_{1}

^{*}

As the fundamental theory of quantum information science, recently I proposed the linguistic interpretation of quantum mechanics, which was characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. This turn from physics to language does not only extend quantum theory to classical theory but also yield the quantum mechanical world view. Although the wave function collapse (or more generally, the post-measurement state) is prohibited in the linguistic interpretation, in this paper I show that the phenomenon like wave function collapse can be realized. That is, the projection postulate is completely clarified in the linguistic interpretation.

Recently in [

As mentioned in a later section (Section 1.3 (C)), the wave function collapse (or more generally, the post- measurement state) is prohibited in the linguistic interpretation. Thus, some asked me “How is the projection postulate?”. This question urges me to write this paper. The reader who would like to know only my answer may skip this section and read from Section 2.

Now we briefly introduce quantum language as follows.

Consider an operator algebra

Hilbert space H with the norm

(called a

The measurement theory (=quantum language = the linguistic interpretation) is classified as follows.

That is, when_{1}) is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when _{2}) is called classical measurement theory.

Also, note that, when

1)

Also, when

2)

Let

A mixed state

which is called a state space. It is well known (cf. [

For instance, in the above 2) we must clarify the meaning of the “value” of

(B) If ^{*} topology of

then

And the value of

According to the noted idea (cf. [

1) [

2) [Countable additivity] F is a mapping from

0-element and the identity in ^{*} topology in

Measurement theory (A) is composed of two axioms (i.e., Axioms 1 and 2) as follows. With any system S, a basic structure

state

The Axiom 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics. And thus, it is a statement without reality.

Now we can present Axiom 1 in the

Axiom 1 [Measurement]. The probability that a measured value x (

Next, we explain Axiom 2. Let ^{*} topology) ^{*} topology) is called a Markov operator, if it satisfies that 1):

It is clear that the dual operator

deterministic. Also note that, for any observable

Now Axiom 2 is presented as follows (For details, see [

Axiom 2 [Causality]. Let

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 (=the manual to use Axioms 1 and 2). However, it is better to know the linguistic interpretation, if we would like to make progress quantum language early.

The essence of the manual is as follows:

(C) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it cannot be measured any longer. Hence, the wave function collapse is prohibited. We are not concerned with the problem: “When is a measurement taken?”. 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.

and so on. For details, see [

From here, I devote myself to quantum system (A_{1}) (and not classical system (A_{2})).

Let

Axiom 1 says:

(D_{1}) The probability that a measured value

Also, the von Neumann-Lüders projection postulate (in the Copenhagen interpretation, cf. [

(D_{2}) When a measured value

And therefore, when a next measurement

Problem 1. In the linguistic interpretation, the phrase: post-measurement state in the (D_{2}) is meaningless. Also, the above (=(D_{1}) + (D_{2})) is equivalent to the simultaneous measurement

not exist in the case that _{2}) is meaningless in general. Therefore, we have the following problem:

(E) Instead of the

In the following section, I answer this problem within the framework of the linguistic interpretation.

Consider two basic structure

or

Thus the Markov operator

Define the observable

Let

Fix a pure state

(F) the probability that a measured value

(In a similar way, the same result is easily obtained in the case of (7)).

Thus, we see the following.

(G_{1}) if

(G_{2}) in case that a measured value

where it should be recalled that

Considering the correspondence: (D) Û (G), that is,

namely,

there is a reason to assume that the true meaning of the (D) is just the (G). Also, note the taboo phrase “post- measurement state” is not used in (G_{2}) but in (D_{2}). Hence, we obtain the answer of Problem 1 (i.e.,

Remark 1. So called Copenhagen interpretation may admit the post-measurement state (cf. [

this idea would not generally be approved. That is because, if the post-measurement state is admitted, a series of problems occur, that is, “When is a measurement taken?”, or “When does the wave function collapse happen?”, which is beyond Axioms 1 and 2. Readers should remember Wittgenstein’s famous word: “The limits of my language mean the limits of my world”, or “What we cannot speak about we must pass over in silence”.

As mentioned in Section 1.3 (C), the wave function collapse (or more generally, the post-measurement state) is prohibited in the linguistic interpretation. Hence, some asked me “How about the projection postulate?”. In this paper I answer this question as follows:

(H) The von Neumann-Lüders projection postulate (D_{2}) concerning the measurement _{2}) is wrong). However, in the linguistic interpretation (i.e., without the phrase: “post-measure- ment state”), the similar result (G_{2}) concerning

As mentioned in Remark 1, the projection postulate (i.e., wave function collapse) is not completely established in so called Copenhagen interpretation, and thus, it is usually regarded as “postulate”. However, in the linguistic interpretation, the projection postulate is completely clarified, and hence, it should be regarded as a theorem. I hope that confusion on the wave function collapse will be calming.

ShiroIshikawa, (2015) Linguistic Interpretation of Quantum Mechanics; Projection Postulate. Journal of Quantum Information Science,05,150-155. doi: 10.4236/jqis.2015.54017