Partial trace

Suppose $V$, $W$ are finite-dimensional vector spaces over a field, with dimensions $m$ and $n$, respectively. For any space {{tmath|1= A }}, let $L(A)$ denote the space of linear operators on $A$. The partial trace over $W$ is then written as {{tmath|1= \operatorname{Tr}_W: \operatorname{L}(V \otimes W) \to \operatorname{L}(V) }}, where $\backslash otimes$ denotes the Kronecker product.

It is defined as follows:

For {{tmath|1= T\in \operatorname{L}(V \otimes W) }}, let {{tmath|1= e_1, \ldots, e_m }}, and {{tmath|1= f_1, \ldots, f_n }}, be bases for V and W respectively; then T has a matrix representation

: $\backslash \{a\_\{k\; \backslash ell,\; i\; j\}\backslash \}\; \backslash quad\; 1\; \backslash leq\; k,\; i\; \backslash leq\; m,\; \backslash quad\; 1\; \backslash leq\; \backslash ell,j\; \backslash leq\; n$

relative to the basis $e\_k\; \backslash otimes\; f\_\backslash ell$ of $V\; \backslash otimes\; W$.

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

: $b\_\{k,\; i\}\; =\; \backslash sum\_\{j=1\}^n\; a\_\{k\; j,\; i\; j\}$

This gives a matrix b_k,i. The associated linear operator on V is independent of the choice of bases and is by definition the partial trace.

Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V 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

: $\backslash operatorname\{Tr\}\_W:\; \backslash operatorname\{L\}(V\; \backslash otimes\; W)\; \backslash rightarrow\; \backslash operatorname\{L\}(V)$

such that

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

To see that the conditions above determine the partial trace uniquely, let $v\_1,\; \backslash ldots,\; v\_m$ form a basis for $V$, let $w\_1,\; \backslash ldots,\; w\_n$ form a basis for $W$, let $E\_\{ij\}\; :\; V\; \backslash to\; V$ be the map that sends $v\_i$ to $v\_j$ (and all other basis elements to zero), and let $F\_\{kl\}\; \backslash colon\; W\; \backslash to\; W$ be the map that sends $w\_k$ to {{nowrap|$w\_l$.}} Since the vectors $v\_i\; \backslash otimes\; w\_k$ form a basis for $V\; \backslash otimes\; W$, the maps $E\_\{ij\}\; \backslash otimes\; F\_\{kl\}$ form a basis for {{nowrap|$\backslash operatorname\{L\}(V\; \backslash otimes\; W)$.}}

From this abstract definition, the following properties follow:

: $\backslash operatorname\{Tr\}\_W\; (I\_\{V\; \backslash otimes\; W\})\; =\; \backslash dim\; W\; \backslash \; I\_\{V\}$

: $\backslash operatorname\{Tr\}\_W\; (T\; (I\_V\; \backslash otimes\; S))\; =\; \backslash operatorname\{Tr\}\_W\; ((I\_V\; \backslash otimes\; S)\; T)\; \backslash quad\; \backslash forall\; S\; \backslash in\; \backslash operatorname\{L\}(W)\; \backslash quad\; \backslash forall\; T\; \backslash in\; \backslash operatorname\{L\}(V\; \backslash otimes\; W).$

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

: $\backslash operatorname\{Tr\}^U\_\{X,Y\}\; :\; \backslash operatorname\{Hom\}\_C(X\backslash otimes\; U,\; Y\backslash otimes\; U)\; \backslash to\; \backslash operatorname\{Hom\}\_C(X,Y)$

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,U and bijection $X+U\backslash cong\; Y+U$ there exists a corresponding "partially traced" bijection {{nowrap|$X\backslash cong\; Y$.}}

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

let

: $\backslash \{f\_i\backslash \}\_\{i\; \backslash in\; I\}$

be an orthonormal basis for W. Now there is an isometric isomorphism

: $\backslash bigoplus\_\{\backslash ell\; \backslash in\; I\}\; (V\; \backslash otimes\; \backslash mathbb\{C\}\; f\_\backslash ell)\; \backslash rightarrow\; V\; \backslash otimes\; W$

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

of operators on V

:

T_{21} & T_{22} & \ldots & T_{2 j} & \ldots \\

\vdots & \vdots & & \vdots \\

T_{k1}& T_{k2} & \ldots & T_{k j} & \ldots \\

\vdots & \vdots & & \vdots

\end{bmatrix},

where $T\_\{k\; \backslash ell\}\; \backslash in\; \backslash operatorname\{L\}(V)$.

First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum

: $\backslash sum\_\{\backslash ell\}\; T\_\{\backslash ell\; \backslash ell\}$

converges in the strong operator topology of L(V), it is independent of the chosen basis of W. 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.

Suppose W has an orthonormal basis, which we denote by ket vector notation as {{tmath|1= \{ \vert \ell \rangle\}_\ell }}. Then

: $\backslash operatorname\{Tr\}\_W\backslash left(\backslash sum\_\{k,\backslash ell\}\; T^\{(k\; \backslash ell)\}\; \backslash ,\; \backslash otimes\; \backslash ,\; |\; k\; \backslash rangle\; \backslash langle\; \backslash ell\; |\backslash right)\; =\; \backslash sum\_j\; T^\{(j\; j)\}\; .$

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

Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then

: $\backslash int\_\{\backslash operatorname\{U\}(W)\}\; (I\_V\; \backslash otimes\; U^*)\; T\; (I\_V\; \backslash otimes\; U)\; \backslash \; d\; \backslash mu(U)$

commutes with all operators of the form $I\_V\; \backslash otimes\; S$ and hence is uniquely of the form $R\; \backslash otimes\; I\_W$. The operator R is the partial trace of T.

The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product $H\_A\; \backslash otimes\; H\_B$ of Hilbert spaces. 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 $H\_A\; \backslash otimes\; H\_B\; .$

The partial trace of ρ with respect to the system B, denoted by $\backslash rho\; ^A$, is called the reduced state of ρ on system A. In symbols,{{r|sh}}

$$\backslash rho^A\; =\; \backslash operatorname\{Tr\}\_B\; \backslash rho.$$

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

: $\backslash operatorname\{Tr\}\_A\; (\; M\; \backslash cdot\; \backslash rho^A)\; =\; \backslash operatorname\{Tr\}\; (\; M\; \backslash otimes\; I\; \backslash cdot\; \backslash rho).$

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

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

: $\backslash operatorname\{Tr\}\_B\; :\; T(H\_A\; \backslash otimes\; H\_B)\; \backslash rightarrow\; T(H\_A)$

is completely positive and trace-preserving.

The density matrix ρ is Hermitian, positive semi-definite, and has a trace of 1. It has a spectral decomposition:

: $\backslash rho=\backslash sum\_\{m\}p\_m|\backslash Psi\_m\backslash rangle\backslash langle\; \backslash Psi\_m|;\backslash \; 0\backslash leq\; p\_m\backslash leq\; 1,\backslash \; \backslash sum\_\{m\}p\_m=1$

Its easy to see that the partial trace $\backslash rho\; ^A$ also satisfies these conditions. For example, for any pure state $|\backslash psi\_A\backslash rangle$ in $H\_A$, we have

: $\backslash langle\backslash psi\_A|\backslash rho^A|\backslash psi\_A\backslash rangle=\backslash sum\_\{m\}p\_m\backslash operatorname\{Tr\}\_B[\backslash langle\backslash psi\_A|\backslash Psi\_m\backslash rangle\backslash langle\; \backslash Psi\_m|\backslash psi\_A\backslash rangle]\backslash geq\; 0$

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

The partial trace map as given above induces a dual map $\backslash operatorname\{Tr\}\_B\; ^*$ between the C*-algebras of bounded operators on $\backslash ;\; H\_A$ and $H\_A\; \backslash otimes\; H\_B$ given by

: $\backslash operatorname\{Tr\}\_B\; ^*\; (A)\; =\; A\; \backslash otimes\; I.$

$\backslash operatorname\{Tr\}\_B\; ^*$ maps observables to observables and is the Heisenberg picture representation of {{tmath|1= \operatorname{Tr}_B }}.

Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply

: $C(X)\; \backslash otimes\; C(Y)\; =\; C(X\; \backslash times\; Y).$

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

{{reflist|refs=

{{cite book

| last1 = Steeb | first1 = Willi-Hans

| last2 = Hardy | first2 = Yorick

| year = 2006

| title = Problems and Solutions in Quantum Computing and Quantum Information

| url = https://books.google.com/books?id=HGMy_dSmfbkC&pg=PA80

| page = 80

| publisher = World Scientific

}}

}}

• {{cite journal

| last1 = Filipiak | first1 = Katarzyna

| last2 = Klein | first2 = Daniel

| last3 = Vojtková | first3 = Erika

| title = The properties of partial trace and block trace operators of partitioned matrices

| journal = Electronic Journal of Linear Algebra

| volume = 33 | year = 2018

| doi = 10.13001/1081-3810.3688

}}

• {{cite book

| last = Johnston | first = Nathaniel

| year = 2021

| title = Advanced Linear and Matrix Algebra

| publisher = Springer

| url = https://books.google.com/books?id=w24vEAAAQBAJ&pg=PA367

| page = 367

| doi = 10.1007/978-3-030-52815-7

| isbn = 978-3-030-52815-7

}}

{{DEFAULTSORT:Partial Trace}}

Category:Linear algebra

Category:Functional analysis