In our macroscopic world, we know that determining your height will not make you taller nor smaller and merely staring at an object will not change it either. The moon is up there and would stay the same whether you look at it or not. But these are acts of measurement and according to quantum measurement theory, the state of a system changes when it is measured. Here we will construct a measurement according to the postulates of quantum mechanics that does not change the state of a system.
In our macroscopic world, we know that determining your height will not make you taller nor smaller and merely staring at an object will not change it either. The moon is up there and would stay the same whether you look at it or not. But these are acts of measurement and according to quantum measurement theory, the state of a system changes when it is measured. Here we will construct a measurement according to the postulates of quantum mechanics that does not change the state of a system.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.
According to orthodox quantum mechanics, state vectors change in two incompatible ways: “deterministically” in accordance with Schr0̈dinger's time-dependent equation, and probabilistically if and only if a measurement is made. It is argued here that the problem of measurement arises because the precise, mutually exclusive conditions for these two types of transitions to occur are not specified within orthodox quantum mechanics. Fundamentally, this is due to an inevitable ambiguity in the notion of “measurement” itself. Hence, if the problem of measurement is to be resolved, a new, fully objective version of quantum mechanics needs to be developed which does not incorporate the notion of measurement in its basic postulates at all.
This chapter is a refresher on the fundamental concepts of quantum mechanics. One essential difference between classical and quantum mechanics is that in classical mechanics, we are used to deterministic and precise laws stemming from the fact that the act of measurement can be ignored. Indeed, for all practical purposes, the interaction with the macroscopic system to be measured can be rendered negligibly small. However, at very small scales, the situation changes radically. The assumption that the measurement is ignorable is not true for micro-objects. In order to perform a measurement, we must interact with the system we are measuring. We discovered experimentally that nature has an objective scale, encoded by Planck's constant, that limits our ability to perform perturbation-free measurements. All we can do is perform many measurements on identically prepared systems, which would allow us to determine the probability of a given outcome. Micro-objects behave non-intuitively, described by new laws, those of quantum mechanics. In this chapter, we will briefly cover the postulates of quantum mechanics and introduce the Schrodinger and Heisenberg pictures of quantum mechanics.
Quantum mechanics has predicted many counterintuitive phenomena in daily life, and has changed our view of the world. Among such predictions, the existence of a macroscopic object in superposition is especially unbelievable. As Einstein asked, "Do you really believe that the moon exists only when you look at it?". However, recent experimental results on a mesoscopic scale will ultimately require us to dismiss commonsense so-called macroscopic reality. Leggett and Garg applied the Bell scheme for testing local realism to the time evolution of a macroscopic two-state system, and proposed a temporal version of the Bell inequality (the Leggett-Garg (LG) inequality) for testing macroscopic realism. However, as with the Bell inequality, the statistical approach behind this scheme may be less effective in showing clear incompatibility. Here we propose a temporal version of the Greenberger-Horne-Zeilinger (GHZ) scheme without statistical treatment for testing Einstein's moon using three-time correlations.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.