It can be show that distinguish an entangled or non-entangled density matrix is a NP-hard problem known as Quantum Separability Problem. This entropy measures the quantum correlations between the two subsystems and are always different from zero if the state you are considering is somehow interacting with something else (another subsystem or an environment).įrom the classical and the maximally entangled scenario you can see that there are not differences between the two density matrices and so, given ad arbitary density matrix you do not have a criterion to distinguish the two cases and the Von Neumann entropy will result the same. In this case the Von Neumann entropy will be $S=-Tr(\rho\log\rho)=\log2$. In that case your density matrix will be $$\rho=\frac~.$$ As a result, different von Neumann entropies can be associated with the same state. The density matrix can be written as $$\rho=\sum w_i|\alpha_i\rangle\langle\alpha_i|$$ where the $w_i$ can be the classical probabilities, if you can't say exactly where the state is in the Hilbert space, or the quantum ones, if you don't want (or can't) write the state as a definite vector of the Hilbert space.Īs an example take a two level system and consider the case in which you classically can't say in which one of the two states the system is. Entropy of quantum states Paolo Facchi, Giovanni Gramegna, Arturo Konderak Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. Neumann entropy: the quantum version of Shannon entropy. Additive property of the Von Neumann entropy. How is entropy increase in an isolated thermodynamic system consistent with the unitary invariance of Von Neumann entropy 2. First of all you have to distinguish the classical uncertainty form the quantum one. In this series we will derive some useful properties of the von. Von Neumann entropy vs Shannon entropy for a quantum state vector.
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