Measurement in quantum mechanics

In quantum physics, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis. Quantum physics has proven to be an empirical success and to have wide-ranging applicability. However, on a more philosophical level, debates continue about the meaning of the measurement concept.

Quotes

• What we have learnt from this chapter is that we cannot have a direct evidence of, i.e. directly measure, a quantum state of a single system. Our experience is only connected with the experimental values of observables, and any time we measure an observable we can only have a partial experience of a system under a certain perspective but we can never have a complete experience that would be represented by an observation of the state vector, which is – in a quantum-mechanical sense – a complete description of the system. In other words, the quantum state is not an observable in the classical sense. However, since this feature of the quantum state is not due to subjective ignorance but rather to an intrinsic characteristic of the microscopic world, there are no definitive reasons to deny the reality of a quantum state.
  • Gennaro Auletta, Mauro Fortunato and Giorgio Parisi, Quantum Mechanics (2009)

• By now the reader will have realized that measurement in quantum physics is fundamentally different from that in classical physics. In classical physics, a measurement reveals a pre-existing property of the physical system that is tested. If a car is driving at 180 km h⁻¹ on the highway, the measurement of its speed by radar determines a property that exists prior to the measurement, which gives the police the legitimacy to give a ticket to the driver. On the contrary, the measurement of the S_x component of a spin-1/2 particle in the state |+〉 does not reveal a value of S_x existing before the measurement. The spread in the results of measuring S_x in this case is sometimes attributed to “uncontrollable perturbation of the spin due to the measurement process,” but the value of S_x does not exist before the measurement, and that which does not exist cannot be perturbed.
  • Michel Le Bellac, Quantum physics (2006)

• When we measure a real dynamical variable ξ, the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second. Hence, after the first measurement has been made, the system is in an eigenstate of the dynamical variable ξ, the eigenvalue it belongs to being equal to the result of the first measurement. This conclusion must still hold if the second measurement is not actually made. In this way we see that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement. We can infer that, with the dynamical system in any state, any result of a measurement of a real dynamical variable is one of its eigenvalues. Conversely, every eigenvalue is a possible result of a measurement of the dynamical variable for some state of the system, since it is certainly the result if the state is an eigenstate belonging to this eigenvalue. This gives us the physical significance of eigenvalues. The set of eigenvalues of a real dynamical variable are just the possible results of measurements of that dynamical variable and the calculation of eigenvalues is for this reason an important problem.
  • P. A. M. Dirac, The Principles of Quantum Mechanics (4th ed., 1958)

• Hence coherence survives to the extent to which an experiment fails to distinguish between different eigenvalues of the observable being measured, and distinct final states of the object display interference if the final states of the apparatus overlap. […] When the apparatus ends up in one member of a set of orthogonal states, the experiment unambiguously determines whether or not the object is then in a particular eigenstate of the observable of interest, but the result becomes increasingly ambiguous with increasing overlap between the states of the apparatus; furthermore, states of the object with distinct eigenvalues interfere with a visibility that is a measure of the ambiguity with which the states assumed by the apparatus determine the states of the object.
  • Kurt Gottfried and Tung-Mow Yan, Quantum Mechanics: Fundamentals (2nd ed., 2003)

• We see that the measuring process in quantum mechanics has a "two- faced" character: it plays different parts with respect to the past and future of the electron. With respect to the past, it "verifies" the probabilities of the various possible results predicted from the state brought about by the previous measurement. With respect to the future, it brings about a new state. Thus the very nature of the process of measurement involves a far-reaching principle of irreversibility.
  • Lev Landau and Evgeny Lifshitz, Quantum Mechanics: Non-relativistic theory (3rd ed., 1977)

• As we apply the results obtained in this section, we should remember that common terms like "measurement" and "information" are being used here with a specific technical meaning. In particular, this is not the place for a detailed analysis of real experimental measurements and their relation to the theoretical framework. We merely note that, in the information theoretic view of quantum mechanics, the probabilities and the related density operators and entropies, which are employed to assess the properties of quantum states and the outcomes of measurement, provide a coherent and consistent basis for understanding and interpreting the theory.
  • Eugen Merzbacher, Quantum Mechanics (1998)

• We now consider measurements of A and B when they are compatible observables. Suppose we measure A first and obtain result a'. Subsequently, we may measure B and get result b'. Finally we measure A again. It follows from our measurement formalism that the third measurement always gives a' with certainty; that is, the second (B) measurement does not destroy the previous information obtained in the first (A) measurement. This is rather obvious when the eigenvalues of A are nondegenerate:

  $|\alpha \rangle {\xrightarrow {\text{A measurement}}}|a',b'\rangle {\xrightarrow {\text{B measurement}}}|a',b'\rangle {\xrightarrow {\text{A measurement}}}|a',b'\rangle$.

  • J. J. Sakurai and Jim J. Napolitano, Modern Quantum Mechanics (2nd ed., 2011)

• This strange fact—that the system evolves one way between measurements and another way during a measurement—has been a source of contention and confusion for decades. It raises a question: Shouldn’t the act of measurement itself be described by the laws of quantum mechanics? The answer is yes. The laws of quantum mechanics are not suspended during measurement. However, to examine the measurement process itself as a quantum mechanical evolution, we must consider the entire experimental setup, including the apparatus, as part of a single quantum system.
  • Leonard Susskind and Art Friedman, Quantum Mechanics: The Theoretical Minimum (2014)