"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 \{F_i\} on a Hilbert space \mathcal{H} that sum to the
identity matrix, :\sum_{i=1}^n F_i = \operatorname{I}. In quantum mechanics, the POVM element F_i is associated with the measurement outcome i, such that the probability of obtaining it when making a measurement on the
quantum state \rho is given by :\text{Prob}(i) = \operatorname{tr}(\rho F_i) , where \operatorname{tr} is the
trace operator. When the quantum state being measured is a pure state |\psi\rangle this formula reduces to :\text{Prob}(i) = \operatorname{tr}(|\psi\rangle\langle\psi| F_i) = \langle\psi|F_i|\psi\rangle.
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, A_{ij}, such that \textstyle \sum_j A_{ij} A^\dagger_{ij} = E_i. 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 \textstyle \sum_j A^\dagger_{ij} \rho A_{ij}. This can be regarded as representing \textstyle E_i 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: :\rho \to \rho' = \frac{\Pi_i \rho \Pi_i}{\operatorname{tr} (\rho \Pi_i)}. If the initial state \rho is pure, and the projectors \Pi_i have rank 1, they can be written as projectors onto the vectors |\psi\rangle and |i\rangle, respectively. The formula simplifies thus to :\rho = |\psi\rangle\langle\psi| \to \rho' = \frac{|\langle i |\psi \rangle|^2} = |i\rangle\langle i|. Lüders rule has historically been known as the "reduction of the wave packet" or the "
collapse of the wavefunction". The pure state |i\rangle implies a probability-one prediction for any von Neumann observable that has |i\rangle 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. on the states |\psi\rangle=|0\rangle and |\varphi\rangle=(|0\rangle+|1\rangle)/\sqrt2. Note that on the Bloch sphere orthogonal states are antiparallel. 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 |0 \rangle and |1 \rangle with complex coefficients: : | \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle A measurement in the (|0\rangle, |1\rangle) basis will yield outcome |0 \rangle with probability | \alpha |^2 and outcome |1 \rangle with probability | \beta |^2, so by normalization, : | \alpha |^2 + | \beta |^2 = 1. An arbitrary state for a qubit can be written as a linear combination of the
Pauli matrices, which provide a basis for 2 \times 2 self-adjoint matrices: After a measurement in the computational basis, the outcome of a \sigma_x or \sigma_y 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 :{H} = \frac{{p}^2}{2m} + \frac{1}{2}m\omega^2 {x}^2, where {H}, the
momentum operator {p} and the
position operator {x} are self-adjoint operators on the Hilbert space of square-integrable functions on the
real line. The energy eigenstates solve the time-independent
Schrödinger equation: :{H} |n\rangle = E_n |n\rangle. These eigenvalues can be shown to be given by :E_n = \hbar\omega\left(n + \tfrac{1}{2}\right), and these values give the possible numerical outcomes of an energy measurement upon the oscillator. The set of possible outcomes of a
position measurement on a harmonic oscillator is continuous, and so predictions are stated in terms of a
probability density function P(x) that gives the probability of the measurement outcome lying in the infinitesimal interval from x to x + dx. ==History of the measurement concept==