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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.
In this case, even entangled states can be written as a mixture of tensor products of single-party aphysical states, very similar to the form of separable states. In the qubit case, M k {\displaystyle M_{k}} are physical density matrices, which is consistent with the fact that for two qubits all PPT states are separable.
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.
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. [2] For instance, on space forms (i.e. spaces of constant sectional curvature) and two-dimensional smooth Riemannian manifolds, the Riemann curvature tensor has a simple expression in terms of the Kulkarni–Nomizu product of the metric = with itself; namely, if we denote by
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.
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 .
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.
For more properties and a generalization of the partial trace, see traced monoidal categories. If A is a general associative algebra over a field k, then a trace on A is often defined to be any map tr : A ↦ k which vanishes on commutators; tr([a,b]) = 0 for all a, b ∈ A. Such a trace is not uniquely defined; it can always at least be ...