Remote operations and their application to measurement of nonlocal variables: Stateoperator approach

1
Remote operations and their applicationto
measurement of nonlocal variablesState-operator
approach

• Berry Groisman, Benni Reznik and Yakir Aharonov
• School of Physics and Astronomy, Tel Aviv
• University, Tel Aviv 69978, Israel

We present a systematic method for constructing
deterministic remote operations on single and
multiple systems of arbitrary discrete
dimensionality. These operations include remote
rotations, remote interactions and measurements.
We further apply our method of remote rotations
to the measurement of nonlocal operators. We have
reconsidered the measurability problem for
nonlocal operators which have been previously
considered to be in conflict with relativistic
causality and hence unmeasurable. We argue that
by weakening the preparation role of ideal
measurements, many of these observables become
measurable.
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How can Alice perform a Remote Unitary
Transformation on Bob's system?
t
?
Device performing unitary transformation
system in unknown state
x
t
Method 1 Teleportation
Back teleportation
Alice gets the Bobs system in her hands
Unitary transformation
Forward teleportation
x
3
Method 2 Remote rotation
Alice knows the angle
Bob knows the axis.
z
y
x
z
2a
y
x
4
The State-Operator ( Stator)
is a hybrid linear construction of states in
Alice's Hilbert space and operators acting on
Bob's system
Stator describes quantum correlations between
states on one side and operators on the other
side
2-level stator
A
act on
A
B
S always goes in pair with B
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Preparation of Stator
t
sz
B
If
1 cbit
Meas.
a
b
b
a
a
B
b
x
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What can we do with Stators?

• Remote unitary operations
• Quantum cryptography

Stators minimize the resources
entanglement and classical
communication

• Remote operations on N systems
• Distributed quantum computation

• Remote interactions and
• 
  measurements-
• Quantum computer

• Instantaneous measurements of
• nonlocal
  variables

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Why is the Stator useful?
S satisfies the eigenoperator equation
Stator simplify considerably the construction
of remote unitary operations and interactions
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How does the remote rotation work?
t
a
x
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How to construct a remote unitary transformation
on n-level system ?
3-level system
The generator of rotation around z-axis Lz
An eigenoperator equation for Lz ASLzS
?
?
0
V
2
1
A2SLz2S
2 eigenoperators Lz , Lz2
Resources 2 etrits 2 nits
The most general
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Remote operation on N distributed systems
2 spin-half particles
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Remote interactions and measurements
Remote CNOT
t
B
sza
Meas.
x
12
Instantaneous probabilistic rotations
t
No final correction syB
?
a
classical communication
x
t
1/2
1/2
a
x
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Instantaneous measurement
c?tltltL
Stage III Interpretation of local outcomes
IB
Stage II Local interactions, measurements and
classical recording of local outcomes
local outcomes
t0?t
t0
Stage I Preparation of measurement devices
L
Space-like regions
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A measurement of an observable prepare
eigenstate of that same observable.
1. Nondemolition measurement
Measurable
Bell (maximally entangled) eigenstates
Unmeasurable
nonlocality without entanglement
non-maximally entangled eigenstates
Bennett C H, DiVincenzo D P, Fuchs C A, Mor T,
Rains E, Shor P W, Smolin J A and Wootters W K,
(1999).
B. Groisman and L. Vaidman (2001)
J. Walgate, L. Hardy (2002)
4x4 twisted basis
S. Popescu and L. Vaidman (1994)
D. Beckman, D. Gottesman, M.A. Nielsen and J.
Preskill (2001)
A measurement of an observable does not
necessarily prepare eigenstates of that same
observable.
2. Demolition measurement
All nonlocal variables are measurable !
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How does it work?
Instantaneous demolition measurement c?tltltL
The result of nonlocal measurement
t1
IB
local outcomes
Step II Trivial local measurements
Step I instantaneous transformation of
original nonmeasurable set of eigenstates to
measurable set
Classical information is splite between A and B
t0
L
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2 x2 twisted product basis
Step II Bob performs a trivial measurement in
z-basis
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2 x2 twisted product basis
Step II Bob performs a trivial measurement in
z-basis
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General2 x2 twisted product basis
More complicated, because
probability 1/2
probability 1/2
Need to be corrected
J. I. Cirac, W. D?r, B. Kraus and M.
Lewenstein, Phys. Rev. Lett, 86, 544 (2001).
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References

• Benni Reznik, Yakir Aharonov, and Berry
  Groisman, Remote
• operations and interactions for systems of
  arbitrary-dimensional
• Hilbert space State-operator approach,
• Phys. Rev. A 65, 032312 (2002).
• Berry Groisman and Benni Reznik, Measurements of
  semi-local
• and non-maximally entangled states,
  quant-ph/0111012
• to be published in Phys. Rev. A.