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How is the entropy of a von Neumann system calculated?
The von Neumann entropy is then given by Since, for a pure state, the density matrix is idempotent, ρ = ρ2, the entropy S ( ρ) for it vanishes. Thus, if the system is finite (finite-dimensional matrix representation), the entropy S ( ρ) quantifies the departure of the system from a pure state.
Why is the vanishing entropy of a pure state important?
In other words, it codifies the degree of mixing of the state describing a given finite system. Measurement decoheres a quantum system into something noninterfering and ostensibly classical; so, e.g., the vanishing entropy of a pure state
How is entropy cancelled by an equal amount?
The left-hand inequality can be roughly interpreted as saying that entropy can only be cancelled by an equal amount of entropy. If system A and system B have different amounts of entropy, the smaller can only partially cancel the greater, and some entropy must be left over.
When is the entropy of a composite system maximized?
Likewise, the right-hand inequality can be interpreted as saying that the entropy of a composite system is maximized when its components are uncorrelated, in which case the total entropy is just a sum of the sub-entropies. This may be more intuitive in the phase space formulation]
Why was the density matrix introduced by von Neumann?
Von Neumann introduced the density matrix in the context of states and operators in a Hilbert space. The knowledge of the statistical density matrix operator would allow us to compute all average quantities in a conceptually similar, but mathematically different way.
Can a composite system be lower than its entropy?
While in Shannon’s theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case, i.e., it is possible that S(ρAB) = 0, while S(ρA) = S(ρB) > 0 .
How did John von Neumann contribute to quantum mechanics?
John von Neumann established a rigorous mathematical framework for quantum mechanics in his 1932 work Mathematical Foundations of Quantum Mechanics. In it, he provided a theory of measurement, where the usual notion of wave-function collapse is described as an irreversible process (the so-called von Neumann or projective measurement).