Quantum Reference Frames and the Classification of RotationallyInvariant Maps

1
Quantum Reference Frames and theClassification
ofRotationally-Invariant Maps

• Jean Christian Boileau (University of Toronto)
• Lana Sheridan (University of Waterloo)
• Martin Laforest (University of Waterloo)
• Stephen Bartlett (University of Sydney)
• Reference arXiv0709.0142
• Presented at QIP08
• December 17th 2007

2
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

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Part IThe classification of Rotationally-Invari
ant Maps
4
Background
Math Physics
SU(2) set of 2 by 2 unitary complex matrices
with determinant one
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 (2j1)-dimensional Hilbert space.
Consider rotations on a single spin-j.
Angular momentum operators for the x-, y- and
z-axis.
Generators of the Lie algebra of SU(2)
Jx, Jy and Jz.
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Covariance With Respect to SU(2)

• We say that a map is covariant with respect
  to SU(2) iff

for all
i.e. R is the irrep of SU(2) for some
(2j1)-dimensional Hilbert space.
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Representation for maps which are covariant with
respect to an irreducible representation of SU(2)

• Define
• Theorem 1 Any map which is covariant with
  respect to a irreducible representation of SU(2)
  acting on an Hilbert space of dimension 2j1 has
  the form

where is a 2j1 by 2j1 density matrix.
for some real coefficients .
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Sketch of Proof

• First show that
  is covariant with respect to SU(2).
• 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 2j1 independent that describe
  maps of the form
• It is known that there is no more free
  parameters.

.
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Open Questions

• What about covariance with respect to other Lie
  Group?
• What are the other restrictions on the qn
  parameters?

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Part IIDynamics of Quantum Directional
Reference Frame
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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 is invariant under rotations
  about the -axis.
• Therefore, is diagonal in the basis
  consisting of the eigenvectors of .

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Scenario
Reservoir contains many identical subsystems of
dimension d.
Quantum Reference Frame
...
1
2
3
apply the map
1
1
1
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Scenario
Reservoir contains many identical subsystems of
dimension d.
Quantum Reference Frame
X
...
1
2
3
n-1
2
3
...
X
X
n
X
apply the map
X discarded
n
n
n
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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-invari
  ant.
• Which implies that for all k is diagonal
  in the basis given by the eigenvectors of

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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).
• The joint operator (?) considered is restricted
  to measurements
• d2
• the quality function is fixed (somewhat
  arbitrarily).

In 2), the states of the subsystems of the
reservoir are not necessarily rotationally-invaria
nt.
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• 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

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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.

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General Recursion Formula for the Moments
where the s are real coefficients.
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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).
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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.
• 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.

BRST06 Longevity scales as O(j2). (specific
F)
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• 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 qn0 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.
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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.

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Measurement of spin-1/2

• In term of the first moment, we can rewrite the
  quality function
• Theorem 2 tells us
• Simple calculation give us

z
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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.

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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 As coefficients.

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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.

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Gate Fidelity

• We can pick the quality function to be
• where E(?)?are the Kraus operators of the
  approximate gate.

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Different Methods To Implement the Gate

• 1. Projective Measurement

where
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Different Methods To Implement the Gate

• 2. Filtering operation

(not unitary)
where ?k is the projector into the subspace of
total angular momentum k.
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Different Methods To Implement the Gate

• 3. Use coupling between the spins of the quantum
  reference frame and of the reservoir

Use
(unitary)
to implement
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Different Results
A plot of the gate fidelity with number of
repetitions, n, for j8 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
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Longevity
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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.

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Acknowledgements

• Funding agencies
• CIFAR
• CIPI
• MITACS
• PREA
• CFI
• OIT

• Perimeter Institute
• University of Sydney
• Rob Spekkens
• Terry Rudolph
• CQIQC

Reference arXiv0709.0142