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Inlinear algebra andfunctional analysis, thepartial trace is a generalization of thetrace. Whereas the trace is ascalar-valued function on operators, the partial trace is anoperator-valued function. The partial trace has applications inquantum information anddecoherence which is relevant forquantum measurement and thereby to the decoherent approaches tointerpretations of quantum mechanics, includingconsistent histories and therelative state interpretation.

Suppose${\displaystyle V}$,${\displaystyle W}$ are finite-dimensionalvector spaces over afield, withdimensions${\displaystyle m}$ and${\displaystyle n}$, respectively. For any space⁠${\displaystyle A}$⁠, let${\displaystyle L(A)}$ denote the space oflinear operators on${\displaystyle A}$. The partial trace over${\displaystyle W}$ is then written as⁠${\displaystyle {\mathrm{Tr}}_W:\mathrm{L}(V\otimes W)\to \mathrm{L}(V)}$⁠, where${\displaystyle \otimes }$ denotes theKronecker product.

It is defined as follows: For⁠${\displaystyle T\in \mathrm{L}(V\otimes W)}$⁠, let⁠${\displaystyle e_1,\dots ,e_m}$⁠, and⁠${\displaystyle f_1,\dots ,f_n}$⁠, be bases forV andW respectively; thenT has a matrix representation

${\displaystyle \{a_{k\ell ,ij}\}1\le k,i\le m,1\le \ell ,j\le n}$

relative to the basis${\displaystyle e_k\otimes f_{\ell }}$ of${\displaystyle V\otimes W}$.

Now for indicesk,i in the range 1, ...,m, consider the sum

${\displaystyle b_{k,i}=\sum _{j=1}^n{a}_{kj,ij}}$

This gives a matrixb_k,i. The associated linear operator onV is independent of the choice of bases and is by definition thepartial trace.

Among physicists, this is often called "tracing out" or "tracing over"W to leave only an operator onV in the context whereW andV are Hilbert spaces associated with quantum systems (see below).

The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear map

${\displaystyle {\mathrm{Tr}}_W:\mathrm{L}(V\otimes W)\to \mathrm{L}(V)}$

such that

${\displaystyle {\mathrm{Tr}}_W(R\otimes S)=\mathrm{Tr}(S)R\mathrm{\forall }R\in \mathrm{L}(V)\mathrm{\forall }S\in \mathrm{L}(W).}$

To see that the conditions above determine the partial trace uniquely, let${\displaystyle v_1,\dots ,v_m}$ form a basis for${\displaystyle V}$, let${\displaystyle w_1,\dots ,w_n}$ form a basis for${\displaystyle W}$, let${\displaystyle E_{ij}:V\to V}$ be the map that sends${\displaystyle v_i}$ to${\displaystyle v_j}$ (and all other basis elements to zero), and let${\displaystyle F_{kl}:W\to W}$ be the map that sends${\displaystyle w_k}$ to${\displaystyle w_l}$. Since the vectors${\displaystyle v_i\otimes w_k}$ form a basis for${\displaystyle V\otimes W}$, the maps${\displaystyle E_{ij}\otimes F_{kl}}$ form a basis for${\displaystyle \mathrm{L}(V\otimes W)}$.

From this abstract definition, the following properties follow:

${\displaystyle {\mathrm{Tr}}_W(I_{V\otimes W})=\dim W{I}_V}$

${\displaystyle {\mathrm{Tr}}_W(T(I_V\otimes S))={\mathrm{Tr}}_W((I_V\otimes S)T)\mathrm{\forall }S\in \mathrm{L}(W)\mathrm{\forall }T\in \mathrm{L}(V\otimes W).}$

It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion ofTraced monoidal category. A traced monoidal category is a monoidal category${\displaystyle (C,\otimes ,I)}$ together with, for objectsX,Y,U in the category, a function of Hom-sets,

${\displaystyle {\mathrm{Tr}}_{X,Y}^U:{\mathrm{Hom}}_C(X\otimes U,Y\otimes U)\to {\mathrm{Hom}}_C(X,Y)}$

satisfying certain axioms.

Another case of this abstract notion of partial trace takes place in the category of finite sets and bijections between them, in which the monoidal product is disjoint union. One can show that for any finite sets,X,Y,U and bijection${\displaystyle X+U\cong Y+U}$ there exists a corresponding "partially traced" bijection${\displaystyle X\cong Y}$.

The partial trace generalizes to operators on infinite dimensional Hilbert spaces. SupposeV,W are Hilbert spaces, and let

${\displaystyle \{f_i{\}}_{i\in I}}$

be anorthonormal basis forW. Now there is an isometric isomorphism

${\displaystyle \underset{\ell \in I}{⨁}(V\otimes \mathbb{C}{f}_{\ell })\to V\otimes W}$

Under this decomposition, any operator${\displaystyle T\in \mathrm{L}(V\otimes W)}$ can be regarded as an infinite matrixof operators onV

where${\displaystyle T_{k\ell }\in \mathrm{L}(V)}$.

First supposeT is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators onV. If the sum

${\displaystyle \sum _{\ell }{T}_{\ell \ell }}$

converges in thestrong operator topology of L(V), it is independent of the chosen basis ofW. The partial trace Tr_W(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined.

SupposeW has an orthonormal basis, which we denote byket vector notation as⁠${\displaystyle \{|\ell ⟩{\}}_{\ell }}$⁠. Then

${\displaystyle {\mathrm{Tr}}_W\left(\sum _{k,\ell }{T}^{(k\ell )}\otimes |k⟩⟨\ell |\right)=\sum _j{T}^{(jj)}.}$

The superscripts in parentheses do not represent matrix components, but instead label the matrix itself.

In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) ofW. Suitably normalized means thatμ is taken to be a measure with total mass dim(W).

Theorem. SupposeV,W are finite dimensional Hilbert spaces. Then

${\displaystyle {\int }_{\mathrm{U}(W)}(I_V\otimes U^{\ast })T(I_V\otimes U)d\mu (U)}$

commutes with all operators of the form${\displaystyle I_V\otimes S}$ and hence is uniquely of the form${\displaystyle R\otimes I_W}$. The operatorR is the partial trace ofT.

The partial trace can be viewed as aquantum operation. Consider a quantum mechanical system whose state space is the tensor product${\displaystyle H_A\otimes H_B}$ of Hilbert spaces. A mixed state is described by adensity matrixρ, that is a non-negative trace-class operator of trace 1 on the tensor product${\displaystyle H_A\otimes H_B.}$The partial trace ofρ with respect to the systemB, denoted by${\displaystyle {\rho }^A}$, is called the reduced state ofρ on systemA. In symbols,^[1]$${\displaystyle {\rho }^A={\mathrm{Tr}}_B\rho .}$$

To show that this is indeed a sensible way to assign a state on theA subsystem to ρ, we offer the following justification. LetM be an observable on the subsystemA, then the corresponding observable on the composite system is${\displaystyle M\otimes I}$. However one chooses to define a reduced state${\displaystyle {\rho }^A}$, there should be consistency of measurement statistics. The expectation value ofM after the subsystemA is prepared in${\displaystyle {\rho }^A}$ and that of${\displaystyle M\otimes I}$ when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:

${\displaystyle {\mathrm{Tr}}_A(M\cdot {\rho }^A)=\mathrm{Tr}(M\otimes I\cdot \rho ).}$

We see that this is satisfied if${\displaystyle {\rho }^A}$ is as defined above via the partial trace. Furthermore, such operation is unique.

LetT(H) be theBanach space of trace-class operators on the Hilbert spaceH. It can be easily checked that the partial trace, viewed as a map

${\displaystyle {\mathrm{Tr}}_B:T(H_A\otimes H_B)\to T(H_A)}$

is completely positive and trace-preserving.

The density matrix ρ isHermitian,positive semi-definite, and has a trace of 1. It has aspectral decomposition:

${\displaystyle \rho =\sum _m{p}_m|{\mathrm{\Psi }}_m⟩⟨{\mathrm{\Psi }}_m|;0\le p_m\le 1,\sum _m{p}_m=1}$

Its easy to see that the partial trace${\displaystyle {\rho }^A}$ also satisfies these conditions. For example, for any pure state${\displaystyle |{\psi }_A⟩}$ in${\displaystyle H_A}$, we have

${\displaystyle ⟨{\psi }_A|{\rho }^A|{\psi }_A⟩=\sum _m{p}_m{\mathrm{Tr}}_B[⟨{\psi }_A|{\mathrm{\Psi }}_m⟩⟨{\mathrm{\Psi }}_m|{\psi }_A⟩]\ge 0}$

Note that the term${\displaystyle {\mathrm{Tr}}_B[⟨{\psi }_A|{\mathrm{\Psi }}_m⟩⟨{\mathrm{\Psi }}_m|{\psi }_A⟩]}$ represents the probability of finding the state${\displaystyle |{\psi }_A⟩}$ when the composite system is in the state${\displaystyle |{\mathrm{\Psi }}_m⟩}$. This proves the positive semi-definiteness of${\displaystyle {\rho }^A}$.

The partial trace map as given above induces a dual map${\displaystyle {\mathrm{Tr}}_B^{\ast }}$ between theC*-algebras of bounded operators on${\displaystyle H_A}$ and${\displaystyle H_A\otimes H_B}$ given by

${\displaystyle {\mathrm{Tr}}_B^{\ast }(A)=A\otimes I.}$

${\displaystyle {\mathrm{Tr}}_B^{\ast }}$ maps observables to observables and is theHeisenberg picture representation of⁠${\displaystyle {\mathrm{Tr}}_B}$⁠.

Suppose instead of quantum mechanical systems, the two systemsA andB are classical. The space of observables for each system are then abelian C*-algebras. These are of the formC(X) andC(Y) respectively for compact spacesX,Y. The state space of the composite system is simply

${\displaystyle C(X)\otimes C(Y)=C(X\times Y).}$

A state on the composite system is a positive elementρ of the dual of C(X ×Y), which by theRiesz–Markov theorem corresponds to a regular Borel measure onX ×Y. The corresponding reduced state is obtained by projecting the measureρ toX. Thus the partial trace is the quantum mechanical equivalent of this operation.

• Filipiak, Katarzyna; Klein, Daniel; Vojtková, Erika (2018). "The properties of partial trace and block trace operators of partitioned matrices".Electronic Journal of Linear Algebra.33.doi:10.13001/1081-3810.3688.
• Johnston, Nathaniel (2021).Advanced Linear and Matrix Algebra. Springer. p. 367.doi:10.1007/978-3-030-52815-7.ISBN 978-3-030-52815-7.

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