Measurement in quantum mechanics

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic.

The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about local hidden variables within quantum systems.

Measuring a quantum system generally changes the quantum state that describes that system. This is a central feature of quantum mechanics, one that is both mathematically intricate and conceptually subtle. The mathematical tools for making predictions about what measurement outcomes may occur, and how quantum states can change, 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.

On a more philosophical level, debates continue about the meaning of the measurement concept. The different interpretations of quantum mechanics, concern of solving what is known as the measurement problem.

Mathematical formalism

"Observables" as self-adjoint operators

In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system. The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". These observables play the role of measurable quantities familiar from classical physics: position, momentum, energy, angular momentum and so on. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. Many treatments of the theory focus on the finite-dimensional case, as the mathematics involved is somewhat less demanding. Indeed, introductory physics texts on quantum mechanics often gloss over mathematical technicalities that arise for continuous-valued observables and infinite-dimensional Hilbert spaces, such as the distinction between bounded and unbounded operators; questions of convergence (whether the limit of a sequence of Hilbert-space elements also belongs to the Hilbert space), exotic possibilities for sets of eigenvalues, like Cantor sets; and so forth. These issues can be satisfactorily resolved using spectral theory; The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it.

The Born rule associates a probability with each unit vector in the Hilbert space, in such a way that these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis. Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like a choice of basis for that vector to be embedded in. Gleason's theorem establishes the converse: all assignments of probabilities to unit vectors (or, equivalently, to the operators that project onto them) that satisfy these conditions take the form of applying the Born rule to some density operator.

Generalized measurement (POVM)

In functional analysis and quantum measurement theory, a positive-operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVMs) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see Schrödinger–HJW theorem); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory. They are extensively used in the field of quantum information.

In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices <math>\{F_i\} </math> on a Hilbert space <math> \mathcal{H} </math> that sum to the identity matrix,

:<math>\sum_{i=1}^n F_i = \operatorname{I}.</math>

In quantum mechanics, the POVM element <math>F_i</math> is associated with the measurement outcome <math>i</math>, such that the probability of obtaining it when making a measurement on the quantum state <math>\rho</math> is given by

:<math>\text{Prob}(i) = \operatorname{tr}(\rho F_i) </math>,

where <math>\operatorname{tr}</math> is the trace operator. When the quantum state being measured is a pure state <math>|\psi\rangle</math> this formula reduces to

:<math>\text{Prob}(i) = \operatorname{tr}(|\psi\rangle\langle\psi| F_i) = \langle\psi|F_i|\psi\rangle</math>.

State change due to measurement

A measurement upon a quantum system will generally bring about a change of the quantum state of that system. Writing a POVM does not provide the complete information necessary to describe this state-change process. originally introduced operators with two indices, <math>A_{ij}</math>, such that <math>\textstyle \sum_j A_{ij} A^\dagger_{ij} = E_i</math>. The extra index does not affect the computation of the measurement outcome probability, but it does play a role in the state-update rule, with the post-measurement state being now proportional to <math>\textstyle \sum_j A^\dagger_{ij} \rho A_{ij}</math>. This can be regarded as representing <math>\textstyle E_i</math> as a coarse-graining together of multiple outcomes of a more fine-grained POVM. Kraus operators with two indices also occur in generalized models of system-environment interaction. If the POVM is itself a PVM, then the Kraus operators can be taken to be the projectors onto the eigenspaces of the von Neumann observable:

:<math>\rho \to \rho' = \frac{\Pi_i \rho \Pi_i}{\operatorname{tr} (\rho \Pi_i)}.</math>

If the initial state <math>\rho</math> is pure, and the projectors <math>\Pi_i</math> have rank 1, they can be written as projectors onto the vectors <math>|\psi\rangle</math> and <math>|i\rangle</math>, respectively. The formula simplifies thus to

:<math>\rho = |\psi\rangle\langle\psi| \to \rho' = \frac{|i\rangle\langle i | \psi\rangle\langle\psi | i \rangle\langle i|}{|\langle i |\psi \rangle|^2} = |i\rangle\langle i|.</math>

Lüders rule has historically been known as the "reduction of the wave packet" or the "collapse of the wavefunction". The pure state <math>|i\rangle</math> implies a probability-one prediction for any von Neumann observable that has <math>|i\rangle</math> as an eigenvector. Introductory texts on quantum theory often express this by saying that if a quantum measurement is repeated in quick succession, the same outcome will occur both times. This is an oversimplification, since the physical implementation of a quantum measurement may involve a process like the absorption of a photon; after the measurement, the photon does not exist to be measured again.

The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. A pure state for a qubit can be written as a linear combination of two orthogonal basis states <math>|0 \rangle </math> and <math>|1 \rangle </math> with complex coefficients:

: <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle </math>

: <math>| \alpha |^2 + | \beta |^2 = 1.</math>

An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for <math>2 \times 2</math> self-adjoint matrices: After a measurement in the computational basis, the outcome of a <math>\sigma_x</math> or <math>\sigma_y</math> measurement is maximally uncertain.

A pair of qubits together form a system whose Hilbert space is 4-dimensional. One significant von Neumann measurement on this system is that defined by the Bell basis, This system is defined by the Hamiltonian

:<math>{H} = \frac