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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.
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 ...
A quantum state for an imperfectly isolated system will generally evolve to be entangled with the quantum state for the environment. Consequently, even if the system's initial state is pure, the state at a later time, found by taking the partial trace of the joint system-environment state, will be mixed. This phenomenon of entanglement produced ...
The effect of decoherence on density matrices is essentially the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the trace, with respect to any environmental basis, of the density matrix of the combined system and its environment.
In quantum mechanics, and especially quantum information theory, the purity of a normalized quantum state is a scalar defined as ... (see partial trace).
MPT MONO: was shown by Lindblad, [20] using a representation of stochastic maps as a partial trace over an auxiliary system; MPT ⇒ {\displaystyle \Rightarrow } SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";