3. The main kinds of interpretation for quantum measurement

There are more than 10 kinds of interpretations for quantum measurement in quantum mechanics, such as Copenhagen interpretation, quantum logic, many worlds interpretation, stochastic interpretation, many-minds interpretation, etc. In this chapter, we just choose four of them to expound. According this section, we will know how difficult for physicists to solve these problems in one theory.

#### 3.1. The Copenhagen interpretation

The Copenhagen interpretation was formed in 1925 to 1927 by Niels Bohr and Werner Heisenberg. In fact, it is still the most commonly taught interpretations of quantum mechanics today.

According to the Copenhagen interpretation, the physical law that microscopic objects obey are different from that the macroscopic objects obey. Microscopic objects can be in superposition states, but the macroscopic objects are forbidden. According to the Copenhagen interpretation, the statuses of macroscopic objects are definite. We can say a macroscopic object is in this status or not, but cannot say this macroscopic object is both in this status and not. Now that the laws in microscopic world and macroscopic world are different, then the Copenhagen interpretation assumes the existence of macroscopic measurement apparatuses that obey classical physics to make measurement for microscopic objects that obey quantum mechanics.

However, this assumption does not solve the problems of quantum measurement. It throws all the problems to the macroscopic apparatuses, but it even cannot answer how to distinguish the macroscopic object that obeys the classical laws and microscopic ensemble that obeys the quantum mechanics. Moreover it also cannot answer how the nonlocality produces in quantum measurement process because, there is no seed for nonlocality growing no matter in classical physics or quantum mechanics.

#### 3.2. Many worlds interpretation

function of single particle. We still take the experiment of two-slit interference of electrons, for example. If the detector behind the slits has detected the signal and we can distinguish which slit the electrons pass, then the interference phenomenon will disappear. In language of quantum mechanics, the diffused wave function ψð Þ x; t of the electron will collapse into δð Þ x0; t immediately after this measurement. This process is very fast and does not seem to need to cost time. How this process happens and whether this process violates the law of

The third problem is the basis-preferred problem. The basis-preferred problem refers to a quantum system that is measured which prefers to collapse to a set of eigenstates. For example, a spin system with an initial state j i ψ ⟩ ¼ aj i↑ ⟩ þ bj i↓ ⟩ can collapse into the state of the set f g j i ↑⟩ ; j i ↓⟩ ,

a certain measurement, this state prefers one of these sets. Why the state prefers some basis set

Without any exaggeration, quantum measurement is one the most interesting and fascinating topics in quantum theory. There are too many unsolved mysteries in quantum measurement, and these spur us to further understand the quantum measurement and find the answers.

There are more than 10 kinds of interpretations for quantum measurement in quantum mechanics, such as Copenhagen interpretation, quantum logic, many worlds interpretation, stochastic interpretation, many-minds interpretation, etc. In this chapter, we just choose four of them to expound. According this section, we will know how difficult for physicists to solve

The Copenhagen interpretation was formed in 1925 to 1927 by Niels Bohr and Werner Heisenberg. In fact, it is still the most commonly taught interpretations of quantum mechanics today. According to the Copenhagen interpretation, the physical law that microscopic objects obey are different from that the macroscopic objects obey. Microscopic objects can be in superposition states, but the macroscopic objects are forbidden. According to the Copenhagen interpretation, the statuses of macroscopic objects are definite. We can say a macroscopic object is in this status or not, but cannot say this macroscopic object is both in this status and not. Now that the laws in microscopic world and macroscopic world are different, then the Copenhagen interpretation assumes the existence of macroscopic measurement apparatuses that obey classical physics to make measurement for microscopic objects that obey quantum mechanics.

However, this assumption does not solve the problems of quantum measurement. It throws all the problems to the macroscopic apparatuses, but it even cannot answer how to distinguish the macroscopic object that obeys the classical laws and microscopic ensemble that obeys the

3. The main kinds of interpretation for quantum measurement

2

<sup>p</sup> ð Þ j i<sup>↑</sup> ⟩ <sup>þ</sup> j i<sup>↓</sup> ⟩ ; <sup>1</sup><sup>=</sup> ffiffiffi

<sup>2</sup> � <sup>p</sup> ð Þg j i<sup>↑</sup> ⟩ � j i<sup>↓</sup> ⟩ , but under

causation of relativity theory are still unclear for us.

138 Advanced Technologies of Quantum Key Distribution

and it can also collapse into the state of the set 1= ffiffiffi

these problems in one theory.

3.1. The Copenhagen interpretation

under quantum measurement? Does it have awareness?

Many worlds interpretation was proposed by Hugh Everett in 1952. It supposes that there are a large, perhaps infinite, number of universes and every alternate state is in one of these universes [2, 3]. Many worlds interpretation denies the wave function collapse under quantum measurement. It asserts that the object that will be measured and the observer that will do the measurement are in a relative state. Each measurement will be a branch point and makes observer enter a universe. According to the thought of many worlds interpretation, the Schrödinger cat is alive in a universe and dead in the other universe. After the measurement, the observer will enter one of these two universes.

The advantage of this interpretation is that the discussion of collapse mechanism is avoided. However, the basis-preferred problem is still the big issue in many worlds interpretation although the quantum decoherence had been introduced into in the period of "post-Everett". Some researchers still think the many worlds interpretation of quantum theory exists only to the extent that the associated basis problem is solved [4–6]. Using the decoherence to define the Everett branches will lead to an approximate specification of a preferred basis and contradicts with the "exact" definition of the Everett branches.

#### 3.3. Many-minds interpretation

Many-minds interpretation is the extension of many worlds interpretation. It was proposed by Heinz-Dieter Zeh in 1970 to solve the "branch determining problem" and the puzzling concept of observers being in a superposition with themselves in many worlds interpretation [7–9]. The thought of this interpretation is when an observer measures a quantum system, then a state that is consistent with minds which produced by the observer brain, called mental states, will entangle with this quantum system. The mental state of the brain corresponding with this system is involving, and ultimately, only one mind is experienced, leading the others to branch off and become inaccessible. In this way, every sentient being is attributed with an infinity of minds, whose prevalence corresponds to the amplitude of the wave function. As an observer checks a measurement, the probability of realizing a specific measurement directly correlates to the number of minds they have where they see that measurement.

However, like the many worlds interpretation, the many-minds interpretation is still a local theory. Although the correlations of individual minds and objects could be the violation of Bell's inequality, the interactions between them that only take place are local, and only the separated events that are space-like separated could influence the minds of observers. Additionally, it tosses the basis-preferred problem to the mentality of observer and makes this physical problem fall into an endless discussion of mental state of human.

### 3.4. Dynamical reduction models

The theory of dynamical reduction models is a nonlinear and stochastic modification of the Schrödinger equation. It is proposed by Bassia and Ghirardia [10]. They integrated the master equation and linear Schrödinger equation and proposed a new nonlinear differential equation. This theory successfully solves the problems of "stochastic output" and "preferred basis" in quantum measurement and deduced the Born probability rule basing on the white noise model. However, it is still a nonrelativistic theory and remains the nonlocality problem.

K rð Þ¼ ;r0; t; t<sup>0</sup> C

S tð Þ¼ <sup>0</sup>; t<sup>1</sup>

K rð Þ ;r0; t; t<sup>0</sup> in Eq. (1) is the propagator and defined into

into a more general formulation in the following:

a. The formulation should be simple and concise.

W pð Þ chosen in extended Feynman path integral are

The H<sup>0</sup> in Eq. (5) is the main Hamiltonian:

<sup>R</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>iπℏc<sup>2</sup> <sup>p</sup>

F rð Þ¼ ;r0; t; t<sup>0</sup> R

b. It should obey the combination rule because the propagator is linear.

c. It is consisted by the four-dimension scalars, vectors and tensors.

form

this be?

condition.

and

X

where the coefficient C is a constant independent of paths and S is the action with classical

ðt1 t0

K rð Þ¼ ;r0; t; t<sup>0</sup> ⟨ r U t b ð Þ ; t<sup>0</sup>

Eq. (1) reveals an important assumption in Feynman path integral: the weights of different paths for propagator are the same. This assumption makes Feynman path integral very successful in nonrelativistic quantum theory, but it is also the top offender that impedes the integration between Feynman path integral and relativity in non-field theory. Why should

For the extension, it is necessary to break up this assumption, and Eq. (1) should be written

where R is the parameter that is independent of paths and Wð Þ ℘ is the weight function with paths [13]. Additionally, some rules should be set to limit the range of choices for R and Wð Þ ℘ :

d. It should be transformed into Feynman path integral in low-energy and low-velocity

Under these four limitations, the forms of R and W pð Þ are very few. The final forms of R and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m<sup>2</sup>c<sup>4</sup> þ ð Þ p � A<sup>0</sup>

Pð Þ ℘

2 c2

H0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mc<sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>0</sup> <sup>p</sup> ; Wð Þ¼ <sup>℘</sup>

q

H<sup>0</sup> ¼

X

� � � � � �r0⟩

allpathsexp ð Þ iS=<sup>ℏ</sup> (1)

http://dx.doi.org/10.5772/intechopen.74253

141

Stochastic Quantum Potential Noise and Quantum Measurement

Lð�r tð Þ;r tð ÞÞdt (2)

D E (3)

allpathsWð Þ <sup>℘</sup> exp ð Þ iS=<sup>ℏ</sup> (4)

<sup>P</sup>ð Þ <sup>℘</sup> ð Þ Δτ �1=<sup>2</sup> (5)

(6)
