Density matrices and mixed preparations

If preparation j occurs with probability wj and produces a normalized ket, average the rank-one projectors weighted by those probabilities. Averaging kets directly is incorrect: independent preparation labels are classical alternatives, so no controlled relative phase links them. The resulting operator is positive because every vector has a nonnegative quadratic form against each projector. Its trace is one because the recipe weights sum to one. These properties characterize finite-dimensional density operators.

For an observable A, expectation is the trace of rho times A. For a projective outcome, replace A by its projector to obtain a Born probability. In infinite dimension, rho must be a positive trace-class unit-trace operator, and unbounded-observable moments need additional finiteness conditions. A matrix that is Hermitian with trace one but has a negative eigenvalue is not a valid density matrix: some possible measurement would receive a negative probability.

Off-diagonal matrix entries encode phase-sensitive predictions relative to the selected basis. A coherent superposition can have nonzero off-diagonal entries even when its diagonal probabilities match those of an incoherent mixture. Diagonalizing any density operator removes those entries in its eigenbasis, so having a diagonal matrix does not by itself mean a preparation lacks coherence relative to every physically relevant basis. State properties and basis-specific resources must be described separately.

Purity is the trace of rho squared. For a finite d-dimensional normalized state it lies between one divided by d and one. Equality to one means a rank-one projector, which describes a pure state ray; smaller purity means a mixed state. In a qubit, the Bloch-vector length provides the same test. Pure states lie on the unit sphere, mixed states inside it, and the maximally mixed state sits at its centre.

A maximally mixed qubit can be prepared by selecting the two computational states equally, or by selecting the plus and minus X eigenstates equally. Both recipes produce the same operator and therefore identical statistics for every measurement on the qubit alone. Their discarded classical labels are different records, but those records are not recoverable from rho without additional access to a preparation register. This is ensemble nonuniqueness rather than an uncertainty about which matrix entries to use.

The same mathematical density operator can also describe a subsystem entangled with another system. Locally, its measurement predictions match those of an appropriate classical preparation mixture, even though the global correlations differ. Calling every mixed state merely ignorance of a unique pre-existing pure state therefore adds information that the operator does not contain. The next lessons introduce the tensor-product description needed to distinguish a local state from its possible global extensions.