Richard Cleve

1
DENSITY MATRICES, traces, Operators and
Measurements
Lectures 10 ,11 and 12

• Richard Cleve

Michael A. Nielsen
Sources
Michele Mosca
2
Review Density matrices of pure states
We have represented quantum states as vectors
(e.g. ???, and all such states are called pure
states)
An alternative way of representing quantum states
is in terms of density matrices (a.k.a. density
operators)
The density matrix of a pure state ??? is the
matrix ? ??? ???
Example the density matrix of ??0? ??1? is
3
Reminder Trace of a matrix
The trace of a matrix is the sum of its diagonal
elements
e.g.
Some properties
Orthonormal basis
4
Example Notation of Density Matrices and traces
Notice that ?0?0??, and ?1?1??.
So the probability of getting 0 when measuring
?? is
where ? ???? is called the density matrix for
the state ??
5
Review Mixture of pure states
A state described by a state vector ?? is called
a pure state.
What if we have a qubit which is known to be in
the pure state ?1? with probability p1, and in
?2? with probability p2 ? More generally,
consider probabilistic mixtures of pure states
(called mixed states)
6
Density matrices of mixed states
A probability distribution on pure states is
called a mixed state ( (??1?, p1), (??2?, p2),
, (??d?, pd))
The density matrix associated with such a mixed
state is
Example the density matrix for ((?0?, ½ ), (?1?,
½ )) is
Question what is the density matrix of ((?0?
?1?, ½ ), (?0? - ?1?, ½ )) ?
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Density matrix of a mixed state (use of trace)
then the probability of measuring 0 is given by
conditional probability
Density matrices contain all the useful
information about an arbitrary quantum state.
8
Recap operationally indistinguishable states
Since these are expressible in terms of density
matrices alone (independent of any specific
probabilistic mixtures), states with identical
density matrices are operationally
indistinguishable
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Applying Unitary Operator to a Density Matrix of
a pure state
If we apply the unitary operation U to the
resulting state is with density matrix
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Density Matrix
Applying Unitary Operator to a Density Matrix of
a mixed state
How do quantum operations work for these mixed
states?
If we apply the unitary operation U to the
resulting state is with density matrix
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Operators on Density matrices of mixed states.
Thus this is true always
Effect of a unitary operation on a density
matrix applying U to ? still yields U? U
Effect of a measurement on a density matrix
measuring state ? with respect to the basis
??1?, ??2?,..., ??d?, still yields the k th
outcome with probability ??k?? ??k?
Why?
12
How do quantum operations work using density
matrices?
Effect of a measurement on a density matrix
measuring state ? with respect to the basis
??1?, ??2?,..., ??d?, yields the k th outcome
with probability ??k?? ??k?
(this is because ??k?? ??k? ??k??? ????k?
???k????2 )
and the state collapses to ??k? ??k?
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More examples of density matrices
The density matrix of the mixed state ((??1?,
p1), (??2?, p2), ,(??d?, pd)) is
Examples (from previous lecture)
1. 2. ?0? ?1? and -?0? - ?1? both have
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More examples of density matrices
Examples (continued)
has
...? (later)
7. The first qubit of ?01? - ?10?
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To Remember Three Properties of Density Matrices

• Three properties of ?
• Tr? 1 (Tr M M11 M22 ... Mdd )
• ? ? (i.e. ? is Hermitian)
• ???? ??? ? 0, for all states ???

Moreover, for any matrix ? satisfying the above
properties, there exists a probabilistic mixture
whose density matrix is ?
Exercise show this
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Use of Density Matrix and Trace to Calculate the
probability of obtaining state in measurement
If we perform a Von Neumann measurement of the
state wrt a basis containing
, the probability of obtaining
is
This is for a pure state. How it would be for a
mixed state?
17
Density Matrix
Use of Density Matrix and Trace to Calculate the
probability of obtaining state in measurement
(now for measuring a mixed state)
If we perform a Von Neumann measurement of the
state wrt a basis containing
the probability of obtaining
is
The same state
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Conclusion Density Matrix Has Complete
Information
In other words, the density matrix contains all
the information necessary to compute the
probability of any outcome in any future
measurement.
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Spectral decomposition can be used to represent a
useful form of density matrix

• Often it is convenient to rewrite the density
  matrix as a mixture of its eigenvectors
• Recall that eigenvectors with distinct
  eigenvalues are orthogonal
• for the subspace of eigenvectors with a common
  eigenvalue (degeneracies), we can select an
  orthonormal basis

20
Continue - Spectral decomposition used to
diagonalize the density matrix

• In other words, we can always diagonalize a
  density matrix so that it is written as

where is an eigenvector with eigenvalue
and forms an orthonormal basis
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Taxonomy of various normal matrices
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Normal matrices
Definition A matrix M is normal if MM MM
Theorem M is normal iff there exists a unitary U
such that M UDU, where D is diagonal (i.e.
unitarily diagonalizable)
Examples of abnormal matrices
is not even diagonalizable
is diagonalizable, but not unitarily
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Unitary and Hermitian matrices
with respect to some orthonormal basis
Normal
Unitary MM I which implies ?k 2 1, for
all k
Hermitian M M which implies ?k ? R, for all k
Question which matrices are both unitary and
Hermitian?
Answer reflections (?k ? 1,?1, for all k)
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Positive semidefinite matrices
Positive semidefinite Hermitian and ?k ? 0, for
all k
Theorem M is positive semidefinite iff M is
Hermitian and, for all ???, ??? M ??? ? 0
(Positive definite ?k gt 0, for all k)
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Projectors and density matrices
Projector Hermitian and M 2 M, which implies
that M is positive semidefinite and ?k ? 0,1,
for all k
Density matrix positive semidefinite and Tr M
1, so
Question which matrices are both projectors and
density matrices?
Answer rank-one projectors (?k 1 if k k0 and
?k 0 if k ? k0 )
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Taxonomy of normal matrices
If Hermitian then normal
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Review Bloch sphere for qubits
Consider the set of all 2x2 density matrices ?
They have a nice representation in terms of the
Pauli matrices
Note that these matricescombined with Iform a
basis for the vector space of all 2x2 matrices
We will express density matrices ? in this basis
Note that the coefficient of I is ½, since X, Y,
Z have trace zero
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Bloch sphere for qubits polar coordinates
We will express
First consider the case of pure states ??? ???,
where, without loss of generality, ???
cos(?)?0? e2i?sin(?)?1? (?, ? ? R)
Therefore cz cos(2?), cx cos(2?)sin(2?), cy
sin(2?)sin(2?)
These are polar coordinates of a unit vector (cx
, cy , cz) ? R3
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Bloch sphere for qubits location of pure and
mixed states
Note that orthogonal corresponds to antipodal here
Pure states are on the surface, and mixed states
are inside (being weighted averages of pure
states)
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General quantum operations
Decoherence, partial traces, measurements.
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General quantum operations (I)
General quantum operations are also called
completely positive trace preserving maps, or
admissible operations
condition
Let A1, A2 , , Am be matrices satisfying
Example 1 (unitary op) applying U to ?
yields U? U
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General quantum operations Decoherence
Operations
Example 2 (decoherence) let A0 ?0??0? and A1
?1??1?
This quantum op maps ? to ?0??0???0??0?
?1??1???1??1?
For ??? ??0? ??1?,
Corresponds to measuring ? without looking at
the outcome
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General quantum operations measurement operations
Example 3 (trine state measurement)
Let ??0? ?0?, ??1? ?1/2?0? ?3/2?1?, ??2?
?1/2?0? ? ?3/2?1?
Then
Condition satisfied
We apply the general quantum mapping operator

• The probability that state ??k? results in
  outcome state Ak is 2/3.
• This can be adapted to actually yield the value
  of k with this success probability

34
General quantum operations Partial trace
discards the second of two qubits
Example 4 (discarding the second of two qubits)
Let A0 I??0? and A1
I??1?
We apply the general quantum mapping operator
State ? ?? becomes ?
State
becomes
Note 1 its the same density matrix as for
((?0?, ½), (?1?, ½))
Note 2 the operation is the partial trace Tr2 ?
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Distinguishing mixed states
Several mixed states can have the same density
matrix we cannot distinguish between them.
How to distinguish by two different density
matrices?
Try to find an orthonormal basis ??0?, ??1? in
which both density matrices are diagonal
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Distinguishing mixed states (I)
Whats the best distinguishing strategy between
these two mixed states?
?1 also arises from this orthogonal mixture
as does ?2 from
?/8180/822.5
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Distinguishing mixed states (II)
Densi?ty matrices ?1 and ?2 are simultaneously
diagonalizable
Weve effectively found an orthonormal basis
??0?, ??1? in which both density matrices are
diagonal
??1?
?1?
??
??0?
Rotating ??0?, ??1? to ?0?, ?1? the scenario can
now be examined using classical probability
theory
?0?
Distinguish between two classical coins, whose
probabilities of heads are cos2(?/8) and ½
respectively (details exercise)
Question what do we do if we arent so lucky to
get two density matrices that are simultaneously
diagonalizable?
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Reminder Basic properties of the trace
The trace of a square matrix is defined as
It is easy to check that
and
The second property implies
Calculation maneuvers worth remembering are
and
Also, keep in mind that, in general,
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Partial Trace

• How can we compute probabilities for a partial
  system?
• E.g.

Partial measurement
40
Partial Trace

• If the 2nd system is taken away and never again
  (directly or indirectly) interacts with the 1st
  system, then we can treat the first system as the
  following mixture
• E.g.

From previous slide
41
Partial Trace we derived an important formula to
use partial trace
Derived in previous slide
42
Why?

• the probability of measuring e.g. in the
  first register depends only on

43
Partial Trace can be calculated in arbitrary basis

• Notice that it doesnt matter in which
  orthonormal basis we trace out the 2nd system,
  e.g.

• In a different basis

44
Partial Trace
(cont) Partial Trace can be calculated in
arbitrary basis
Which is the same as in previous slide for other
base
45
Methods to calculate the Partial Trace

• Partial Trace is a linear map that takes
  bipartite states to single system states.
• We can also trace out the first system
• We can compute the partial trace directly from
  the density matrix description

46
Partial Trace using matrices

• Tracing out the 2nd system

Tr 2
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Examples Partial trace (I)
Two quantum registers (e.g. two qubits) in states
? and ? (respectively) are independent if then
the combined system is in state ? ? ??
In such circumstances, if the second register
(say) is discarded then the state of the first
register remains ?
In general, the state of a two-register system
may not be of the form ? ?? (it may contain
entanglement or correlations)
We can define the partial trace, Tr2 ? , as the
unique linear operator satisfying the identity
Tr2(? ??) ?
For example, it turns out that
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Examples Partial trace (II)
Weve already seen this defined in the case of
2-qubit systems discarding the second of two
qubits Let A0 I??0? and
A1 I??1?
For the resulting quantum operation, state ? ??
becomes ?
For d-dimensional registers, the operators are Ak
I???k? , where ??0?, ??1?, , ??d?1? are an
orthonormal basis
As we see in last slide, partial trace is a
matrix. How to calculate this matrix of partial
trace?
49
Examples Partial trace (III) calculating
matrices of partial traces
For 2-qubit systems, the partial trace is
explicitly
and
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Unitary transformations dont change the local
density matrix

• A unitary transformation on the system that is
  traced out does not affect the result of the
  partial trace
• I.e.

51
Distant transformations dont change the local
density matrix

• In fact, any legal quantum transformation on the
  traced out system, including measurement (without
  communicating back the answer) does not affect
  the partial trace
• I.e.

52
Why??

• Operations on the 2nd system should not affect
  the statistics of any outcomes of measurements on
  the first system
• Otherwise a party in control of the 2nd system
  could instantaneously communicate information to
  a party controlling the 1st system.

53
Principle of implicit measurement

• If some qubits in a computation are never used
  again, you can assume (if you like) that they
  have been measured (and the result ignored)
• The reduced density matrix of the remaining
  qubits is the same

54
POVMs (I)
Positive operator valued measurement (POVM) Let
A1, A2 , , Am be matrices satisfying
Then the corresponding POVM is a stochastic
operation on ? that, with probability
produces the outcome j
(classical information)
(the collapsed quantum state)
Example 1 Aj ??j???j? (orthogonal projectors)
This reduces to our previously defined
measurements
55
POVMs (II) calculating the measurement outcome
and the collapsed quantum state
When Aj ??j???j? are orthogonal projectors and
? ??????,
Tr??j???j???????j???j? ??j???????j???j??j?
???j????2
probability of the outcome
Moreover,
(the collapsed quantum state)
56
The measurement postulate formulated in terms of
observables
This is a projector matrix
57
The measurement postulate formulated in terms of
observables
The same
58
An example of observables in action
59
An example of observables in action
60
What can be measured in quantum mechanics?
Computer science can inspire fundamental
questions about physics.
We may take an informatic approach to physics.
(Compare the physical approach to information.)
Problem What measurements can be performed in
quantum mechanics?
61
What can be measured in quantum mechanics?

• Traditional approach to quantum measurements
• A quantum measurement is described by an
  observable M
• M is a Hermitian operator acting on the state
  space
• of the system.

Measuring a system prepared in an eigenstate of M
gives the corresponding eigenvalue of M as
the measurement outcome.
The question now presents itself Can every
observable be measured? The answer theoretically
is yes. In practice it may be very awkward, or
perhaps even beyond the ingenuity of the
experimenter, to devise an apparatus which could
measure some particular observable, but the
theory always allows one to imagine that the
measurement could be made. - Paul A. M. Dirac
62
Von Neumann measurement in the computational
basis

• Suppose we have a universal set of quantum
  gates, and the ability to measure each qubit in
  the basis
• If we measure we get with probability

63
In section 2.2.5, this is described as follows

• We have the projection operators and
  satisfying
• We consider the projection operator or
  observable
• Note that 0 and 1 are the eigenvalues
• When we measure this observable M, the
  probability of getting the eigenvalue is
  and we are in that case left with the state

64
What is an Expected value of an observable

• If we associate with outcome the
  eigenvalue then the expected outcome is

65
Von Neumann measurement in the computational
basis

• Suppose we have a universal set of quantum
  gates, and the ability to measure each qubit in
  the basis
• Say we have the state
• If we measure all n qubits, then we obtain with
  probability
• Notice that this means that probability of
  measuring a in the first qubit equals

66
Partial measurements

• If we only measure the first qubit and leave the
  rest alone, then we still get with probability
• The remaining n-1 qubits are then in the
  renormalized state

• (This is similar to Bayes Theorem)

67
Most general measurement
68
In section 2.2.5

• This partial measurement corresponds to measuring
  the observable

69
Von Neumann Measurements

• A Von Neumann measurement is a type of projective
  measurement. Given an orthonormal basis , if we
  perform a Von Neumann measurement with respect
  to of the state then we measure with
  probability

70
Von Neumann Measurements

• E.x. Consider Von Neumann measurement of the
  state with respect to the orthonormal
  basis
• Note that

• We therefore get with probability

71
Von Neumann Measurements

• Note that

72
How do we implement Von Neumann measurements?

• If we have access to a universal set of gates and
  bit-wise measurements in the computational basis,
  we can implement Von Neumann measurements with
  respect to an arbitrary orthonormal basis as
  follows.

73
How do we implement Von Neumann measurements?

• Construct a quantum network that implements the
  unitary transformation

• Then conjugate the measurement operation with
  the operation

74
Another approach
These two approaches will be illustrated in next
slides
75
Example Bell basis change

• Consider the orthonormal basis consisting of the
  Bell states

• Note that

We discussed Bell basis in lecture about
superdense coding and teleportation.
76
Bell measurements destructive and non-destructive

• We can destructively measure

• Or non-destructively project

77
Most general measurement
78
Simulations among operations general quantum
operations
Fact 1 any general quantum operation can be
simulated by applying a unitary operation on a
larger quantum system
U
?
?
output
input
?0?
discard
?0?
discard
zeros
?0?
Example decoherence
79
Simulations among operations simulations of POVM
Fact 2 any POVM can also be simulated by
applying a unitary operation on a larger quantum
system and then measuring
U
?
?
quantum output
input
?0?
j
classical output
?0?
?0?
80
Separable states
A bipartite (i.e. two register) state ? is a

• product state if ? ???
• separable state if

( p1 ,, pm ? 0)
(i.e. a probabilistic mixture of product states)
Question which of the following states are
separable?
81
Continuous-time evolution
Although weve expressed quantum operations in
discrete terms, in real physical systems, the
evolution is continuous
?1?
Let H be any Hermitian matrix and t ? R
Then eiHt is unitary why?
?0?
H UDU, where
Therefore eiHt U eiDt U
(unitary)
82
Partially covered in 2007

• Density matrices and indistinguishable states
• Taxonomy of normal operators
• General Quantum Operations
• Distinguishing states
• Partial trace
• POVM
• Simulations of operators
• Separable states
• Continuous time evolution