We generalize protective measurement for protective joint measurement of several observables. The merit of joint protective measurement is the determination of the eigenstates of an unknown Hamiltonian rather than the determination of features of an unknown quantum state. As an example, we precisely determine the two eigenstates of an unknown Hamiltonian by a single joint protective measurement of the three Pauli matrices on a qubit state. Introduction Protective measurement is one of the unexpected consequences of the strange structure of quantum mechanics. According to general wisdom, we cannot gain information on the unknown state ρ of a single quantum system unless we distort the state itself. In particular, we cannot learn the unknown state of a single system whatever test we apply to it. It came as a surprise that in weak measurements [1] the expectation value 〈  〉 of an observable  can be tested on a large ensemble of identically prepared unknown states in such a way that the distortions per single systems stay arbitrarily small (see [2], too). An indirectly related surprise came with the so-called protective measurements [3, 4, 5] capable of testing 〈  〉 at least in an unknown eigenstate of the Hamiltonian Ĥ at arbitrarily small distortion of the state itself. Interesting debates followed the proposal as to the merit of protective measurement in the interpretation of the wave function of a single system instead of a statistical ensemble (see, e.g., [6] and references therein). My work investigates an alternative merit of protective measurement. First I construct joint protective measurements of several observables  1 ,  2 , … and re-state the original equations for them in a general form. Then I show that the straightforward task that a single joint protective measurement solves on a single system is the determination of the eigenstates of an otherwise unknown Hamiltonian.
We generalize protective measurement for protective joint measurement of several observables. The merit of joint protective measurement is the determination of the eigenstates of an unknown Hamiltonian rather than the determination of features of an unknown quantum state. As an example, we precisely determine the two eigenstates of an unknown Hamiltonian by a single joint protective measurement of the three Pauli matrices on a qubit state. Introduction Protective measurement is one of the unexpected consequences of the strange structure of quantum mechanics. According to general wisdom, we cannot gain information on the unknown state ρ of a single quantum system unless we distort the state itself. In particular, we cannot learn the unknown state of a single system whatever test we apply to it. It came as a surprise that in weak measurements [1] the expectation value 〈  〉 of an observable  can be tested on a large ensemble of identically prepared unknown states in such a way that the distortions per single systems stay arbitrarily small (see [2], too). An indirectly related surprise came with the so-called protective measurements [3, 4, 5] capable of testing 〈  〉 at least in an unknown eigenstate of the Hamiltonian Ĥ at arbitrarily small distortion of the state itself. Interesting debates followed the proposal as to the merit of protective measurement in the interpretation of the wave function of a single system instead of a statistical ensemble (see, e.g., [6] and references therein). My work investigates an alternative merit of protective measurement. First I construct joint protective measurements of several observables  1 ,  2 , … and re-state the original equations for them in a general form. Then I show that the straightforward task that a single joint protective measurement solves on a single system is the determination of the eigenstates of an otherwise unknown Hamiltonian.
Protective measurement, in the language of standard quantum mechanics, is a method to measure the expectation value of an observable on a single quantum system (Aharonov and Vaidman, 1993; Aharonov, Anandan and Vaidman, 1993). For a conventional impulsive measurement, the state of the measured system is strongly entangled with the state of the measuring device during the measurement, and the measurement result is one of the eigenvalues of the measured observable. By contrast, during a protective measurement, the measured state is protected by an appropriate procedure so that it neither changes nor becomes entangled with the state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables on a single quantum system, and in particular, the wave function of the system can also be measured as expectation values of certain observables. It is expected that protective measurements can be performed in the near future with the rapid development of weak measurement technologies (e.g. Kocsis et al., 2011; Lundeen et al., 2011). In this chapter, we will give a clear introduction to protective measurement in quantum mechanics. Standard quantum mechanics and impulsive measurement The standard formulation of quantum mechanics, which was first developed by Dirac (1930) and von Neumann (1955), is based on the following basic principles. 1 Physical states The state of a physical system is represented by a normalized wave function or unit vector | ψ ( t )〉 in a Hilbert space. The Hilbert space is complete in the sense that every possible physical state can be represented by a state vector in the space. 2 Physical properties Every measurable property or observable of a physical system is represented by a Hermitian operator on the Hilbert space associated with the system. A physical system has a determinate value for an observable if and only if it is in an eigenstate of the observable (this is often called the eigenvalue–eigenstate link).
Protective measurement is a new measuring method introduced by Aharonov, Anandan and Vaidman (1993). By a protective measurement, one can measure the expectation value of an observable on a single quantum system, even if the system is initially not in an eigenstate of the measured observable. This remarkable feature of protective measurements was challenged by Uffink (1999, 2012). He argued that only observables that commute with the system's Hamiltonian can be protectively measured, and a protective measurement of an observable that does not commute with the system's Hamiltonian does not actually measure the observable, but measure another related observable that commutes with the system's Hamiltonian. In this paper, we show that there are several errors in Uffink's arguments, and his alternative interpretation of protective measurements is untenable.
Based on an analysis of protective measurements, we show that the quantum state represents the physical state of a single quantum system. This result is more definite than the PBR theorem [Pusey, Barrett, and Rudolph, Nature Phys. 8, 475 (2012)]. The physical meaning of the quantum state is an important interpreta-tive problem of quantum mechanics. A long-standing question is whether a pure state relates only to an ensemble of identically prepared systems or di-rectly to the state of a single system. Recently, Pusey, Barrett and Rudolph (PBR) demonstrated that under an independence assumption, the quantum state is a representation of the physical state of a single quantum system [1]. This poses a further interesting question, namely whether ψ-ontology can be argued without resorting to nontrivial assumptions such as the in-dependence assumption (cf. Ref. [2-4]). In this Letter, we will show that protective measurements [5,6] already provide such an argument. The meaning of the quantum state is usually analyzed in the context of conventional impulsive measurements of an ensemble of identically prepared systems. However, it has been known that the quantum state of a single prepared system can be protectively measured [5-9]. During a protective measurement, the measured system is protected by an appropriate procedure (e.g. via the quantum Zeno effect) so that its quantum state neither changes nor becomes entangled with the quantum state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables from a single quantum system, even if the system is initially not in an eigenstate of the measured observable, and in particular, the quantum state of the system can also be measured as expectation values of certain observables. It is expected that protective measurements will be realized in the near future with the rapid development of quantum technologies.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.
Making measurements on single quantum systems is considered difficult, almost impossible if the state is a-priori unknown. Protective measurements suggest a possibility to measure single quantum systems and gain some new information in the process. Protective measurement is described, both in the original and generalized form. The degree to which the system and the apparatus remain entangled in a protective measurement, is assessed. A possible experimental test of protective measurements is discussed.
Quantum theory has traditionally – and not altogether unreasonably – been taken as a challenge to “realism” about physical theories. At the very least, the ways in which it is often formulated, presented and used suggest a non-realist understanding of the theory because the significance of the “state” of a system is described in terms of a catalogue of predictions. While protective measurement does not force one to give up on this standard view, it does support an alternative contention, namely, that the state of a physical system (a wave function, for example) has a somewhat more direct physical significance. I conclude with what I take to be the upshot of these observations for approaches to interpreting quantum theory and evaluating those interpretations. Introduction In 1993 I wrote an article (Dickson, 1995) about the scheme for protective measurement first described (to my knowledge) by Aharonov, Anandan and Vaidman (1993). There I claimed that protective measurement makes “realism” about quantum theory more attractive than it might otherwise have been. I still believe some form of that claim to be true, and I am grateful to the editor and to Cambridge University Press for the opportunity to say so in the manner that I'm inclined to now, 20 years later. Of course, the debates about the interpretation of quantum theory have moved on quite a bit in the intervening time. We've seen a rise to prominence of “subjectivist” interpretations, largely based on developments in quantum information theory (but with its own historical precedents, as Timpson (2010) has pointed out – see Fuchs (2002) for a landmark paper in the modern development, and Timpson (2013) for a thorough overview and evaluation). The various “many-somethings” (worlds, minds, perspectives, whatever) interpretations have, arguably, seen a rise in fortune as well.