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Quantum Reference Frames
and the
Classification of
Rotationally-Invariant Maps
Jean Christian Boileau (University of Toronto)
Lana Sheridan (University of Waterloo)
Martin Laforest (University of Waterloo)
Stephen Bartlett (University of Sydney)
Reference : arXiv:0709.0142
Presented at QIP’08
December 17th 2007
Topics:
 Introduce a representation for any map which
is covariant with respect to an irreducible
representation of SU(2).
 Use this representation to study the dynamics
of quantum directional reference frame
 Introduce and analyze the moments, quality and longevity
of a QDRF.
 Examples
 Conclusion
Part I:
The classification of
Rotationally-Invariant Maps
Background
Math
SU(2): set of 2 by 2
unitary complex matrices
with determinant one
Physics
Rotations in Space
 SU(2) is isomorphic (up to a sign) to the group of space rotations SO(3)
Consider an irreducible
representation of SU(2)
acting on a (2j+1)dimensional Hilbert
space.
Generators of the Lie
algebra of SU(2):
Jx, Jy and Jz.
Consider rotations on a
single spin-j.
Angular momentum
operators for the x-, yand z-axis.
Covariance With Respect to SU(2)
 We say that a map
respect to SU(2) iff
is covariant with
for all
i.e. R is the irrep of SU(2) for some (2j+1)-dimensional Hilbert space.
Representation for maps which are
covariant with respect to an irreducible
representation of SU(2)
 Define
where
is a 2j+1 by 2j+1 density matrix.
Theorem 1: Any map which is covariant with respect to a
irreducible representation of SU(2) acting on an Hilbert
space of dimension 2j+1 has the form
for some real coefficients
.
Sketch of Proof
• First show that
respect to SU(2).
is covariant with
 Easy to show by using the fact that any SU(2) element can be
decomposed into rotations around the Y and Z axes.
• Therefore any map of the form
must also
be covariant.
• To prove that every covariant maps can be written as
above, we use the Liouville representation where density
matrices are represented by vectors and
• Show that there is 2j+1 independent that describe maps of
the form
.
• It is known that there is no more free parameters.
Open Questions
 What about covariance with respect to other
Lie Group?
 What are the other restrictions on the qn
parameters?
Part II:
Dynamics of Quantum
Directional Reference Frame
What is a Quantum Directional
Reference Frame?
 Consider the initial state of a spin-j:
 Suppose that
depends only on some
“classical” direction
The state
about the
is invariant under rotations
-axis.
 Therefore,
is diagonal in the basis
consisting of the eigenvectors of .
Scenario
Quantum
Reference Frame
Reservoir: contains many
identical subsystems of
dimension d.
1
2
3
apply the map
1
1
1
...
Scenario
Quantum
Reference Frame
Reservoir: contains many
identical subsystems of
dimension d.
2
X
1
X
3X
3
... n-1
X
n
...
apply the map
X: discarded
n
n
n
Extra Assumptions:
 The joint map
is rotationally-invariant.
 The state of the subsystems in the reservoir are
invariant under space-rotations.
 This implies that the back-action map
on the
quantum reference frame is rotationally-invariant.
 Which implies that
for all k is diagonal in the basis
given by the eigenvectors of
Previous Related Works
•
S. Bartlett, T. Rudolph, R. Spekkens, and P. Turner, Degradation
of a Quantum Reference Frame, New J. Phys. 8, 58 (2006).
D. Poulin and J. Yard, Dynamics of a Quantum Reference
Frame, New J. Phys. 9, 156 (2007).
•
1)
2)
3)
The joint operator () considered is restricted to
measurements
d=2
the quality function is fixed (somewhat arbitrarily).
In 2), the states of the subsystems of the reservoir are not necessarily
rotationally-invariant.
 We use the term quality function for any
function F that is meant to quantify the ability of
the reference frame to perform a particular task.
 The quality measure should not be biased such
that it favors a quantum reference frame that is
pointed in any particular direction relative to some
external frame. All directions must be equally
valid. Therefore, F does not depend on the
direction of
, but only on the eigenvalues of
Moments
 An equivalent set of parameters to the
eigenvalues of
are the moments of
:
 Any quality function F depends only on
those moments.
 To analyze the behavior of F, it is sufficient
to study the evolution of the moments.
General Recursion Formula for
the Moments
where the
’s are real coefficients.
Get ride of some of the
coefficients
Theorem 2:
If
is even, then
and if
is odd, then
Proof: Corollary of Theorem 1 (by induction using commutator relations).
Longevity
 We are interested in the scaling, with respect to
Hilbert space dimension, of how many times a
quantum reference frame can be used before the
value of its quality function F falls below a certain
threshold.
BRST06:
Longevity scales as O(j2). (specific F)
 Because we consider any quality function F,
the longevity of the reference frame can be
arbitrary (in general).
 But we can study the scaling of the moments.
Theorem 3: Consider a quantum reference frame
with initial state
, which is used for performing
a rotationally-invariant joint operation . If this
operation induces a disturbance map
that satisfies the following assumptions:
 there exists some nmax such that qn=0 for all n
nmax,
 qn O(1) and

.
then the number of times that such a quantum
reference frame can be used before its moment
falls below a certain threshold value scales as j2.
The proof is based on Theorem 2.
Example A.1: Measurements of
spin-1/2
 Suppose that the reservoir consists of spin-1/2
systems. Each spin is either parallel or anti-parallel to
(with the same probability).
 The goal is to use the quantum reference frame to
guess the direction of each spin-1/2.
 The optimal joint measurement  is a projection onto
the subspaces corresponding to different values of the
total angular momentum.

Measurement of spin-1/2
 In term of the first moment, we can rewrite the
quality function:
 Theorem 2 tells us
z
 Simple calculation give us
Example A.2: Measurements of
spin-1
 Suppose that the reservoir consists of spin-1 systems.
Each spin has either 1, 0 or -1 angular momentum in
the direction (with the same probability).
 The goal is to use the quantum reference frame to
guess the angular momentum in the direction of
each spin-1.
 The optimal joint measurement  is a projection onto
the subspaces corresponding to different values of the
total angular momentum.

Measurements of spin-1
 In terms of moments:
 We can use again Theorem 2 to evaluate the
moments. Simple calculations give us the
values of the three A’s coefficients.
Example B: Pauli Operator
 Suppose, we want to implement a Pauli Z
operation on a qubit:
using a quantum reference frame to define the z-axis.
 Cannot be implemented without errors if the
reference frame is quantum and restricted to a
finite space.
Gate Fidelity
 We can pick the quality function to be
where E() are the Kraus operators of the
approximate gate.
Different Methods To Implement
the Gate
1. Projective Measurement
where
Different Methods To Implement
the Gate
2. Filtering operation:
=
(not unitary)
where k is the projector into the subspace of total
angular momentum k.
Different Methods To Implement
the Gate
3. Use coupling between the spins of the
quantum reference frame and of the
reservoir :
Use
to implement
(unitary)
Different Results
A plot of the gate fidelity with number of repetitions, n,
for j=8 for the three methods, (2.1) (dot-dashed line),
(2.2) (dashed line), and 2.3) (solid line). This behavior
of this value of j is representative
Longevity
Conclusion
 We provide a “polynomial” representation (Theorem 1) for any
map which is covariant with respect to an irreducible
representation of SU(2).
 We generalize the concept of quality function and introduce the
moments of a quantum reference frame.
 We give recursive equations (Theorem 2) for how the moments
evolve with the number of uses of the quantum reference frame.
 We derive sufficient conditions (Theorem 3) for the longevity of a
quantum reference frame to scale by a factor proportional to
square the dimension of the quantum reference frame.
Acknowledgements





Perimeter Institute
University of Sydney
Rob Spekkens
Terry Rudolph
CQIQC
 Funding agencies:






CIFAR
CIPI
MITACS
PREA
CFI
OIT
Reference : arXiv:0709.0142