Mathematical Formulation of Quantum Mechanics - The Problem of Measurement

The Problem of Measurement

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement. The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain):

• Let A have spectral resolution

where E_A is the resolution of the identity (also called projection-valued measure) associated to A. Then the probability of the measurement outcome lying in an interval B of R is |E_A(B) ψ|2. In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure

• If the measured value is contained in B, then immediately after the measurement, the system will be in the (generally non-normalized) state E_A(B) ψ. If the measured value does not lie in B, replace B by its complement for the above state.

For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues λ_i, with corresponding eigenvectors ψ_i. The projection-valued measure associated with A, E_A, is then

where B is a Borel set containing only the single eigenvalue λ_i. If the system is prepared in state

Then the probability of a measurement returning the value λ_i can be calculated by integrating the spectral measure

over B_i. This gives trivially

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate.

A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

by a finite set of positive operators

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {λ₁ ... λ_n} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λ_i. Instead of collapsing to the (unnormalized) state

after the measurement, the system now will be in the state

Since the F_i F_i* 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general mixed states.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).