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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 ...
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.
Since the trace is only over a subspace, it is technically called a partial trace. Key to this step is that the (partial) trace adequately summarizes the system from Bob's point of view. That is, everything that Bob has access to, or could ever have access to, measure, or detect, is completely described by a partial trace over H A of the system ...
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
Quantum tomography is a process by which, given a set of data representing the results of quantum measurements, a density matrix consistent with those measurement results is computed. [ 25 ] [ 26 ] When analyzing a system with many electrons, such as an atom or molecule , an imperfect but useful first approximation is to treat the electrons as ...
In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.
A quantum depolarizing channel is a model for quantum noise in quantum systems. The d {\displaystyle d} -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map Δ λ {\displaystyle \Delta _{\lambda }} , depending on one parameter λ {\displaystyle \lambda } , which maps a state ρ {\displaystyle \rho } onto ...
Quantum state tomography is a process by which, given a set of data representing the results of quantum measurements, a quantum state consistent with those measurement results is computed. [50] It is named by analogy with tomography , the reconstruction of three-dimensional images from slices taken through them, as in a CT scan .