. That is, in the above 2), the countable additivity (i.e.,
) is naturally discussed in the W*-algebraic formulation. However, the C*-algebraic formulation has a merit such that we can use it without sufficient mathematical preparation. For the C*-algebraic version of this paper, see my preprint [10] .
1.3. Axiom 1 [Measurement] and Axiom 2 [Causality]
With any system S, a basic structure
can be associated in which the measurement theory (A) of that system can be formulated. A state of the system S is represented by an element
and an observable is represented by an observable
in
. Also, the measurement of the observable
for the system
with the state
is denoted by
(or more precisely,
). An observer can obtain a measured value x (
) by the measurement
.
The Axiom 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics (cf. ref. [22] ). And thus, it is a statement without reality.
Now we can present Axiom 1 in the W*-algebraic formulation as follows.
Axiom 1 [ Measurement ]. The probability that a measured value
obtained by the measurement
belongs to a set
is given by
if
is essentially continuous at
.
Next, we explain Axiom 2. Let
and
be basic structures. A continuous linear operator
(with weak* topology)
(with weak* topology) is called a Markov operator, if it satisfies that 1)
for any non-negative element
in
, 2)
, where
is the identity in
,
. In addition to the above 1) and 2), in this paper we assume that
and
.
It is clear that the dual operator
satisfies that
. If it holds that
, the
is said to be deterministic. If it is not deterministic, it is said to be non-deterministic or decoherence. Here note that, for any observable
in
, the
is an observable in
.
Now Axiom 2 in the measurement theory (A) is presented as follows:
Axiom 2 [Causality]. Let
. The causality is represented by a Markov operator
.
1.4. The Linguistic Interpretation (=The Manual to Use Axioms 1 and 2)
In the above, Axioms 1 and 2 are kinds of spells, (i.e., incantation, magic words, metaphysical statements), and thus, it is nonsense to verify them experimentally. Therefore, what we should do is not “to understand” but “to use”. After learning Axioms 1 and 2 by rote, we have to improve how to use them through trial and error.
We can do well even if we do not know the linguistic interpretation. However, it is better to know the linguistic interpretation (=the manual to use Axioms 1 and 2), if we would like to make progress quantum language early.
The essence of the manual is as follows:
(D) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it cannot be measured any longer. Thus, the collapse of the wavefunction is prohibited (cf. [7] ). We are not concerned with anything after measurement. That is, any statement including the phrase after the measurement is wrong. Also, the causality should be assumed only in the side of system, however, a state never moves. Thus, the Heisenberg picture should be adopted, and thus, the Schrödinger picture should be prohibited. Also, it is added that there is no probability without a measurement.
and so on. For details, see [8] .
1.5. Generalized Simultaneous Measurement, Parallel Measurement
Definition 2. [Generalized simultaneous observable, Generalized simultaneous measurement] Let
be a basic structure. Consider observables
in
. Let
be the product measurable space, i.e., the product space
and the product s-field
, which is defined by the smallest s-field that contains a family
. An observable
in
is called the generalized simultaneous observable (or, quasi-product observable, combined observable, etc.) of
, if it holds that
(2)
Also, the measurement
is called a generalized simultaneous measurement of measurements
. A generalized simultaneous observable is called a simultaneous observable, if it holds:
Note that the existence and the uniqueness of a generalized simultaneous observable
in
are not assured in general, however the simultaneous observable always exists if observables
commute, i.e.,
(3)
Definition 3. [Parallel observable, Parallel measurement] For each
, consider a basic structure
and a measurement
. We consider the spatial tensor W*-algebra
, and consider the product measurable space
. Consider the observable
in
such that
which is called the parallel observable of
. And let
. Then the measurement
(which is also denoted by
) is called a parallel measurement of
. Note that the parallel measurement always exists uniquely.
2. Bell’s Inequality Always Holds in Classical and Quantum Systems
Our Main Assertion about Bell’s Inequality
In this paper, I assert that Bell’s inequality should be studied in the framework of quantum language (i.e., quantum theory with the linguistic Copenhagen interpretation). Let us start from the following definition, which is a slight modification of the generalized simultaneous observable in Definition 2. That is, Definitions 2 - 4 are due to the linguistic Copenhagen interpretation (D), “Only one measurement is permitted”.
Definition 4 [Combined observable (cf. ref. [12] )] Let
be a basic structure. Put
. Consider four observables:
,
,
,
in
. The four observables are said to be combinable if there exists an observable
in
such that
(4)
for any
. The observable
is said to be a combined observable of
. Also, the measurement
is called the combined measurement of
,
,
and
.
Remark 5. 1) Note that the Formula (4) implies that
for all
.
2) Syllogism (i.e.,
) does not hold in quantum systems but in classical systems (cf. ref. [8] ). A certain combined observable plays an important role in the proof of the classical syllogism (cf. ref. [12] ).
The following theorem is all of our insistence concerning Bell’s inequality. We assert that this is the true Bell’s inequality.
Theorem 6. [Bell’s inequality in quantum language] Let
be a basic structure. Put
. Fix the pure state
. And consider the four measurements
,
,
and
. Or equivalently, consider the parallel measurement
. Define four correlation functions
such that
Assume that four observables
,
,
and
are combinable, that is, we have the combined observable
in
such that it satisfies (4). Then we have a combined measurement
of
,
,
and
. And further, we have Bell’s inequality in quantum language as follows.
(5)
Proof. Clearly we see,
,
(6)
(for example,
). Therefore, we see that
This completes the proof.
As the corollary of this theorem, we have the followings:
Corollary 7. Consider the parallel measurement
as in Theorem 6. Let
be a measured value of the parallel measurement
. Let N be sufficiently large natural number. Consider N-parallel measurement
. Let
be the measured value. That is,
Here, note that the law of large numbers says: for sufficiently large N,
Then, it holds, by the Formula (5), that
(7)
which is also called Bell’s inequality in quantum language.
Remark 8. [The conventional Bell’s inequality (cf. refs. [17] [16] [18] )] The mathematical Bell’s inequality is as follows: Let
be a probability space. Let
be a measurable functions. Define the correlation functions
by
. Then, the following mathematical Bell’s inequality (or precisely, CHSH inequality (cf. ref. [16] )) holds:
(8)
(E) This is easily proved as follows.
This completes the proof.
Recall Theorem 6 (Bell’s inequality in quantum language), in which we have, by the combinable condition, the probability space
. Therefore the proof of Theorem 6 and the above proof (E) are, from the mathematical point of view, the same.
3. “Bell’s Inequality” Is Violated in Classical Systems as Well as Quantum Systems
In the previous section, we show that Theorem 6 (or Corollary 7) says
(F1) Under the combinable condition (cf. Definition 4), Bell’s Inequality (5) (or, (7)) holds in both classical systems and quantum systems.
Or, equivalently,
(F2) If Bell’s Inequality (5) (or (7)) is violated, then the combined observable does not exist, and thus, we cannot obtain the measured value (by the combined measurement).
This is similar to the following elementary statement in quantum mechanics:
(
) We have no (generalized) simultaneous measurement of the position observable Q and the momentum observable P, and thus we cannot obtain the measured value (by the generalized simultaneous measurement),
which may be, from Einstein’s point of view, represented that “true value (or, hidden variable) of the position and momentum” does not exist. Since the error
is usually defined by
, it is not easy to define the errors
and
in Heisenberg’s uncertainty principle
.
This definition was completed and Heisenberg’s uncertainty principle was proved in ref. [11] . Also, according to the maxim of dualism: “To be is to be perceived” due to G. Berkeley, we think that it is not necessary to name that does not exist (or equivalently, that is not measured).
The above statement (F2) makes us expect that
(G) Bell’s inequality (5) (or (7)) is violated in classical systems as well as quantum systems without the combinable condition.
This (G) was already shown in my previous paper [2] . However, I received a lot of questions concerning (G) from the readers. Thus, in this section, we again explain the (G) precisely.
Bell Test Experiment
In order to show the (G), three steps ([Step: I] - [Step: III]) are prepared in what follows.
[Step: I].
Put
. Define complex numbers
(
: the complex field)
such that
. Define the probability space
such that
(9)
The correlation
is defined as follows:
(10)
Now we have the following problem:
( H) Find a measurement
such that
(11)
and
which is the same as the condition in Remark 5.
[Step: II].
Let us answer this problem (H) in the two cases (i.e., classical case and quantum case), that is,
•
1) the case of quantum system:
Put
For each
, define the observable
in
such that
where
. Then, we have four observable:
(12)
and further,
(13)
in
, where it should be noted that
is separated by
and
.
Further define the singlet state
, where
Thus we have the measurement
in
. The followings are clear: for each
,
(14)
For example, we easily see:
Therefore, the measurement
satisfies the condition (H).
2) the case of classical systems:
Put
,
(
, i.e., the point measure at
)). Define the observable
in
such that
Thus, we have four observables
(15)
in
(though the variables are not separable (cf. the formula (13)). Then, it is clear that the measurement
satisfies the condition (H).
2)’ the case of classical systems:
It is easy to show a lot of different answers from the above 2). For example, as a slight generalization of (9), define the probability measure
such that
(16)
And consider the real-valued continuous function
such that
. And assume that
for some
,
(
, i.e., the point measure at
)). Define the observable
in
such that
(17)
Thus, we have four observables
in
(though the variables are not separable (cf. the Formula (13)). Then, it is clear that the measurement
satisfies the condition (H).
[Step: III].
As defined by (9), consider four complex numbers
such that
. Thus we have four observables
in
. Thus, we have the parallel measurement
in
.
Thus, putting
we see, by (10), that
(18)
Further, assume that the measured value is
. That is,
Let N be sufficiently large natural number. Consider N-parallel measurement
. Assume that its measured value is
. That is,
Then, the law of large numbers says that
This and the Formula (18) say that
(19)
Therefore, Bell’s Inequality (5) (or (7)) is violated in classical systems as well as quantum systems.
Remark 9. For completeness, note that the observables
in the classical
are not combinable in spite that these commute. Also, note that the Formulas (16) and (17) imply that
which is similar as in Remark 5; 1) or in (H).
4. Conclusions
In Bohr-Einstein debates (refs. [13] [14] ), Einstein’s standing-point (that is, “the moon is there whether one looks at it or not” (i.e., physics holds without observers)) is on the side of the realistic world view in Figure 1. On the other hand, we think that Bohr’s standing point (that is, “to be is to be perceived” (i.e., there is no science without measurements)) is on the side of the linguistic world view in Figure 1.
In this paper, contrary to Bell’s spirit (which inherits Einstein’s spirit), we try to discuss Bell’s inequality in Bohr’s spirit (i.e., in the framework of quantum language). And we show Theorem 6 (Bell’s inequality in quantum language), which says the statement (F2), that is,
(I1) (º(F2)): If Bell’s Inequality (5) (or (7)) is violated, then the combined observable does not exist, and thus, we cannot obtain the measured value (by the measurement of the combined observable).
Also, recall that Bell’s original argument says, roughly speaking, that
(I2) If the mathematical Bell’s Inequality (8) is violated in Bell test experiment (the quantum case of Section 3.1), then hidden variables do not exist.
It should be note that the concept of “hidden variable” is independent of measurements, thus, the (I2) is a philosophical statement in Einstein’s spirit, on the other hand, the (I1) is a statement in Bohr’s spirit (i.e., there is no science without measurements). It is sure that Bell’s answer (I2) is attractive philosophically, however, we believe in the scientific superiority of our answer (I1). That is, we think that our (I1) is a scientific representation of the philosophical (I2). If so, we can, for the first time, understand Bell’s inequality in science. That is, Theorem 6 is the true Bell’s inequality. And we conclude that whether or not Bell’s inequality holds does not depend on whether classical systems or quantum systems (in Section 3), but depend on whether the combined measurement exists or not (in Section 2).
We hope that our proposal will be examined from various points of view1.
NOTES
1For the further information of quantum language, see http://www.math.keio.ac.jp/~ishikawa/indexe.html