maximal verschränkter Zustand

Entropy

Quelle: https://en.wikipedia.org/w/index.php?title=Quantum_entanglement&action=edit&section=14

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

Definition

In classical information theory H, the Shannon entropy, is associated to a probability distribution, $p_1,\cdots ,p_n$, in the following way:^[92]

$H(p_1,\cdots ,p_n)=-\sum _i{p}_i{\log }_2{p}_i.$

Since a mixed state ρ is a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy:^[55]: 264

$S(\rho )=-\text{Tr}\left(\rho {\log }_2\rho \right),$

which can be expressed in terms of the eigenvalues of ρ:

$S(\rho )=-\text{Tr}\left(\rho {\log }_2\rho \right)=-\sum _i{\lambda }_i{\log }_2{\lambda }_i$.

Since an event of probability 0 should not contribute to the entropy, and given that

$\underset{p\to 0}{lim}p\log p=0,$

the convention 0 log(0) = 0 is adopted. When a pair of particles is described by the spin singlet state discussed above, the von Neumann entropy of either particle is log(2), which can be shown to be the maximum entropy for 2 × 2 mixed states.^[52]: 15

As a measure of entanglement

Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.^[93][94] If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.^[95] It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n, ..., 1/n}.^[46]: 505 Therefore, a bipartite pure state ρ ∈ H_A ⊗ H_B is said to be a maximally entangled state if the reduced state of each subsystem of ρ is the diagonal matrix^[96]

$\left[\begin{array}{ccc}\frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{array}\right].$

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.^[48]: 471 Rényi entropy can be used as a measure of entanglement.^{[48]: 447, 480 [97]}

Rényi entropy also can be used as a measure of entanglement.