Transcript Slide 1

Information Processing by Single Particle
Hybrid Entangled States
Archan S. Majumdar
S. N. Bose National Centre for Basic Sciences
Kolkata, India
Collaborators:
S. Adhikari
D. Home
A. K. Pan
T. Pramanik
International Program on Quantum Information, IOP Bhubaneswar, 2010
Perspective
Hybrid entangled states:
The theoretical framework of Q.M. allows for the existence of entangled states involving
spaces with distinctly different properties, e.g., spin & linear momentum; polarization
Hilbert
& angular momentum; energy level & spin. etc… [c.f. Zukowski, Zeilinger, et. al. (2009), etc.]
Single particle (intra-particle) entanglement:
Entanglement between different degrees of freedom of the same particle. [c.f. Blasone et. al.
(2009) for neutrino oscillations; Dunningham & Vedral (2007) nonlocality for photons ; Home et al
(2001) & Hasegawa et. al. (2003) for neutron path and spin, etc..]
Motivations:
Is it possible to perform information processing with such states ?
These states could be more robust against noise and dissipative effects.
Could have applications in complex systems. e.g., CM physics, BH physics ?
Difficulty: Entanglement NOT shared between spatially separated entities
Plan


Setting up path-spin entangled state of a single particle
Swapping path-spin entanglement with spin-spin entanglement of two
qubits

Teleportation using path-spin entanglement of a single particle as
resource

A demonstration of contextuality using single-particle energy-spin
entangled states

Encoding information using the superposition of path and spin degrees
of freedom

Summary and Conclusions
Setting up a path-spin entangled state
of a single particle

Initial state:

Beam-splitter

Spin-flipper (or CNOT)
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Emergent path-spin entangled state
Path-spin hybrid entangled state of a single particle
Salient features:

Dichotomic path and spin variables – essentially two qubits carried by a
single particle

Intra-particle entanglement (as distinct from the entanglement of spins of
two different particles, or even hybrid entanglement of say the polarization
of one photon & spin of another spatially separated photon

Entanglement NOT of similar modes, e.g., as in single particle neutrino
flavour states

This state is a true example of a single particle hybrid entangled state where
the entanglement is between the path and spin of the same particle
Swapping path-spin entanglement onto
spin-spin entanglement of two qubits
Aim: To swap the entanglement of the created path-spin
entangled state with that of two spatially separated qubits.

The two qubits are disentangled initially, and have no
interaction among them.
Protocol: Alice has the path-spin entangled state “1”, and a
Second particle “2” in a spin (up) state. Bob has particle “3”
also in spin (up) state. Through some LOCC, the state of “2”
And “3” get entangled, while the path-spin entangled state “1”
is destroyed.
Protocol for entanglement swapping

Alice performs CNOT with
“2” (target) & “1” (source).

“1” is sent to Bob, and
“2” to Charlie.

Bob performs CNOT with
“1” (source) & “3” (target).

Bob combines the two channels
of “1” using BS2, and performs
measurements with SG1 & SG2

Bob applies unitary operations
on “3”.

Result: “2” (with Charlie) and “3” (with Bob) get entangled.
Protocol for entanglement swapping
Alice possesses path-spin entangled state, and another particle in spin up state
After CNOT operation:
Particle “1” is sent to Bob. His CNOT:
Particle “2” is sent to Charlie.
Bob uses a 50-50 beam splitter:
SG1 & SG2 are placed using which the
qubit “s” undergoes the transformation:
Depending upon the following possibilities,
Bob makes the corresponding unitary operations:
Measurement outcomes & Unitary operations
Final outcome:
Bob and Charlie share two-qubit entangled state with the
same information content as the original path-spin entangled state.
| 23   |01   |10
Intra-particle entanglement of “1” is swapped onto the spin-spin inter-particle
entanglement of “2” & “3”.
“2” & “3” never interact with each other, but entanglement is set-up by their
separate interactions with “1”.
S. Adhikari, A. S. Majumdar, D. Home, A. K. Pan, EPL 89, 10005 (2010)
Teleportation through single particle
path-spin entangled state
Perspective: Path-spin entangled state is NOT shared non-locally
between two distant parties. How can such entanglement be used
as a resource ?
Aim: To transfer the state of an unknown qubit
to a distant location using particle “1” (path-spin entangled) state as
resource.
Method: Since entanglement is not shared a-priori between the two
distant parties, particle “1” has to be transported at some stage.
Protocol for information transfer using
single particle path-spin entangled state
Alice: “1” & “2” (unknown)
& auxiliary “a”.
Performs two CNOTs
Sends particle “1”
to Bob.
Bob: Performs
CNOT
Alice: Measures “1” & “a”,
communicates with Bob.
Bob: Unitary transformation
on “3” to recover teleported state.
Steps of the teleportation protocol
Alice possesses “1”:
“2”:
Performs two CNOTS:
Alice sends “1” to Bob. After
Bob confirms receipt, Alice
Measures “2” & “a”.
and “a”:
Teleportation protocol…. {Case i}
Bob performs suitable unitary transformations based on Alice’s measurement
results communicated to him, e.g., if the outcome is
&
Bob combines the two channels of “1” using BS2, and performs a CNOT
involving “1” & “3”:
Bob measures spin of “1” by SG2/SG3. The possible outcomes are:
occurring with equal probability ¼.
Bob performs the corresponding unitary operations
Fidelity of teleportation
Unitary operations performed by Bob:
Average Fidelity:
If the path-spin state “1” is maximally entangled, i.e.,
teleportation
is perfect. Bob possesses the same state “3” as the one “2” originally
held by Alice.
T.Pramanik, S.Adhikari, A.S.Majumdar, D.Home, A.K.Pan, Phys. Lett. A 374, 1121 (2010)
Security of the Protocol
What happens when particle “1” is lost in transit ?
“1” is held by Eve (now), and “2” & “a” are possessed by Alice.
Eve can perform a spin measurement on “1”. However, she fails to uncover
information about the state “2”:
When Bob does NOT receive particle “1”, Alice performs the following operations
to recover the state to be teleported:
Alice makes a spin measurement on her particle “a” in the basis
Then she makes (i) unitary operation
(ii) does nothing
to retrieve the unknown state to be teleported.
Information in “2” is never lost even if “1” is lost in transit.
T. Pramanik, S. Adhikari, A. S. Majumdar, D. Home, A. K. Pan, Phys. Lett. A (2010)
Energy-spin entangled state of a
single atom in cavity-QED
Two-(energy) level atom in (energy & spin) state
Apply
in vacuum Cavity:
Stern-Gerlach along x-axis:
Place
Place
in
in
state in the spin up path
state in the spin down path
With suitable chosen interaction times, energy-spin entangled state of atom:
Proposal for testing contextuality with
single atom energy-spin entangled state
Energy-spin entangled state
Emerging from C0 & C1
pulses in
Measurements through
SG and four detectors
,
C0
Violation of Bell-CHSH inequality
Non-contextual model:
Q.M. correlations:
Encoding information using the quantum
superposition principle
Perspective: Dense coding requires quantum entanglement;
Is it possible to encode information using the superposition principle ?
We propose a scheme using the path and spin degrees of freedom
of a single particle.

Operational difficulty of Bell State Analysis in standard quantum dense
coding scheme leads to maximum channel capacity of about 1.6 bits in
stead of the theoretically attainable 2 bits.

Is it possible to do better using the spin and path variables of a spin-1/2
particle ?
Set-up for encoding information
Alice prepares state by
choosing particular values
of the phase of SR & PS

Bob obtains particle in
channel corresponding to
Alice’s unitary operation

Protocol for encoding information
Initial state:
Phase-shifter:
Final state on emerging from BS2:
SR:
The scheme
Alice has two possible choices for the parameters
and


The four possible combination of choices (unitary operations):
U1 (
  0 ,  0
) U2
This leads to the following distinct possibilities for the channel where Bob finds
the particle.
Thus, two bits of information can be communicated by Alice to Bob without
using quantum entanglement. This scheme is solely based on the
superposition principle.
D.Home, A.S.Majumdar, S. Adhikari, A.K.Pan, M.Whitaker, arXiv:0906.0270
Summary & Conclusions

Information processing through single particle hybrid entangled states

Generation of path-spin entangled state by BS and SF (or CNOT)

Swapping of path-spin intra-particle entanglement onto spin-spin interparticle entanglement

Teleportation of a spin qubit using single particle path-spin entanglement
as resource. Secure protocol with ideal fidelity.

Proposal to test quantum contextuality using single atom energy-spin
entangled state. Other forms of hybrid entanglement ?

Encoding information through the superposition of path & spin of a single
particle
Perspectives & future directions
A.S.M., D.H., S.A., A.K.P., T.P., EPL (2010); PLA (2010); arXiv:0906.0270

Nature of quantum entanglement independent of the particular
realization of Hilbert space – single particle hybrid entangled states
could be useful information processing resource.

Path (or linear momentum) degrees of freedom ubiquitous in
experiments. Our protocols are based on exploiting these hitherto
unexplored resources.

Entanglement at the level of a single particle could be less
susceptible to decoherence effects – this feature needs further
development and testing in real experiments.

Reliance of information processing protocols on entanglement vis-àvis the superposition principle ?