Partial trace

Left hand side shows a full density matrix$\rho _{AB}$of a bipartite qubit system. The partial trace is performed over a subsystem of 2 by 2 dimension (single qubit density matrix). The right hand side shows the resulting 2 by 2 reduced density matrix$\rho _{A}$.

Inlinear algebraandfunctional analysis,thepartial traceis a generalization of thetrace.Whereas the trace is ascalarvalued function on operators, the partial trace is anoperator-valued function. The partial trace has applications inquantum informationanddecoherencewhich is relevant forquantum measurementand thereby to the decoherent approaches tointerpretations of quantum mechanics,includingconsistent historiesand therelative state interpretation.

Details

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

It is defined as follows: For$T\in \operatorname {L} (V\otimes W)$,let$e_{1},\ldots,e_{m}$,and$f_{1},\ldots,f_{n}$,be bases forVandWrespectively; thenT has a matrix representation

$\{a_{k\ell,ij}\}\quad 1\leq k,i\leq m,\quad 1\leq \ell,j\leq n$

relative to the basis$e_{k}\otimes f_{\ell }$of$V\otimes W$.

Now for indicesk,iin the range 1,...,m,consider the sum

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

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

Among physicists, this is often called "tracing out" or "tracing over"Wto leave only an operator onVin the context whereWandVare Hilbert spaces associated with quantum systems (see below).

Invariant definition

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

$\operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\rightarrow \operatorname {L} (V)$

such that

$\operatorname {Tr} _{W}(R\otimes S)=\operatorname {Tr} (S)\,R\quad \forall R\in \operatorname {L} (V)\quad \forall S\in \operatorname {L} (W).$

To see that the conditions above determine the partial trace uniquely, let$v_{1},\ldots,v_{m}$form a basis for$V$,let$w_{1},\ldots,w_{n}$form a basis for$W$,let$E_{ij}\colon V\to V$be the map that sends$v_{i}$to$v_{j}$(and all other basis elements to zero), and let$F_{kl}\colon W\to W$be the map that sends$w_{k}$to$w_{l}$.Since the vectors$v_{i}\otimes w_{k}$form a basis for$V\otimes W$,the maps$E_{ij}\otimes F_{kl}$form a basis for$\operatorname {L} (V\otimes W)$.

From this abstract definition, the following properties follow:

$\operatorname {Tr} _{W}(I_{V\otimes W})=\dim W\ I_{V}$

$\operatorname {Tr} _{W}(T(I_{V}\otimes S))=\operatorname {Tr} _{W}((I_{V}\otimes S)T)\quad \forall S\in \operatorname {L} (W)\quad \forall T\in \operatorname {L} (V\otimes W).$

Category theoretic notion

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$(C,\otimes,I)$together with, for objectsX,Y,Uin the category, a function of Hom-sets,

$\operatorname {Tr} _{X,Y}^{U}\colon \operatorname {Hom} _{C}(X\otimes U,Y\otimes U)\to \operatorname {Hom} _{C}(X,Y)$

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,Uand bijection$X+U\cong Y+U$there exists a corresponding "partially traced" bijection$X\cong Y$.

Partial trace for operators on Hilbert spaces

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

$\{f_{i}\}_{i\in I}$

be anorthonormal basisforW.Now there is an isometric isomorphism

$\bigoplus _{\ell \in I}(V\otimes \mathbb {C} f_{\ell })\rightarrow V\otimes W$

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

${\begin{bmatrix}T_{11}&T_{12}&\ldots &T_{1j}&\ldots \\T_{21}&T_{22}&\ldots &T_{2j}&\ldots \\\vdots &\vdots &&\vdots \\T_{k1}&T_{k2}&\ldots &T_{kj}&\ldots \\\vdots &\vdots &&\vdots \end{bmatrix}},$

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

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

$\sum _{\ell }T_{\ell \ell }$

converges in thestrong operator topologyof 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.

Computing the partial trace

SupposeWhas an orthonormal basis, which we denote byketvector notation as$\{|\ell \rangle \}_{\ell }$.Then

$\operatorname {Tr} _{W}\left(\sum _{k,\ell }T^{(k\ell )}\,\otimes \,|k\rangle \langle \ell |\right)=\sum _{j}T^{(jj)}.$

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

Partial trace and invariant integration

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 that μ is taken to be a measure with total mass dim(W).

Theorem.SupposeV,Ware finite dimensional Hilbert spaces. Then

$\int _{\operatorname {U} (W)}(I_{V}\otimes U^{*})T(I_{V}\otimes U)\ d\mu (U)$

commutes with all operators of the form$I_{V}\otimes S$and hence is uniquely of the form$R\otimes I_{W}$.The operatorRis the partial trace ofT.

Partial trace as a quantum operation

The partial trace can be viewed as aquantum operation.Consider a quantum mechanical system whose state space is the tensor product$H_{A}\otimes H_{B}$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$H_{A}\otimes H_{B}.$ The partial trace of ρ with respect to the systemB,denoted by$\rho ^{A}$,is called the reduced state of ρ on systemA.In symbols,

$\rho ^{A}=\operatorname {Tr} _{B}\rho.$

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

$\operatorname {Tr} _{A}(M\cdot \rho ^{A})=\operatorname {Tr} (M\otimes I\cdot \rho ).$

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

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

$\operatorname {Tr} _{B}:T(H_{A}\otimes H_{B})\rightarrow T(H_{A})$

is completely positive and trace-preserving.

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

$\rho =\sum _{m}p_{m}|\Psi _{m}\rangle \langle \Psi _{m}|;\ 0\leq p_{m}\leq 1,\ \sum _{m}p_{m}=1$

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

$\langle \psi _{A}|\rho ^{A}|\psi _{A}\rangle =\sum _{m}p_{m}\operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]\geq 0$

Note that the term$\operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]$represents the probability of finding the state$|\psi _{A}\rangle$when the composite system is in the state$|\Psi _{m}\rangle$.This proves the positive semi-definiteness of$\rho ^{A}$.

The partial trace map as given above induces a dual map$\operatorname {Tr} _{B}^{*}$between theC*-algebrasof bounded operators on$\;H_{A}$and$H_{A}\otimes H_{B}$given by

$\operatorname {Tr} _{B}^{*}(A)=A\otimes I.$

$\operatorname {Tr} _{B}^{*}$maps observables to observables and is theHeisenberg picturerepresentation of$\operatorname {Tr} _{B}$.

Comparison with classical case

Suppose instead of quantum mechanical systems, the two systemsAandBare 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

$C(X)\otimes C(Y)=C(X\times Y).$

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