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The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product H A ⊗ H B {\displaystyle H_{A}\otimes H_{B}} of Hilbert spaces. A mixed state is described by a density matrix ρ , that is a non-negative trace-class operator of trace 1 on the tensor product H A ⊗ H B ...
the reduced state of ρ on system A, ρ A, is obtained by taking the partial trace of ρ with respect to the B system: =. The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. [5] In the Heisenberg picture, the dual map of this channel is
The quantum mechanical counterpart of classical probability distributions are modeled with density matrices. Consider a quantum system that can be divided into two parts, A and B, such that independent measurements can be made on either part. The state space of the entire quantum system is then the tensor product of the spaces for the two parts.
Moreover, a mixed quantum state on a given quantum system described by a Hilbert space can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system for a sufficiently large Hilbert space .
In quantum mechanics, and especially quantum information theory, the purity of a normalized quantum state is a scalar defined as ... (see partial trace).
This form of writing the entropy makes it explicitly clear that the entanglement entropy is the same regardless of whether one computes partial trace over the or subsystem. Many entanglement measures reduce to the entropy of entanglement when evaluated on pure states. Among those are: Distillable entanglement
Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are | |. In other words, the Schmidt decomposition shows that the reduced states of ρ {\displaystyle \rho } on either subsystem have the same spectrum.
In quantum mechanics, ... is the partial transpose of ... is the trace norm or the sum of the singular values of the operator . An alternative and ...