analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued...

of the orthonormal basis. The partial trace is another generalization of the trace that is operator-valued. The trace of a linear operator Z which lives...

not exist and the trace operator T {\textstyle T} presented here must be used to give meaning to u | ∂ Ω {\textstyle u|_{\partial \Omega }} . Then u...

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,...

subspace, it is technically called a partial trace. Key to this step is the assumption that the (partial) trace adequately summarizes the system from...

{\partial ^{2}f}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f'_{x})'_{y}=f''_{xy}=\partial _{yx}f=\partial...

\right|\right)\right],} where tr 2 {\displaystyle \operatorname {tr} _{2}} denotes the partial trace over the Hilbert space H 2 {\displaystyle {\mathcal {H}}_{2}} . This...

the rank 1 matrix ρ = w w ∗ {\displaystyle \rho =ww^{*}} . Then the partial trace of ρ {\displaystyle \rho } , with respect to either system A or B, is...

state of the part H 1 {\displaystyle H_{1}} is expressed then as the partial trace over H 2 {\displaystyle H_{2}} . A mixed state cannot be described with...

the partial trace of ρ with respect to the B system: ρ A = Tr B ρ . {\displaystyle \rho ^{A}=\operatorname {Tr} _{B}\;\rho .} The partial trace operation...

and contraction operations make sense in this context. Tensor product Partial trace Interior product Raising and lowering indices Musical isomorphism Ricci...

larger system and the state of the original system is given by the partial trace of the state of the larger system. In quantum mechanics, dynamical variables...

2 {\displaystyle {\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}} as a partial trace: ρ 12 = T r H 3 ρ 123 {\displaystyle \rho ^{12}={\rm {Tr}}_{{\mathcal...

topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Let H {\displaystyle...

which is the projection operator onto this state. The state of A is the partial trace of ρT over the basis of system B: ρ A   = d e f   ∑ j N B ( I A ⊗ ⟨...

{\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots...

variables x or y. The corresponding operation for density matrices is the partial trace. So one can assign to ρ a state on the subsystem A by ρ A = Tr B ρ A...

needed] Another important property of the GHZ state is that taking the partial trace over one of the three systems yields Tr 3 ⁡ [ ( | 000 ⟩ + | 111 ⟩ 2...

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entanglement entropy is the same regardless of whether one computes partial trace over the A {\displaystyle A} or B {\displaystyle B} subsystem. Many...

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represented by this equation. The trace operation ( T r A {\displaystyle {Tr}_{A}} ) indicates that we are considering the partial trace over the subsystem outside...

state over A ( M ) {\displaystyle {\mathcal {A}}(M)} , we can take the "partial trace" to get states associated with A ( U ) {\displaystyle {\mathcal {A}}(U)}...

=\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot (F_{x},F_{y},F_{z})={\frac {\partial F_{x}}{\partial...

_{12}\circ \Phi )(\rho \otimes \omega )=\rho \,,} where Tr12 is the partial trace operation with respect systems 1 and 2, and ∘ {\displaystyle \circ }...

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system-bath interaction. The state of the system can then be obtained from a partial trace over the combined system and bath: ρ S ( t ) = t r B { ρ S B ( t ) }...

_{d=1}^{n}\left({\frac {\partial g_{bd}}{\partial x^{a}}}+{\frac {\partial g_{ad}}{\partial x^{b}}}-{\frac {\partial g_{ab}}{\partial x^{d}}}\right)g^{cd}\\R_{ij}&:=\sum...

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...

wild branch before it takes effect, but also provide a complete or partial trace of its history. An autorelative pointer is a pointer whose value is...