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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 . The partial trace of ρ with respect to the system B, denoted by , is called the reduced state of ρ on system A.
The trace operator can be defined for functions in the Sobolev spaces , with <, see the section below for possible extensions of the trace to other spaces. Let Ω ⊂ R n {\textstyle \Omega \subset \mathbb {R} ^{n}} for n ∈ N {\textstyle n\in \mathbb {N} } be a bounded domain with Lipschitz boundary.
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
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra.
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. :=. Let ρ AB be a density matrix acting on states in H AB.
The normalization condition that the trace of be equal to 1 defines the partition function to be () = (). If the number of particles involved in the system is itself not certain, then a grand canonical ensemble can be applied, where the states summed over to make the density matrix are drawn from a Fock space .
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory, for example in entanglement characterization and in state purification, and plasticity.