Editing Partial trace

{{Short description|Function over linear operators}}
{{More citations needed|date=December 2024}}
[[File:Partial Trace.svg|thumb|right|Left-hand side shows a full density matrix <math>\rho_{AB}</math> 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 <math>\rho_A</math>.]]

In [[linear algebra]] and [[functional analysis]], the '''partial trace''' is a generalization of the [[trace (linear algebra)|trace]].  Whereas the trace is a [[scalar (mathematics)|scalar]]-valued function on operators, the partial trace is an [[operator (mathematics)|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 <math>V</math>, <math>W</math> are finite-dimensional [[vector space]]s over a [[field (mathematics)|field]], with [[dimension]]s <math>m</math> and <math>n</math>, respectively. For any space {{tmath|1= A }}, let <math>L(A)</math> denote the space of [[linear operators]] on <math>A</math>. The partial trace over <math>W</math> is then written as {{tmath|1= \operatorname{Tr}_W: \operatorname{L}(V \otimes W) \to \operatorname{L}(V) }}, where <math>\otimes</math> 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
: <math> \{a_{k \ell, i j}\}  \quad 1 \leq k, i \leq m, \quad 1 \leq \ell,j \leq n </math>
relative to the basis  <math> e_k \otimes f_\ell </math>  of  <math> V \otimes W</math>.

Now for indices ''k'', ''i'' in the range 1, ..., ''m'', consider the sum
: <math> b_{k, i} = \sum_{j=1}^n a_{k j, i j} </math>

This gives a matrix ''b''<sub>''k'',''i''</sub>. 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).

=== 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 
: <math> \operatorname{Tr}_W: \operatorname{L}(V \otimes W) \rightarrow \operatorname{L}(V) </math> 
such that
: <math> \operatorname{Tr}_W(R \otimes S) = \operatorname{Tr}(S) \, R \quad \forall R \in \operatorname{L}(V) \quad \forall S \in \operatorname{L}(W). </math>
To see that the conditions above determine the partial trace uniquely, let <math>v_1, \ldots, v_m</math> form a basis for <math>V</math>, let <math>w_1, \ldots, w_n</math> form a basis for <math>W</math>, let <math>E_{ij} : V \to V</math> be the map that sends <math>v_i</math> to <math>v_j</math> (and all other basis elements to zero), and let <math>F_{kl} \colon W \to W</math> be the map that sends <math>w_k</math> to {{nowrap|<math>w_l</math>.}} Since the vectors <math>v_i \otimes w_k</math> form a basis for <math>V \otimes W</math>, the maps <math>E_{ij} \otimes F_{kl}</math> form a basis for {{nowrap|<math>\operatorname{L}(V \otimes W)</math>.}}

From this abstract definition, the following properties follow:
: <math> \operatorname{Tr}_W (I_{V \otimes W}) = \dim W \ I_{V} </math>
: <math> \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).</math>

=== Category theoretic notion ===

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 <math>(C,\otimes,I)</math> together with, for objects ''X'', ''Y'', ''U'' in the category, a function of Hom-sets,
: <math>\operatorname{Tr}^U_{X,Y} : \operatorname{Hom}_C(X\otimes U, Y\otimes U) \to \operatorname{Hom}_C(X,Y)</math>
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 <math>X+U\cong Y+U</math> there exists a corresponding "partially traced" bijection {{nowrap|<math>X\cong Y</math>.}}

== 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
: <math> \{f_i\}_{i \in I} </math>
be an [[orthonormal basis]] for ''W''.  Now there is an isometric isomorphism
: <math>  \bigoplus_{\ell \in I} (V \otimes \mathbb{C}   f_\ell) \rightarrow V \otimes W</math>

Under this decomposition, any operator <math> T \in \operatorname{L}(V \otimes W)</math> can be regarded as an infinite matrix
of operators on ''V''
: <math> \begin{bmatrix} T_{11} & T_{12} & \ldots & T_{1 j} & \ldots \\
                        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},</math>
where <math> T_{k \ell} \in \operatorname{L}(V) </math>.

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
: <math> \sum_{\ell} T_{\ell \ell} </math>
converges in the [[strong operator topology]] of L(''V''), it is independent of the chosen basis of ''W''.  The partial trace Tr<sub>''W''</sub>(''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 ===

Suppose ''W'' has an orthonormal basis, which we denote by [[bra-ket|ket]] vector notation as {{tmath|1= \{ \vert \ell \rangle\}_\ell }}.  Then
: <math> \operatorname{Tr}_W\left(\sum_{k,\ell} T^{(k \ell)} \, \otimes \, | k \rangle \langle \ell |\right) = \sum_j T^{(j j)} .</math>

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'') 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
: <math> \int_{\operatorname{U}(W)} (I_V \otimes U^*) T (I_V \otimes U) \ d \mu(U) </math>
commutes with all operators of the form <math> I_V \otimes S </math> and hence is uniquely of the form <math> R \otimes I_W </math>. 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 <math>H_A \otimes H_B</math> 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 <math> H_A \otimes H_B .</math>
The partial trace of ''ρ'' with respect to the system ''B'', denoted by <math>\rho ^A</math>, is called the reduced state of ''ρ'' on system ''A''. In symbols,{{r|sh}}
<math display="block"> \rho^A = \operatorname{Tr}_B \rho.</math>

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 <math>M \otimes I</math>. However one chooses to define a reduced state <math>\rho^A</math>, there should be consistency of measurement statistics. The expectation value of ''M'' after the subsystem ''A'' is prepared in <math>\rho ^A</math> and that of <math>M \otimes I</math> when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:
: <math>\operatorname{Tr}_A ( M \cdot \rho^A) = \operatorname{Tr} ( M \otimes I \cdot \rho).</math>

We see that this is satisfied if <math>\rho ^A</math> 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 
: <math>\operatorname{Tr}_B : T(H_A \otimes H_B) \rightarrow T(H_A)</math>
is completely positive and trace-preserving.

The density matrix ρ is [[Hermitian matrix|Hermitian]], [[Positive semi-definite matrix|positive semi-definite]], and has a trace of 1. It has a [[Spectral decomposition (Matrix)|spectral decomposition]]:
: <math>\rho=\sum_{m}p_m|\Psi_m\rangle\langle \Psi_m|;\ 0\leq p_m\leq 1,\ \sum_{m}p_m=1</math>

Its easy to see that the partial trace <math>\rho ^A</math> also satisfies these conditions. For example, for any pure state <math>|\psi_A\rangle</math> in <math>H_A</math>, we have
: <math>\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</math>
Note that the term <math>\operatorname{Tr}_B[\langle\psi_A|\Psi_m\rangle\langle \Psi_m|\psi_A\rangle]</math> represents the probability of finding the state <math>|\psi_A\rangle</math> when the composite system is in the state <math>|\Psi_m\rangle</math>. This proves the positive semi-definiteness of <math>\rho ^A</math>.

The partial trace map as given above induces a dual map <math>\operatorname{Tr}_B ^*</math> between the [[C*-algebra]]s of bounded operators on <math>\; H_A</math> and <math>H_A \otimes H_B</math> given by
: <math>\operatorname{Tr}_B ^* (A) = A \otimes I.</math>
<math>\operatorname{Tr}_B ^*</math> maps observables to observables and is the [[Heisenberg picture]] representation of {{tmath|1= \operatorname{Tr}_B }}.

=== Comparison with classical case ===

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
: <math>C(X) \otimes C(Y) = C(X \times Y).</math>

A state on the composite system is a positive element ''ρ'' of the dual of C(''X'' &times; ''Y''), which by the [[Riesz–Markov theorem]] corresponds to a regular Borel measure on ''X'' &times; ''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.

== References ==
{{reflist|refs=

<ref name="sh">{{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
}}</ref>

}}
* {{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]]