Transcript single

School of Physics
something
and Astronomy
FACULTY OF MATHEMATICAL
OTHER
AND PHYSICAL SCIENCES
Nonlocality of a single particle
Jacob Dunningham
Vlatko Vedral
Paraty, 16 August 2007
Nonlocality is usually confirmed in an EPR-type experiment
Alice and Bob look for “better than perfect correlations” in their
measurements
Two or more particles in the system
“…. I would not call that one but rather the characteristic trait of quantum
mechanics, the one that enforces its entire departure from classical lines of
thought. By the interaction the two representatives [the quantum states] have
become entangled."
Schrödinger (Cambridge Philosophical Society, 1935)
It would be strange if the characteristic trait of quantum mechanics
did not apply to a single particle, but was an emergent property of
two or more particles.
Single particle nonlocality
• Tan, Walls and Collett (1991) first suggested that a single particle could
exhibit nonlocality
Single particle nonlocality
• Tan, Walls and Collett (1991) first suggested that a single particle could
exhibit nonlocality
• Lucien Hardy (1994) tightens up some assumptions in TWC
Single particle nonlocality
• Tan, Walls and Collett (1991) first suggested that a single particle could
exhibit nonlocality
• Lucien Hardy (1994) tightens up some assumptions in TWC
• Greenberger, Horne, and Zeilinger (GHZ) raise objections: not a real
experiment/ multiparticle effect in disguise (1995)
Single particle nonlocality
• Tan, Walls and Collett (1991) first suggested that a single particle could
exhibit nonlocality
• Lucien Hardy (1994) tightens up some assumptions in TWC
• Greenberger, Horne, and Zeilinger (GHZ) raise objections: not a real
experiment/ multiparticle effect in disguise (1995)
• Other schemes…lots of debate…no clear consensus (1995 - 2007)
Single particle nonlocality
• Tan, Walls and Collett (1991) first suggested that a single particle could
exhibit nonlocality
• Lucien Hardy (1994) tightens up some assumptions in TWC
• Greenberger, Horne, and Zeilinger (GHZ) raise objections: not a real
experiment/ multiparticle effect in disguise (1995)
• Other schemes…lots of debate…no clear consensus (1995 - 2007)
• What is needed is a feasible experiment to resolve the issue….
“The term ‘particle’ survives in modern physics but very little of its classical
meaning remains. A particle can now best be defined as the conceptual
carrier of a set of variates. . . It is also conceived as the occupant of a state
defined by the same set of variates... It might seem desirable to distinguish
the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any
logical basis for such a distinction. ‘Discovering’ a particle means
observing certain effects which are accepted as proof of its existence.”
A. S. Eddington, Fundamental Theory, (Cambridge University Press.,
Cambridge, 1942) pp. 30-31.
What is a beam splitter?
Any physical process that
transforms states in the same
way as a beam splitter.
For atoms, this is equivalent to
Josephson coupling
Hardy Scheme
2
1
Reference: L. Hardy, Phys. Rev. Lett. 73, 2279 (1994)
Hardy Scheme
2
1
Reference: L. Hardy, Phys. Rev. Lett. 73, 2279 (1994)
Hardy Scheme
2
Bob
1
Experiment 1:
Alice and Bob both measure the
number of photons on their path
Alice
They never both detect one
Hardy Scheme
2
Bob
1
Experiment 2:
Alice makes a homodyne detection and Bob
detects the number in path 2
If Bob detects no particles the state at Alice’s
detectors is
So, if Alice detects one, it must be at c1
Alice
Conversely, if Alice detects one particle at d1
and none at c1 then Bob cannot detect none,
i.e. he must detect one!
Hardy Scheme
2
Bob
1
Experiment 3:
The roles of Alice and Bob are
reversed.
Alice
Alice measures the number of
particles on path 1 and Bob
makes a homodyne detection.
If Bob detect one particle at d2 and
nothing at c2 then Alice must
detect one particle.
Hardy Scheme
2
Bob
1
Experiment 4:
Alice and Bob both make
homodyne detections
There is a finite probability that:
Alice detects one particle at d1
and none at c1 AND
Alice
Bob detects one particle at d2 and
none at c2
Recap
Experiment 1: Alice and Bob cannot both detect a particle in their path.
Experiment 2: If Alice detects one particle at d1 and nothing at c1 it
follows that Bob must detect a particle on path 2.
Experiment 3:
If Bob detects one particle at d2 and nothing
at c2 it follows that Alice must detect a
particle on path 1.
Experiment 4:
One possible outcome is that Alice detects
one particle at d1 and nothing at c1 AND
Bob detects one particle at d2 and nothing
at c2.
Recap
Experiment 1: Alice and Bob cannot both detect a particle in their path.
Experiment 2: If Alice detects one particle at d1 and nothing at c1 it
follows that Bob must detect a particle on path 2.
Experiment 3:
If Bob detects one particle at d2 and nothing
at c2 it follows that Alice must detect a
particle on path 1.
CONTRADICTION
Experiment 4:
One possible outcome is that Alice detects
one particle at d1 and nothing at c1 AND
Bob detects one particle at d2 and nothing
at c2.
NONLOCALITY
Objections
Greenberger, Horne, and Zeilinger (1995)
“Partly-cle” states are unobservable - violate superselection rules
Does not correspond to a real experiment
Objections
Greenberger, Horne, and Zeilinger (1995)
“Partly-cle” states are unobservable - violate superselection rules
Does not correspond to a real experiment
They proposed an alternative scheme that required additional particles. These
particles also introduced additional nonlocality
They concluded that ‘apparent’ single particle nonlocality is really a multiparticle
effect
However, they did not disprove single particle nonlocality more generally.
Mixed States
Classical mixture
of number states
where
Convenient to use coherent state inputs and
average over the phases at the end
State truncation
Reference: D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev. Lett. 81 1604 (1998)
State truncation
Input to top beam splitter:
Output from lower beam splitter
Reference: D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev. Lett. 81 1604 (1998)
State truncation
Input to top beam splitter:
Total output:
We need:
Reference: D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev. Lett. 81 1604 (1998)
State creation
State creation
x
State creation
x
State creation
x
State creation
Not Entangled
Will become a mixed state
when we average over
phases at the end
Results
This gives all the same results as Hardy’s scheme
Overall state just before Alice and Bob’s beam splitters is:
Each only makes a local operation
Therefore nonlocality is due to the single particle state
Results
This gives all the same results as Hardy’s scheme
Overall state just before Alice and Bob’s beam splitters is:
Each only makes a local operation
Therefore nonlocality is due to the single particle state
Results are independent of - so we can average over all phases and get
the same result, I.e. a mixed state input also works
This takes care of the GHZ objections
Massive particles
Both Hardy and GHZ said that nonlocality
with single massive particles could not be
observed
Hardy: Superselection rules
GHZ: Not really a single particle effect
This scheme works equally well for
atoms as photons
The only components required are
variable beam splitters and efficient
detectors
Conclusions
• Feasible experimental scheme for demonstrating the
nonlocality of a single particle
• Should work equally well for massive particles as for
photons
•All we need are variable beam splitters and efficient
detectors