partialtrace

Partialtrace is a fundamental operation in quantum information theory used to simplify quantum states. It allows us to remove a subsystem from a composite quantum system, effectively focusing on the remaining parts. When we have a quantum system composed of two or more subsystems, its state can be described by a density matrix. The partial trace operation acts on this density matrix and eliminates the information pertaining to one or more of these subsystems.

For a bipartite system (composed of two subsystems A and B) described by a density matrix ρ_AB, the partial trace over subsystem B, denoted as Tr_B(ρ_AB), results in a density matrix ρ_A that describes the state of subsystem A. This resulting state ρ_A is generally a mixed state, meaning it represents a probabilistic mixture of pure states, even if the original combined state ρ_AB was pure. This is because tracing out a subsystem introduces uncertainty about its state, which in turn affects the apparent state of the remaining subsystem.

The mathematical definition of the partial trace involves summing over a complete orthonormal basis for the subsystem being traced out. If we have a basis {|i>} for subsystem B, then the (m,n)-th element of ρ_A is given by the sum over i of <m,i|ρ_AB|n,i>. The partial trace is essential for calculating properties of subsystems, understanding entanglement, and performing operations in quantum computing. For instance, when analyzing the correlations between two entangled qubits, one might take the partial trace over one qubit to examine the state of the other. This operation is a cornerstone for understanding and manipulating complex quantum states.