Left hand side shows a full density matrix ${\displaystyle \rho _{AB}}$ 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 ${\displaystyle \rho _{A}}$.

In linear algebra and functional 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 function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.

Details

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

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

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

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

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

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

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).

Partial trace for operators on Hilbert spaces

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

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

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

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

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

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

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

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

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

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.

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) 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

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

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 operator R is the partial trace of T.

Partial trace as a quantum operation

The partial trace can be viewed as a quantum 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 a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product ${\displaystyle H_{A}\otimes H_{B}.}$ The partial trace of ρ with respect to the system B, denoted by ${\displaystyle \rho ^{A}}$, is called the reduced state of ρ on system A. In symbols,

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

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 ${\displaystyle M\otimes I}$. However one chooses to define a reduced state ${\displaystyle \rho ^{A}}$, there should be consistency of measurement statistics. The expectation value of M after the subsystem A 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:

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

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

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

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

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:

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

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

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

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

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

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

${\displaystyle \operatorname {Tr} _{B}^{*}}$ maps observables to observables and is the Heisenberg picture representation of ${\displaystyle \operatorname {Tr} _{B}}$.