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Quantum%20Algorithms

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Title: Quantum Algorithms Author: Artur Ekert Last modified by: Michele Mosca Created Date: 6/3/1997 12:46:02 AM Document presentation format: Letter Paper (8.5x11 in) – PowerPoint PPT presentation

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Title: Quantum%20Algorithms


1
Introduction to Quantum Information Processing
Lecture 4
Michele Mosca
2
Overview
  • Von Neumann measurements
  • General measurements
  • Traces and density matrices and partial traces

3
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

4
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

5
Expected value of an observable
  • If we associate with outcome the
    eigenvalue then the expected outcome is

6
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

7
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)

8
In section 2.2.5
  • This partial measurement corresponds to measuring
    the observable

9
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

10
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

11
Von Neumann Measurements
  • Note that

12
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.

13
How do we implement Von Neumann measurements?
  • Construct a quantum network that implements the
    unitary transformation
  • Then conjugate the measurement operation with
    the operation

14
Another approach
15
Ex. Bell basis change
  • Consider the orthonormal basis consisting of the
    Bell states
  • Note that

16
Bell measurement
  • We can destructively measure
  • Or non-destructively project

17
Most general measurement
18
Trace of a matrix
The trace of a matrix is the sum of its diagonal
elements
e.g.
Some properties
Orthonormal basis
19
Density Matrices
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 ??
20
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)
21
Density matrix of a mixed state
then the probability of measuring 0 is given by
conditional probability
Density matrices contain all the useful
information about an arbitrary quantum state.
22
Density Matrix
If we apply the unitary operation U to the
resulting state is with density matrix
23
Density Matrix
If we apply the unitary operation U to the
resulting state is with density matrix
24
Density Matrix
If we perform a Von Neumann measurement of the
state wrt a basis containing
, the probability of obtaining
is
25
Density Matrix
If we perform a Von Neumann measurement of the
state wrt a basis containing
the probability of obtaining
is
26
Density Matrix
In other words, the density matrix contains all
the information necessary to compute the
probability of any outcome in any future
measurement.
27
Spectral decomposition
  • 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

28
Spectral decomposition
  • 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
29
Partial Trace
  • How can we compute probabilities for a partial
    system?
  • E.g.

30
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.

31
Partial Trace
32
Why?
  • the probability of measuring e.g. in the
    first register depends only on

33
Partial Trace
  • Notice that it doesnt matter in which
    orthonormal basis we trace out the 2nd system,
    e.g.
  • In a different basis

34
Partial Trace
35
Distant transformations dont change the local
density matrix
  • Notice that the previous observation implies that
    a unitary transformation on the system that is
    traced out does not affect the result of the
    partial trace
  • I.e.

36
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.

37
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.

38
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

39
Partial Trace
  • This 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

40
Partial Trace using matrices
  • Tracing out the 2nd system

41
Most general measurement
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