Introduction
It was a great realization that information is physical and that a
(classical) Turing machine is not the end of the story of
computation. The physical system in which the information is stored
and manipulated is important and qubits are quite different from
bits.
In this chapter, some background in quantum mechanics is provided.
Not all of this chapter will be directly relevant to our discussion,
but it is included for the sake of completeness of our understanding
of how quantum mechanics from a textbook is related to quantum
computing. The connection is, as of yet, clear but the story seems
incomplete from a physicists perspective and for the subject of error
prevention methods, some of this chapter will be vital. In
particular, the section(s) concerning the density matrix. Not only
is this vital, but not usually covered in most quantum mechanics
classes, either undergraduate or graduate.
It is also worth emphasizing that this chapter is primarily aimed at
physicists and for those others which are interested in the background
physics. It is not necessary for much of what follows.
Schrodinger's Equation
A common starting point in quantum mechanics is Schrodinger's
equation.
This equation is not derived, or justified here, but is
given in a general form:
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(3.1)
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where is the Hamiltonian,
is Planck's constant
(divided by ), and is time. The Hamiltonian contains what
is known about the system's evolution.
Most of the time in these notes, we let .
This equation is (formally) solved by taking the time derivative to be
an ordinary derivative (we assume no explicit time dependence for
), so
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(3.2)
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This means that
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(3.3)
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so
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(3.4)
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Now if is Hermitian, and it is, then the matrix
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(3.5)
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is unitary.
(See Appendix C - Vectors and Linear Algebra, in particular the section entitled Unitary Matrices.) Any
transformation on a closed system can be described by a unitary
transformation and any unitary transformation can be obtained by the
exponentiation of Hermitian matrix.
Aside, for inclusion later in an appendix about Group theory.
It may seem strange to exponentiate a matrix. However, you can define
a function of a matrix according to its Taylor expansion. The details
of this are primarily unimportant here, but just to show how it goes,
it is written out.
The Taylor expansion of an exponential is the following:
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(3.6)
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and this can be used to exponentiate a matrix by letting the matrix
replace in the equation. This can also be used to prove that
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(3.7)
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End Aside
The end result and important point is that the evolution of a quantum
state is, in general, given by a unitary matrix
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(3.8)
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So our objective in quantum information processing is to create a
unitary evolution, and eventual measurement, which will produce a
particular outcome.
Density Matrix for Pure States
Now let us consider the object (a density matrix, or
density operator, of rank one)
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(3.9)
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which is just the outter product of two vectors. For example, if
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(3.10)
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then
However,
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(3.11)
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Again , so . If we
differentiate this with respect to ,
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(3.12)
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which is the Schrodinger equation for the density matrix, with solution,
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(3.13)
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This follows from .
Consider our two-state system
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(3.14)
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A superposition of these two states is
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(3.15)
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where and are complex numbers such that
. The corresponding
pure state, (i.e. rank one) density matrix is given by
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(3.16)
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Note that the superposition in Eq.(3.15) can be obtained
from any pure state by a unitary transformation. Here, the trace of
the density matrix is an important quantity; it is
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(3.17)
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Notice also that the determinant of this matrix is zero:
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(3.18)
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To see this another way, note that the density operator of rank one
can be written as , so that the determinant
is
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(3.19)
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Measurements Revisited
If the state of a quantum system is described by
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(3.20)
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the probability of finding it in the state when measured in
the computational basis is . However, this is a
particular superposition which could be written as
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(3.21)
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In the section entitled Schrodinger’s Equation it was shown that this matrix results
from the exponentiation of a Hermitian matrix and from the section entitled The Pauli Matrices any
Hermitian matrix can be written in terms of the Pauli matrices. To make this explicit using standard conventions,
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(3.22)
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where is a unit vector, and .
One can write this matrix out explicitly
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(3.23)
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Notice this is a special unitary matrix. (See Appendix C - Vectors and Linear Algebra, in particular the subsection Unitary Matrices.)
To see that any state for arbitrary coefficients
, can be obtained by choosing and
appropriately, the state can be chosen as a starting point.
Then
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(3.24)
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For example, choosing gives the original state; choosing
and gives ; and choosing
and gives an equal superposition.
In general, when the system is in the state ,
the probability of finding the state when a measurement is made in the computational basis is given by
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(3.25)
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and the probability of finding is
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(3.26)
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Notice the probabilities add up to one if is a unit vector.
What this shows is that there is a transformation that takes the state
, which has probability of being in the state and
probability 0 of being in the state and transform it
(using a "rotation'' into a state with a different (and generic)
probability of each. This means that the density matrix corresponding
to this system always has determinant zero, meaning (for a two-state system) it has one
eigenvalue 1 and another eigenvalue 0. (The determinant is the
product of the eigenvalues.)
Density Matrix for Mixed States
For a system with dimensions, a mixed state density matrix
(or density operator, see Appendix \ref{app:cohvec}), is a matrix which us used to
describe a more general state of a quantum system and can be written
as
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(3.27)
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where , and the are pure states.
There is also a generalization of the Bloch sphere which is described
in Appendix {app:polvec}.
The mixed state density matrices are important in all descriptions of
physical implementations of quantum information processing.
For this reason, a bit of labor has gone into this section, the rest
of which is devoted to the physical interpretation and properties of
this description of a quantum system.
Density Matrix for a Mixed State: Two States
A mixed state density matrix (for a two-state system) is a rank two
density matrix, , which can be described by
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(3.28)
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where ,
and . The are probabilities and must sum to one.
(Note, if , or if one or one
is zero, this reduces to a pure state.) For
example, the probability of finding the state is
and the probability of finding the state is .
Density Matrix for the Description of Open Quantum Systems: An Example
One example of the utility of a density matrix is the following
statistical problem. Let us consider the collection of two-state
systems, this will be a collection of electrons in a box and their
spin is a two-state system, being either up or down when measured. If
a subset of these electrons was prepared in the state ''up'' before
being put in the box, and the rest ''down,'' then the description of
the system of particles is given by
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(3.29)
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where the fraction of ''up'' particles is and the fraction of ''down'' is . Our system is described by this density matrix because if a particle is chosen at random from the box and measured, the state of the particle is with probability
and with probability . This is known as the statistical
interpretation of the density operator.
There is another example which is more relevant for our purposes. For
a certain system (again a two-state system is take as an example)
if there is some probability for an error to occur, let us say our
example is a unitary operator , then the density matrix for the
system is
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(3.30)
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This is the same form as Eq.(\ref{eq:tsdmatex1}).
Note that in each
case the probabilities associated with the density matrix , and
, (generally, the ) are classical probabilities. That
is, they are associated with a classical probability distribution--the
probability for error/no error and up/down. These are not
probabilities associated with the superposition of the quantum state
in the equation
given by the square of the moduli of the coefficients. This is an
important distinction for the following reason. The state
can be taken to the state with a unitary
transformation. This state is deterministic in the sense that the
result will be obtained from a measurement in the
computational basis since there is no probability for obtaining
. However, for nonzero and a non-identity
operator , the matrix has rank two and thus can never have
probability for either of the two states, or .
Thus, we have maximum knowledge about a pure state since
there is a way to choose a measurement, perhaps after a unitary
transformation, which achieves a certain result with probability one.
For the mixed state density operator this is not possible. The state
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(3.31)
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for which we have the least amount of knowledge is called the
maximally mixed state. The
state could be either up or down with equal probability and neither is
a better guess. If the two eigenvalues are not equal, then there is a
better guess, or bet, as to the result of a measurement and if one
eigenvalue is zero, there is a definite best guess.
To be more specific, independent of basis (unitary transformations),
one always has probability greater than zero of measuring
and probability greater than zero of measuring
. Thus the state described by the density matrix is
a mixed state in the sense
that it can be considered a statistical mixture of the two states
and . This, because classical
probabilities are included separately, is significantly different from
the pure state density matrix, which is a special case of all density
matrices.
To see that mixtures remain after a unitary transformation on the
system, note that a unitary matrix does not change the eigenvalues.
This is because the eigenvalue equation is the same for a Hermitian
matrix and its corresponding diagonal matrix. Let , then
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(3.32)
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Open Quantum Systems
Open quantum systems are those which
interact with an "environment" or have some other outside influence.
Very often only closed quantum systems
are considered in intoductory quantum mechanics since open systems are
more difficult to treat. However, many quantum systems are, in
practice, open quantum systems since they interact with some
environment, or have some outside influence. When quantum systems are
treated as closed, the interacting constituents are considered to be
the whole system in the sense that they do not, to a good
approximation, interact with anything else.
As will be shown later, errors are not restricted to unitary errors.
Furthermore, errors which change the eigenvalues of a density matrix
cannot arise in closed quantum systems since a closed quantum system
evolved under a unitary transformation. Thus, any time an error is
discovered which changes the eigenvalues, it is due to an
environmental interaction. The general description of such errors,
which are not restricted to unitary errors, will be postponed until Chapter {ch:noise}.
General Properties
In general, a density matrix has the following properties:
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(3.33)
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If, in addition, it is a pure state, then
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(3.34)
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Eq.(\ref{eq:dmatpos}) really means that the eigenvalues of the
density matrix are greater than or equal to zero.
Two-State Example: Bloch Sphere
Since our interest is primarily in qubits, which are two-state
systems, we return again to an example.
A very convenient representation of two state density matrices, one
can written in the so-called Bloch sphere
representation given the fact that the density matrix is Hermitian,
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(3.35)
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where, for the density matrix to be positive , and the
are the Pauli matrices
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(3.36)
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The matrix entries on the RHS of this equation are called the Pauli matrices.
The eigenvalues are given by
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(3.37)
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When , the state is pure, i.e., that the matrix
has rank one since it has one eigenvalue one and one zero. If , the density matrix represents a mixed state since rank is
greater than one--there are two non-zero eigenvalues. These leads to
the following picture: the pure states lie on the surface of the
sphere (), and mixed states lie in the interior of
the sphere with the maximally mixed state at the origin. This is
supposedly due to Bloch. Hence the name Bloch sphere.
Using the condition that for a pure
state can also be determined. The square in the Bloch sphere
representation yields
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(3.38)
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and using
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(3.39)
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then if and only if . This technique is
used for higher dimensions.
Two density matrices and
, correspond to orthogonal
states when
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(3.40)
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This implies that
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(3.41)
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Since the magnitudes must be one, the orthogonal states correspond to
pure states on a surface of a sphere which are represented by
antipodal points.
Expectation Values
The expectation value
of an operator , is given by
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(3.42)
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and is the "average value" of the operator. For a pure state
, this reduces to
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(3.43)
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Entangled States
Quantum entanglement is the most uniquely quantum mechanical property
of quantum systems. It is also believed to be responsible for all of
the information-processing advantages quantum systems have over
classical systems. Entangled states puzzled even Einstein and other
founders of quantum theory, so it is not surprising that they have
become a central part of many investigations into quantum theory and
especially quantum information theory.
There are many open problems in this area of research. Some of the
most basic and fundamental questions about entangled states are still
unanswered. For example, given a mixed-state density matrix for a
quantum system, we still do not know how to tell if the systems being
described by that matrix are entangled or not. Also, with few
exceptions, we do not know how to quantify the entanglement in a
system of particles.
What we do understand and what we can explain, is the entanglement
between systems of particles which are describable by pure quantum
states. Thus this section is concerned with pure entangled states of
qubits. Extentions of these ideas will be discussed in this
section.
Entangled States vs. Separable States
Let us consider two particles, one will be called particle A and the
other particle B which are in pure states.
(It is also possible to separate a system into two
subsystems and call the subsystems A and B.) If the particles are
independent and have never interacted, then the state of the system of
the composite system of the two particles can be written as
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(3.44)
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where is the state of particle A, and is
the state of particle B. This is sometimes stated as a postulate of
quantum mechanics \cite{Nielsen/Chuang:book}.
The state, Eq.~(\ref{eq:simpsep}) is not entangled, so it is called separable. More precisely it is referred to as a simply separable state. In general, a state is separable if its density matrix can be written in the form
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(3.45)
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where is a valid density matrix for
subsystem , and .
An entangled state is one that cannot be written in the form of Eq.~(\ref{eq:defsepstate}).
For a pure state, the situation is simpler. A pure state is entangled if and only if it cannot be written in the form of Eq.~(\ref{eq:simpsep}). In other words a pure state is entangled if it cannot be written as the product of two states of the individual
subsystems.
For two particles (or systems) to become entangled, they must interact with each other. They could then be taken far apart and
still share their entanglement. This entanglement cannot increase by acting on an individual particle, or even both particles separately. Only joint measurements on both, or interactions between the two, can increase entanglement. Actions on an individual particle, without involving the other, are called local actions or local operations. Local operations on individual particles can be written as
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(3.46)
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so that
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(3.47)
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A common example, and one often used, is a local unitary transformation
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(3.48)
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Local unitary transformations will not change the entanglement of a system.
Notice that the density matrix for the composite system in
Eq.~(\ref{eq:simpsep}) is
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(3.49)
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where and so the density operator of a product state is
a product of density operators.
Bell States
The simplest examples of entangled states are the entangled states of
two two-state systems. There are four different versions of what the
"maximally entangled state" of two qubits. The "maximally" will
be explained below. These four different versions are called Bell states and are
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(3.50)
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These are an orthonormal ste of states and are all able to be obtained from each other by acting on one particle alone or both individual particles, i.e. acting with local unitary transformations. For example consider the local unitary transformation acting on . The result is . Acting with on gives , etc.
These states certainly cannot be written in the form
If they could, then, letting
and
, and notice that the general
form is
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(3.51)
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so the coefficient of times the coefficient of
minus the coefficient of times the coefficient
of is zero. This is not true for any of the Bell states,
therefore they cannot be written as a tensor product of two 1-particle
states. So, for some 2-particle state,
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(3.52)
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the state is separable, or unentangled, if . Otherwise it is entangled.
Reduced Density Operators and the Partial Trace
The Bell states are maximally entangled states. To understand this, one may consider the fact that these states are pure states, but that information about the individual particles in the system is lacking. In this section, a
more precise meaning of this statement is given.
Let us first consider a tool which will be useful, the partial trace. The partial trace is the
trace over one of the subsystems (particle states) in a composite system. Let us suppose that the density matrix for a composite system
is given by
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(3.53)
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The partial trace is the trace over one of the subsystems. For example the trace over subsystem is
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(3.54)
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since and the trace of a density matrix is one. The matrix is called the reduced density operator, or reduced density matrix.
However, this is a special case. The density matrix for a composite system of two (or more) subsystems cannot be written in this form
unless the two subsystems are not entangled.
For the cases where the two subsystems are entangled, there are at
least two ways to calculate the partial trace. One is to write
the matrix form of the state in terms of sum of tensor products of Pauli
matrices. (See Appendix {app:polvec}, Sec. {sec:2qdmat}.)
The other is to realize that the trace can be calculated by
summing the projections onto the diagonal elements of the subsystem
over which you are tracing. For example, for
a general density matrix , the trace is
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(3.55)
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For the general case, let us consider a density matrix for a bipartite
system, . Let the subsystem have Greek letters as indices and
let the subsystem have Latin indices
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(3.56)
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To calculate the reduced density matrix for subsystem , the trace
over is taken
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(3.57)
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For the partial trace of a density matrix over the
subsystem ,
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(3.58)
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which leaves the part of the matrix corresponding to alone,
and projects the part onto the two diagonal elements and then adds
those. Now let us calculate the partial trace of a Bell state, for example
,
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(3.59)
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which can be written simply as
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(3.60)
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This is quite an interesting and significant find. The density matrix
for the whole system of two qubits is in a pure state indicating
maximal knowledge. However, the reduced density matrix, representing
our knowledge of one of the individual particles, is completely, or
maximally mixed, indicating minimal knowledge. They way this is
interpreted is that the two particles or subsystems, taken together,
are in a definite state, a pure state, whereas taken separately, they
contain as little information as possible. This indicates
entanglement since the two states together make up a definite pure
state, but separately the two states contain as little information as
possible.
It is important to note that for a pure state, the trace over
subsystem produces the same result as the trace over subsystem
. In other words, for a pure state ,
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(3.61)
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How Entangled Is It?
Due to the previous discussion, there is a definite notion of
maximal entanglement for pure states. From the determinant condition,
there is a method for identifying unentangled pure states. One
question is, how entangled is it if it is not separable, nor maximally
entangled? There are now many ways of defining measures of
entanglement which will be explored in a later section and an
appendix. Here two very common ways of measuring the entanglement for
pure states are given. One based on each of the two properties
identified in this section: the partial trace and the determinant of
the pure state's coefficients.
Entanglement: Extensions and Open Problems