Density Matrix - Entropy

Entropy

The von Neumann entropy of a mixture can be expressed in terms of the eigenvalues of or in terms of the trace and logarithm of the density operator . Since is a positive semi-definite operator, it has a spectral decomposition such that where are orthonormal vectors. Therefore the entropy of a quantum system with density matrix is

Also it can be shown that

when have orthogonal support, where is the Shannon entropy. This entropy can increase but never decrease with a projective measurement, however generalised measurements can decrease entropy. The entropy of a pure state is zero, while that of a proper mixture always greater than zero. Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state. Thus the act of measurement induces a fundamental irreversible change on the density matrix; this is analogous to the "collapse" of the state vector, or wavefunction collapse. Perhaps counterintuitively, the measurement actually decreases information by erasing quantum interference in the composite system—cf. quantum entanglement, einselection, and quantum decoherence.

(A subsystem of a larger system can be turned from a mixed to a pure state, but only by increasing the von Neumann entropy elsewhere in the system. This is analogous to how the entropy of an object can be lowered by putting it in a refrigerator: The air outside the refrigerator's heat-exchanger warms up, gaining even more entropy than was lost by the object in the refrigerator. See second law of thermodynamics. See Entropy in thermodynamics and information theory.)