Partial trace

The partial track , even Partialspur or partial track , referred to in the linear algebra and functional analysis , a linear map that the track is used. If a linear operator is defined on the tensor product of two vector spaces, its trace can be determined in two steps that relate to the two factors. In the first step the partial track is created, the second is a track according to the usual definition. The partial track is used in quantum mechanics . With their help, the density operator of any subsystem can be determined from the density operator of an overall system. In other words, the corresponding state of the subsystem is determined from the (pure or incoherently mixed) state of the overall system.

definition

Finite-dimensional case

Let and be finite-dimensional vector spaces,, plus the linear spaces of the linear operators on these, etc. Then the ' partial trace over ' is defined as the linear mapping from to with the identity on and the trace of the operators . ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle W=U\otimes V}$ ${\displaystyle L(U),\,}$ ${\displaystyle L(V),\,}$ ${\displaystyle L(W)}$ ${\displaystyle A^{u}\!:\,U\to U}$ ${\displaystyle \,}$ ${\displaystyle V\,}$ ${\displaystyle \operatorname {tr} _{V}=\operatorname {Id} ^{u}\otimes \operatorname {tr} ^{v}}$ ${\displaystyle L(W)}$ ${\displaystyle L(U)}$ ${\displaystyle \operatorname {Id} ^{u}}$ ${\displaystyle L(U)}$ ${\displaystyle \operatorname {tr} ^{v}}$ ${\displaystyle A_{v}\in L(V)}$

For an operator product with that means ${\displaystyle A=A_{u}\otimes A_{v}}$ ${\displaystyle A_{u}\in L(U),\,A_{v}\in L(V)}$

${\displaystyle \operatorname {tr} _{V}(A)=(\operatorname {Id} ^{u}\otimes \operatorname {tr} ^{v})(A_{u}\otimes A_{v})=A_{u}\cdot \operatorname {tr} ^{v}(A_{v})}$.

Any operator always has representations of the form ${\displaystyle A\in L(W)}$

${\displaystyle A=\textstyle \sum _{i}(A_{ui}\otimes A_{vi})}$  with   ; ${\displaystyle A_{ui}\in L(U),\,A_{vi}\in L(V)\,}$

that continues the linear mapping to all of : ${\displaystyle \operatorname {tr} _{V}}$ ${\displaystyle L(W)}$

${\displaystyle \operatorname {tr} _{V}(A)=\textstyle \sum _{i}\operatorname {tr} ^{v}(A_{vi})\cdot A_{ui}\,.}$

The designation as partial track refers to the fact that the (total) track is the concatenation , as well as analog . ${\displaystyle A\in L(W)}$ ${\displaystyle \operatorname {tr} =\operatorname {tr} ^{u}\circ \operatorname {tr} _{V}}$ ${\displaystyle \operatorname {tr} =\operatorname {tr} ^{v}\circ \operatorname {tr} _{U}}$

Coordinates are usually used for concrete calculations. If vectors and orthonormal bases form in or , the products form such a basis for . An operator is then represented by a four-dimensional matrix , the partial traces by the two-dimensional matrices and obtained by summing over or : for and for . ${\displaystyle e_{i}}$ ${\displaystyle f_{j}}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle e_{i}\otimes f_{j}}$ ${\displaystyle W=U\otimes V}$ ${\displaystyle A\in L(W)}$ ${\displaystyle (a_{ii'jj'})_{i,i',j,j'}}$ ${\displaystyle \operatorname {tr} _{V},\,\operatorname {tr} _{U}}$ ${\displaystyle (b_{ii'})_{i,i'}}$ ${\displaystyle (c_{jj'})_{j,j'}}$ ${\displaystyle j=j'}$ ${\displaystyle i=i'}$ ${\displaystyle b_{ii'}=\textstyle \sum _{j}a_{ii'\!jj}}$ ${\displaystyle \operatorname {tr} _{V}}$ ${\displaystyle c_{jj'}=\textstyle \sum _{i}a_{iijj'}}$ ${\displaystyle \operatorname {tr} _{U}}$

Infinite dimensional case

Like the trace, the partial trace can also be generalized to operators on infinite-dimensional spaces. It is then for trace class operators on Tensorprodukthilberträumen defined in a natural way and for a trace class to be ${\displaystyle \rho }$ ${\displaystyle {\mathcal {H}}\otimes {\mathcal {K}}}$

${\displaystyle \operatorname {tr} _{\mathcal {K}}(\rho )=\sum _{k}{}_{\mathcal {K}}\langle k|\rho |k\rangle _{\mathcal {K}}}$,

where is an orthonormal basis of . Here, too, the result of the construction is independent of the basis for separable Hilbert spaces ( track class, restricted ). ${\displaystyle \left\{|k\rangle _{\mathcal {K}}\right\}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle A}$ ${\displaystyle B}$

Relevance in quantum mechanics

If an observable, represented by the operator , of a quantum mechanical system is measured, the expected value of the measured value is determined by the state of the system in the broad sense, which includes pure and incoherently mixed states. Such a state is fully described by the density operator , a linear operator on the Hilbert space of the system. The expected value we are looking for is . ${\displaystyle X}$ ${\displaystyle \rho }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \operatorname {tr} (\rho X)}$

Is the system of components, subsystems composed , its Hilbert space is the tensor product of Hilbert spaces of the subsystems . For the measurement of an observable component of the density operator is in charge, as well as on for . The relationship then exists between the two ${\displaystyle {\mathcal {S}}={\mathcal {S}}'+{\mathcal {S}}''}$ ${\displaystyle {\mathcal {H}}={\mathcal {H}}'\otimes {\mathcal {H}}''}$ ${\displaystyle X'}$ ${\displaystyle {\mathcal {S}}'}$ ${\displaystyle \rho '}$ ${\displaystyle {\mathcal {H}}'}$ ${\displaystyle \rho }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle X}$

${\displaystyle \rho '=\operatorname {tr} _{{\mathcal {H}}''}(\rho )}$.

The partial trace above 'reduces' the density operator of the overall system to the density operator of the subsystem . Information that affects the complementary subsystem is 'traced out'. In other words: with the help of the density operator, the partial trace determines the state of any subsystem from the state of the system. This is particularly important when information about the complementary subsystem is not accessible or of no interest. ${\displaystyle {\mathcal {H}}''}$ ${\displaystyle {\mathcal {S}}'}$ ${\displaystyle {\mathcal {S}}''}$

Invariance of the partial track

${\displaystyle \rho _{j}}$is invariant under all possible trace- preserving quantum operations ( completely positive maps ) , especially under measurements . The reduced state can therefore also be understood as the state that is obtained when a complete measurement is carried out in the system , but the result is ignored: it is the statistical mean of the conditional states belonging to the various measurement results. For example, in the case of a Von Neumann measurement, the observable applies , with the operator defined, non-normalized having the following properties: is the probability with which the measurement result occurs and is the density operator conditioned on the measurement result . Also is invariant under randomization of the system , e.g. B. below the picture ${\displaystyle {\mathcal {H}}_{j}}$ ${\displaystyle \rho _{j}}$ ${\displaystyle j}$ ${\displaystyle \rho _{j}}$ ${\displaystyle X=\sum _{l}X_{l}|l\rangle \langle l|}$ ${\displaystyle \rho _{j}=\sum _{l}{}_{l}\langle l|\rho |l\rangle }$ ${\displaystyle \bigotimes _{i\not =j}{\mathcal {H}}_{i}}$ ${\displaystyle R_{l}={}_{l}\langle l|\rho |l\rangle }$ ${\displaystyle \operatorname {tr} (R_{l})}$ ${\displaystyle X=l}$ ${\displaystyle R_{l}/\operatorname {tr} (R_{l})}$ ${\displaystyle X=l}$ ${\displaystyle \rho _{j}}$ ${\displaystyle j}$

${\displaystyle R:\rho \mapsto \int _{\mathcal {U}}d\mu (U)(I\otimes U)\rho (I\otimes U^{\dagger }),}$

where the identical map and represents a probability measure on the group of unitary maps . If one chooses for the normalized hair measure over the unitary group, then commutates with all operators of the form and it applies . ${\displaystyle I}$ ${\displaystyle \mu (\cdot )}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle {\mathcal {H}}_{2}}$ ${\displaystyle \mu }$ ${\displaystyle R(\rho )}$ ${\displaystyle I\otimes B}$ ${\displaystyle R(\rho )=\operatorname {tr} _{2}(\rho )\otimes I/\operatorname {dim} {\mathcal {H}}_{2}}$

Partial track as a quantum channel

The mapping is completely positive and thus represents a track-preserving permitted quantum operation (a quantum channel ) whose Kraus representation is through ${\displaystyle Tr_{j}\colon \rho \mapsto \operatorname {tr} _{j}(\rho )}$

${\displaystyle Tr_{j}(\rho )=\sum _{l}(I\otimes |l\rangle \langle l|)\rho (I\otimes |l\rangle \langle l|)}$,

where is an orthonormal basis in the system and the identity on the other subsystems. ${\displaystyle |l\rangle }$ ${\displaystyle j}$ ${\displaystyle I}$

Partial track and entanglement

If one considers a pure state of a composite system, the partial trace can be used as a simple entanglement criterion : is entangled if and only if is not pure. ${\displaystyle \rho =|\psi \rangle \langle \psi |}$ ${\displaystyle |\psi \rangle \in {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}$ ${\displaystyle \operatorname {tr} _{{\mathcal {H}}_{1}}(\rho )}$

literature

• Michael Nielsen and Isaac Chuang: Quantum Computation and Quantum information . 1st edition. Cambridge University Press, Cambridge 2000, ISBN 0-521-63503-9 , pp. 105 (English).
• Michael Wilde: Quantum Information Theory . 1st edition. Cambridge University Press, Cambridge 2013, ISBN 978-1-107-03425-9 , pp. 116 , arxiv : 1106.1445 (English).

Individual evidence

1. ↑ Stephane Attal: Lectures in Quantum Noise Theory . Cape. 2 (English, univ-lyon1.fr [accessed December 19, 2016]).
2. ↑ R., P., M. and K. Horodecki: Quantum Entanglement . In: Rev. Mod. Phys. tape 81 , June 2009, p. 865 , doi : 10.1103 / RevModPhys.81.865 , arxiv : quant-ph / 0702225 (English).