Theory of 'Weak Value' and Quantum Mechanical Measurements

Quantum mechanics provides us many perspectives and insights on Nature and our daily life. However, its mathematical axiom initiated by von Neumann [121] is not satisfied to describe nature phenomena. For example, it is impossible not to explain a non self-adjoint operator, i.e., the momentum operator on a half line (See, e.g., Ref. [154].), as the physical observable. On considering foundations of quantum mechanics, the simple and specific expression is needed. One of the candidates is the weak value initiated by Aharonov and his colleagues [4]. It is remarked that the idea of their seminal work is written in Ref. [3]. Furthermore, this quantity has a potentiality to explain the counter-factual phenomena, in which there is the contradiction under the classical logic, e.g., the Hardy paradox [64]. If so, it may be possible to quantitatively explain quantum mechanics in the particle picture. In this review based on the author thesis [152], we consider the theory of the weak value and construct a measurement model to extract the weak value. See the other reviews in Refs. [12,14,15,20].

Let the weak value for an observable A be defined as

where |i and |f are called a pre-and post-selected state, respectively. As the naming of the “weak value”, this quantity is experimentally accessible by the weak measurement as explained below. As seen in Fig. 1, the weak value can be measured as the shift of a meter of the probe after the weak interaction between the target and the probe with the specific post-selection of the target. Due to the weak interaction, the quantum state of the target is only slightly changed but the information of the desired observable A is encoded in the probe by the post-selection. While the previous studies of the weak value since the seminal paper [4], which will be reviewed in Sec. 3, are based on the measurement scheme, there are few works that the weak value is focused on and is independent of the measurement scheme. Furthermore, in these 20 years, we have not yet understood the mathematical properties of the weak value. In this chapter, we review the historical backgrounds of the weak value and the weak measurement and recent development on the measurement model to extract the weak value.

The time evolution for the quantum state and the operation for the measurement are called a quantum operation. In this section, we review a general description of the quantum operation. Therefore, the quantum operation can describe the time evolution for the quantum state, the control of the quantum state, the quantum measurement, and the noisy quantum system in the same formulation.

Within the mathematical postulates of quantum mechanics [121], the state change is subject to the Schrödinger equation. However, the state change on the measurement is not subject to this but is subject to another axiom, conventionally, von Neumann-Lüders projection postulate [105]. See more details on quantum measurement theory in the books [31,40,194].

Let us consider a state change from the initial state |ψ on the projective measurement 1 for the operator A = j a j |a j a j |. From the Born rule, the probability to obtain the measurement outcome, that is, the eigenvalue of the observable A, is given by

where ρ := |ψ ψ| and P am = |a m a m |. After the measurement with the measurement outcome a m , the quantum state change is given by

which is often called the “collapse of wavefunction” or “state reduction”. This implies that it is necessary to consider the non-unitary process even in the isolated system. To understand the measuring process as quantum dynamics, we need consider the general theory of quantum operations.

Let us recapitulate the general theory of quantum operations of a finite dimensional quantum system [122]. All physically realizable quantum operations can be generally described by a completely positive (CP) map [127,128], since the isolated system of a target system and an auxiliary system always undergoes the unitary evolution according to the axiom of quantum mechanics [121]. Physically speaking, the operation of the target system should be described as a positive map, that is, the map from the positive operator to the positive operator, since the density operator is positive. Furthermore, if any auxiliary system is coupled to the target one, the quantum dynamics in the compound system should be also described as the positive map since the compound system should be subject to quantum mechanics. Given the positive map, the positive map is called a CP map if and only if the positive map is also in the compound system coupled to any auxiliary system. One of the important aspects of the CP map is that all physically realizable quantum operations can be described only by operators defined in the target system. Furthermore, the auxiliary system can be environmental system, the probe system, and the controlled system. Regardless to the role of the auxiliary system, the CP map gives the same description

This content is AI-processed based on open access ArXiv data.