It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical information theory.
For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is[1]
[2] The von Neumann entropy and quantities based upon it are widely used in the study of quantum entanglement.
Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom.
[7] The von Neumann entropy quantifies the extent to which a state is mixed.
An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for
lies on the surface of the unit ball, and it attains its maximum value when
If ρA, ρB are the reduced density matrices of the general state ρAB, then
By symmetry, for any tripartite state ρABC, each of the three numbers S(ρAB), S(ρBC), S(ρAC) is less than or equal to the sum of the other two.
[23] In the simplest case, a system with a finite-dimensional Hilbert space and measurement with a finite number of outcomes, a POVM is a set of positive semi-definite matrices
[25] If ρi are density operators and λi is a collection of positive numbers which sum to unity (
Equality is attained when the supports of the ρi – the spaces spanned by their eigenvectors corresponding to nonzero eigenvalues – are orthogonal.
[26] The time evolution of an isolated system is described by a unitary operator:
Writing a POVM does not provide the complete information necessary to describe this state-change process.
, named for Karl Kraus, provide a specification of the state-change process.
We can define a linear, trace-preserving, completely positive map, by summing over all the possible post-measurement states of a POVM without the normalisation:
[34] A quantum channel will increase or leave constant the von Neumann entropy of every input state if and only if the channel is unital, i.e., if it leaves fixed the maximally mixed state.
[35] The quantum version of the canonical distribution, the Gibbs states, are found by maximizing the von Neumann entropy under the constraint that the expected value of the Hamiltonian is fixed.
A Gibbs state is a density operator with the same eigenvectors as the Hamiltonian, and its eigenvalues are
However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life.
[51] It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n, ..., 1/n}.
For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.
[56] The squashed entanglement is based on the idea of extending a bipartite state
is the quantum version of the Hartley entropy, i.e., the logarithm of the rank of the density matrix.
[58] The density matrix was introduced, with different motivations, by von Neumann and by Lev Landau.
The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.
[60] He introduced the expression now known as von Neumann entropy by arguing that a probabilistic combination of pure states is analogous to a mixture of ideal gases.
[63] His argument was built upon earlier work by Albert Einstein and Leo Szilard.
[64][65][66] Max Delbrück and Gert Molière proved the concavity and subadditivity properties of the von Neumann entropy in 1936.
[67][68] The subadditivity and triangle inequalities were proved in 1970 by Huzihiro Araki and Elliott H.