Quantum Reference Frame Resources and Monotones

Transcription

Quantum Reference Frame
Resources
and
Monotones
Borzu Toloui,
Joint work with Gilad Gour, Barry Sanders
Institute for Quantum Information Science at the University of Calgary
September
23, 2010
1
Contents
1.
2.
3.
4.
5.
6.
Introduction
Quantum Reference Frames
Frameness Measures
Concurrence Monotones
Formation
Distillation
2
Contents
1.
2.
3.
4.
5.
6.
Introduction
Frameness Measures
Formation
Distillation
3
Coherence Controversy
Coherent states:
Assumption: Atoms in the gain medium in energy eigenstates.
Conservation of Energy leads to
Tracing out the gain medium results in
Stephen D. Bartlett, Terry Rudolph, and Robert W. Spekkens,
Dialogue Concerning Two Views on Quantum Coherence: Factist and Fictionist,
Int. J. of Quantum Information 4, 17 (2006) [arXiv:quant-ph/0507214]
4
Experiments to detect coherence
Homodyne detection
5
Homodyne detection
Shows coherence
between states with
different relative
number
6
A classical clock can be used
to generate and detect coherence:
Homodyne detection
Use a classical oscillating current
For the local oscillator
Shows coherence
between states with
different relative
number
7
Contents
1.
2.
3.
4.
5.
6.
Introduction
Frameness Measures
Formation
Distillation
8
1. Introduction
Lack of a shared reference frame
9
1. Introduction
ce
i
l
A
b
o
B
ce
i
l
A
b
Bo
10
1. Introduction
G-Twirling
ce
i
l
A
b
o
B
The relative alignment is unknown.
The channel averages over all alignments.
This imposes a Superselection Rule (SSR).
11
1. Introduction
G-Twirling
ce
i
l
A
b
o
B
This Imposes Superselection Rule (SSR)
on operations as well.
12
1. Introduction
G-Twirling
ce
i
l
A
b
o
B
13
Hilbert Space:
States:
Superoperations:
,
Kraus Operators:
14
1. Introduction
G-Twirling
ce
i
l
A
b
o
B
15
1. Introduction
G-Twirling
ce
i
l
A
b
o
B
X
Resource state
16
1. Introduction
ce
i
l
A
States that do not remain
invariant under twirling,
encode information about
the reference frame.
b
o
B
17
1. Introduction
ce
i
l
A
They are resources.
Resource state
“Quantum frame”
b
o
B
18
Internal Quantum Reference Frame
*S. D. Bartlett, T. Rudolph and R. W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)
19
X
X
20
4. Formation
Example U(1)
The group is associated with phase reference frames.
Unitary representation
Every state can be transformed by the allowed operators into
a standard form
Invariant Operations
And states:
Kraus Operators:
* G. Gour and R. W. Spekkens, NJP 10, 033023 (2008)
21
4. Formation
Example U(1)
22
4. Formation
Example U(1)
Examples:
Allowed by unitary:
Allowed by non-unitary:
Not Allowed:
23
4. Formation
Frameness Cost
Asymptotic transformation rates
Preparation:
is an example of a refbit
24
4. Formation
U(1) Asymptotic Frameness
Phase U(1)
All states with the same symmetry can be transformed to each other
asymptotically in a reversible manner.
States with different symmetries cannot be transformed to each
other.
However we can use catalysts.
25
4. Formation
Example U(1)
Asymptotic transformations:
26
2. Mixed States
Lack of Shared Reference Frame
Mixed states
ce
i
l
A
b
o
B
Noisy G-twirling channel
27
2. Mixed States
Lack of Shared Reference Frame
Mixed states
ce
i
l
A
b
o
B
This is equivalent to Alice transmitting
mixed states in a noiseless
G-twirling channel
28
Contents
1.
2.
3.
4.
5.
6.
Introduction
Frameness Measures
Formation
Distillation
29
2. Mixed States
Measures of Frameness
For any state and any set of
group-invariant operations
30
2. Mixed States
Measures of Frameness for
Mixed states
Convex roof extensions:
1. For any state and any set of
group-invariant operations
2. For any ensemble
31
Contents
1.
2.
3.
4.
5.
6.
Introduction
Frameness Measures
Formation
Distillation
32
3. Concurrence
Concurrence of frameness
Concurrence of frameness for qudits for Abelian groups
A family of concurrence monotones:
33
3. Concurrence
Concurrence
Concurrence of entanglement for qudits
34
3. Concurrence
Concurrence of frameness
Concurrence of frameness for qudits for Abelian groups
Concurrence
of frameness
35
3. Concurrence
Concurrence of Formation of Frameness
This transformation can be generalized to mixed states
The average concurrence of frameness needed to form the
mixed state
36
3. Concurrence
Concurrence of formation
Theorem
For a qubit state
formation of frameness is equal to
where
and
, the concurrence of
are the eigenvalues of the operator
in the standard basis.
37
38
39
40
1. Introduction
ce
i
l
A
They are resources.
Resource state
“Quantum frame”
b
o
B
41
Contents
1.
2.
3.
4.
5.
6.
Introduction
Frameness Measures
Formation
Distillation
42
4. Formation
Frameness Cost
Preparation:
is an example of a refbit
43
4. Formation
Frameness of Formation
Average cost of preparing the state in refbits
44
4. Formation
Concurrence of Formation of Frameness
b
Bo
ce
i
l
A
Supplier of refbits
45
4. Formation
Theorem
For a qubit state
where
and
46
4. Formation
Theorem
For a qubit state
where
and
Corollary
If the frameness
can be expressed as a non-decreasing
and convex function of
, then the frameness of formation of
a qubit state has the same functional dependence on
derived above.
47
4. Formation
Example U(1)
A qubit in the standard form
with
The frameness for this state is
Refbit:
This is also a convex and increasing function of C, so
The frameness of formation of a qubit is
48
4. Formation
Example Z2
The group is associated with parity degree of freedom.
Every pure state can be mapped into a qubit state, as a combination
of standard even (0) and odd (1) parity states
with
The frameness for this standard form is
This is a convex increasing function of C.
So the frameness of formation for a qubit is
49
ContentsContents
1.
2.
3.
4.
5.
6.
Introduction
Frameness Measures
Formation
Distillation
50
5. Distillation
Distillation rates
Distillation:
51
5. Distillation
Distillation protocol for qubits
…
52
5. Distillation
…
X
XX
X
…
53
5. Distillation
…
X
XX
X
…
54
5. Distillation
…
X
XX
X
X
…
…
55
5. Distillation
…
X
XX
X
X
…
…
56
5. Distillation
…
X
XX
X
X
…
…
57
Summary
1.
2.
3.
4.
5.
6.
7.
Lack of shared reference frames between parties imposes
superselection rules on states and operators.
Resources are states that can circumvent the restrictions.
Functions that behave monotonically on average quantify the
strength of mixed state resources.
A family of concurrence monotones exist for pure and mixed
states in an arbitrary but finite dimensional Hilbert space.
The asymptotic rate of preparation and distillation of a state to
refbits quantifies the state’s frameness.
For qubit states, a closed formula exists for the average
frameness cost.
We also have a distillation protocols for qubit states.
58
References
[1] S. D. Bartlett, T. Rudolph and R. W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)
[2] G. Gour and R. W. Spekkens, New Journal of Physics 10, 033023 (2008)
[3] S. D. Bartlett, T. Rudolph, R. W. Spekkens and P. S. Turner, New Journal of Physics
11, 063013 (2009)
[4] N. Schuch, F. Verstraete and J. I. Cirac, Phys. Rev. A. 70, 042310 (2004)
[4] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)
[5] S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997)
[6] G. Gour, Phys. Rev. A 71, 012318 (2005)
[7] G. Gour, Phys. Rev. A 72, 042318 (2005)
[9] G. Vidal, quant-ph/9807077v2
Thank You.
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Quantum Reference Frame Resources and Monotones

Transcription

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