![Partial Trace](https://upload.wikimedia.org/wikipedia/commons/8/85/Partial_Trace.svg)
In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the partial trace is a generalization of the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
. Whereas the trace is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
valued function on operators, the partial trace is an
operator-valued function. The partial trace has applications in
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
and
decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
which is relevant for
quantum measurement
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what ...
and thereby to the decoherent approaches to
interpretations of quantum mechanics
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraord ...
, including
consistent histories
In quantum mechanics, the consistent histories (also referred to as decoherent histories) approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural i ...
and the
relative state interpretation
The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum me ...
.
Details
Suppose
,
are finite-dimensional vector spaces over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, with
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s
and
, respectively. For any space
, let
denote the space of
linear operators
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
on
. The partial trace over
is then written as
.
It is defined as follows:
For
, let
, and
, be bases for ''V'' and ''W'' respectively; then ''T''
has a matrix representation
:
relative to the basis
of
.
Now for indices ''k'', ''i'' in the range 1, ..., ''m'', consider the sum
:
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).
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
:
such that
:
To see that the conditions above determine the partial trace uniquely, let
form a basis for
, let
form a basis for
, let
be the map that sends
to
(and all other basis elements to zero), and let
be the map that sends
to
. Since the vectors
form a basis for
, the maps
form a basis for
.
From this abstract definition, the following properties follow:
:
:
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 In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback.
A traced symmetric monoidal category is a symmetric monoidal category C together with a family of functions
:\math ...
. A traced monoidal category is a monoidal category
together with, for objects ''X, Y, U'' in the category, a function of Hom-sets,
:
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
there exists a corresponding "partially traced" bijection
.
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
:
be an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
for ''W''. Now there is an isometric isomorphism
:
Under this decomposition, any operator
can be regarded as an infinite matrix
of operators on ''V''
:
where
.
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
:
converges in the
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
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.
Computing the partial trace
Suppose ''W'' has an orthonormal basis, which we denote by
ket vector notation as
. Then
:
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
:
commutes with all operators of the form
and hence is uniquely of the form
. 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
In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discusse ...
. Consider a quantum mechanical system whose state space is the tensor product
of Hilbert spaces. A mixed state is described by a
density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
ρ, that is
a non-negative trace-class operator of trace 1 on the tensor product
The partial trace of ρ with respect to the system ''B'', denoted by
, is called the reduced state of ρ on system ''A''. In symbols,
:
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
. However one chooses to define a reduced state
, there should be consistency of measurement statistics. The expectation value of ''M'' after the subsystem ''A'' is prepared in
and that of
when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:
:
We see that this is satisfied if
is as defined above via the partial trace. Furthermore, such operation is unique.
Let ''T(H)'' be the
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
of trace-class operators on the Hilbert space ''H''. It can be easily checked that the partial trace, viewed as a map
:
is completely positive and trace-preserving.
The density matrix ρ is
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
,
positive semi-definite, and has a trace of 1. It has a
spectral decomposition:
:
Its easy to see that the partial trace
also satisfies these conditions. For example, for any pure state
in
, we have
:
Note that the term