Von Neumann entropy

For a pure bipartite state, it is possible to show that the von Neumann entropy of its reduced density matrix, S p ) = —Tr(pjg log2 Pred)> above... 

This result for entanglement is equivalent to the von Neumann entropy of the reduced density matrix p. For our model system of the form AB in the ground state the reduced density matrix pj = TrB(pAB) he basis set (t, 1) is given by... 

As we mentioned before, when a biparticle quantum system AB is in a pure state, there is essentially a unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S = —Tr[p log2 PaI- This approach gives exactly the same formula as the one given in Eq. (26). This is not surprising since all entanglement measures should coincide on pure bipartite states and be equal to the von Neumann entropy of the reduced density matrix (uniqueness theorem). 

Many electron systems such as molecules and quantum dots show the complex phenomena of electron correlation caused by Coulomb interactions. These phenomena can be described to some extent by the Hubbard model [76]. This is a simple model that captures the main physics of the problem and admits an exact solution in some special cases [77]. To calculate the entanglement for electrons described by this model, we will use Zanardi s measure, which is given in Fock space as the von Neumann entropy [78]. 

Figure 5. Lx)cal entanglement given by the von Neumann entropy, Ey, versus V jt in the pure case.

Figure 7. Comparison between the absolute value of the electron correlation = Exact and the von Neumann entropy (S) as a function of the internuclear distance R for the H2 molecule using two Gaussian basis sets STO-3G and 3-21G.

What about an hypothetical experiment yielding a non-Poisson distribution of light-on and light-off states On the basis of the results of Section XV.E, we are led to believe that it would be impossible to make the rate of Gibbs entropy increase to coincide with the KS entropy. We are convinced that for the same reasons the results of the work of Ref. 139 could not be extended to this case, and it would be impossible, in conclusion, to establish a connection between the rate of von Neumann entropy increase and the KS of the experimental sequence of light-on and ligh-off states. 

Note that the entropy as defined in Eq. (46) is not necessarily related to the von Neumann entropy Tr( ) In D ) of the thermal state. The entropy in Eq. (46) estimates only the information contained in the particular decomposition p. 

The eigenvalues of the one- and two-orbital density matrices, w. and w .y, respectively, can then be employed to determine the von Neumann entropy of each spatial orbital. This yields the so-called single-orbital entropy [96],... 

Another perspective is offered by considering Type A nondynamical correlation / left-right strong correlation as a manifestation of quantum entanglement. A physically motivated measure of the latter is the von Neumann entropy [13], or the closely related correlation entropy [14, 15]... 

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