Entropy of Mixed States

Entropy of Mixed States

ByAbdul Wahab Junaid

In quantum mechanics, the concept of entropy extends to mixed states, which are statistical ensembles of quantum states. The entropy of a mixed state provides a measure of the quantum system’s uncertainty or disorder. This concept is fundamental in quantum information theory and quantum statistical mechanics.

Pure vs. Mixed States

  • Pure State: A pure state is described by a single quantum state, represented by a wavefunction ( |\psi\rangle ) or a corresponding density matrix ( \rho = |\psi\rangle \langle\psi| ). The entropy of a pure state is zero because there is no uncertainty about the state of the system.

  • Mixed State: A mixed state represents a probabilistic mixture of different quantum states. It is described by a density matrix ( \rho ) that is not a projector onto a single state, meaning it cannot be written as ( |\psi\rangle \langle\psi| ) for some state ( |\psi\rangle ).

Density Matrix

The density matrix ( \rho ) for a mixed state is given by:

[ \rho = \sum_i p_i |\psi_i\rangle \langle\psi_i| ]

where:

  • ( |\psi_i\rangle ) are the quantum states.
  • ( p_i ) are the probabilities associated with each state, such that ( \sum_i p_i = 1 ).

Von Neumann Entropy

The entropy of a mixed quantum state is quantified by the von Neumann entropy, which is a generalization of Shannon entropy to quantum systems. The von Neumann entropy ( S(\rho) ) of a density matrix ( \rho ) is defined as:

[ S(\rho) = -\text{Tr}(\rho \log_2 \rho) ]

Here:

  • ( \text{Tr} ) denotes the trace of a matrix, which is the sum of its diagonal elements.
  • ( \log_2 \rho ) is the matrix logarithm of the density matrix ( \rho ).

Key Properties of Von Neumann Entropy

  1. Non-Negativity: ( S(\rho) \geq 0 ). The entropy is always non-negative.

  2. Zero Entropy for Pure States: For a pure state, ( \rho = |\psi\rangle \langle\psi| ), the entropy ( S(\rho) = 0 ). This reflects the fact that there is no uncertainty in a pure state.

  3. Maximal Entropy: For a maximally mixed state (where the system is equally likely to be in any of a set of orthogonal states), the entropy is maximal. For a system of dimension ( d ), the maximum entropy is ( \log_2 d ).

  4. Additivity: For two independent systems with density matrices ( \rho_1 ) and ( \rho_2 ), the entropy of the combined system is the sum of their entropies: [ S(\rho_1 \otimes \rho_2) = S(\rho_1) + S(\rho_2) ]

Example: Two-Level System (Qubit)

Consider a qubit system where the mixed state is given by:

[ \rho = p |\psi_1\rangle \langle\psi_1| + (1 - p) |\psi_2\rangle \langle\psi_2| ]

where ( |\psi_1\rangle ) and ( |\psi_2\rangle ) are orthogonal states (e.g., ( |0\rangle ) and ( |1\rangle )) and ( p ) is the probability of the system being in state ( |\psi_1\rangle ).

The density matrix can be diagonalized, and the eigenvalues of ( \rho ) are ( p ) and ( 1 - p ). The von Neumann entropy is then:

[ S(\rho) = -p \log_2(p) - (1 - p) \log_2(1 - p) ]

This is the same as the Shannon entropy of a classical two-state system, reflecting the probabilistic nature of the mixed state.

Summary

The entropy of mixed states in quantum mechanics, measured by the von Neumann entropy, captures the degree of uncertainty or disorder in a quantum system. For pure states, the entropy is zero, indicating no uncertainty. For mixed states, the entropy quantifies the extent to which the system is in a probabilistic mixture of different quantum states. This concept is crucial in understanding quantum information, quantum thermodynamics, and the behavior of quantum systems in general.

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