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TRANSCRIPT
Ivette Fuentes University of Nottingham
QUANTUM INFORMATION PROCESSING
IN SPACETIME
http://rqinottingham.weebly.com/
relativistic quantum information and metrology
postdocs Mehdi Ahmadi Jason Doukas Andrzej Dragan (now in Warsaw) Carlos Sabin Angela White (now in Newcastle) PhD students Nicolai Friis Antony Lee Luke Westwood John Kogias (joint with Adesso) project student Kevin Truong Bartosz Regula (with C. Sabin)
Collaborators Gerardo Adesso (Nottingham) Paul Alsing (AFRL) David Bruschi (Leeds) Tony Downes (Queensland) Marie Ericsson (Upsala) Daniele Faccio (Herriot-Watt) Marcus Huber (Bristol/Barcelona) Göran Johansson (Chalmers) Jorma Louko (Nottingham) Robert Mann (Waterloo) Enrique Solano (Bilbao) Tim Ralph (Queensland) FUNDING: EPSRC (THANKS!!!!)
relativistic quantum information INFORMATION THEORY
superposition
entanglement
CNOT
QUANTUM PHYSICS
computation
communication
RELATIVITY
causality
geometry
relativistic quantum
information
real world experiments
quantum communication
PHOTONS HAVE NO NON-RELATIVISTIC APPROXIMATION
spacebased experiments
Space-QUEST project: distribute entanglement from the International Space Station.
quantum gravity QUANTUM PHYSICS RELATIVITY
Quantum technologies in space
entanglement
entangled pair
OUTLINE
•PART ONE: theoretical considerations
•entanglement basics
•quantum field theory in curved spacetime basics
•PART TWO: entanglement in flat spacetime
•PART THREE: entanglement in curved spacetime
•PART FOUR: entanglement between localized systems
•Quantum gates implemented using relativistic motion
•Teleportation goes relativistic
PART ONE
THEORETICAL CONSIDERATIONS
TELEPORTATION
ENTANGLEMENT AND ITS USES
one particle
two particles entangled
rank 1 rank 2 rank 3
quantum theory classical theory
versus
COMPOSITE SYSTEMS
entangled pair
data
measurement
2 classical bits
manipulation
data teleported!
1
1
1 1
qubit two qubits
HOW DOES THIS WORK IN A RELATIVISTIC QUANTUM THEORY?
Entanglement for mixed states
bipartite system, two particles mixed state
Environment
important object: partial transpose
NECESSARY CONDITION for separability:
partial transpose of ρAB has no negative eigenvalues.
Obtained by taking the trace:
DEFINITION: separable density matrix can be written
Mixed state: density matrix
dual space
quantifying entanglement
Measure of entanglement:
Schmidt basis PURE STATES:
no analogue to Schmidt decomposition
(entropy no longer quantifies entanglement)
MIXED STATES
negativity = sum of negative eigenvalues of
use density matrix
reduced density matrix (subsystem A)
von Neumann entropy
DEFS:
DEF: entanglement between A and B =
but necessary condition for separability (no negative eigenvalues) suggest to use
covariance matrix formalism
covariance matrix: information about the state
symplectic matrix: evolution
computable measures of bipartite and multipartite entanglement
PARTICLES FROM FIELDS Quantum field theory on curved spacetime
Quantum field fundamental • Particles derived notion (if at all)
pos.
neg.
requires classification of modes into
positive ↔ negative frequency
boson Fock space
creation annihilation
PARTICLE INTERPRETATION
timelike Killing vector field
linear field equation
vectorspace of solutions
QUANTUM FIELD THEORY
Spacetime
Killing vector field
Inner product not positive definite!
KILLING OBSERVERS
KILLING OBSERVERS
INSIGHTS
• particles present ill-defined subsystems! • particles well-defined only for killing observers
• particle interpretation may change with
change of Killing vector field
different timelike Killing vectors K and K
different splits of basis in pos/neg
Squeezed states
Bogoliubov transformation
Spacetime
Killing vector field K
Killing vector field K
Minkowski and Rindler coordinates
Rindler coordinates (χ,η): accelerated observers
proper acceleration
Bogo’s
Bogoliubov transformation
Minkowski
Rindler
EXAMPLE: UNRUH EFFECT
trace thermal state
Timelike killing observers
(a) inertial observer
(b) uniformly accelerated observers
k’
Minkowski spacetime in 1+1 dimensions (flat spacetime = no gravity!)
k’
acce
lera
tio
n r
Bob Rob
Rob is causally disconnected from region II
Similar effect in black holes: Hawking radiation
SPACETIME AS A CRISTAL
Curve spacetimes generally do not admit timelike killing vector fields…
Bogoliubov transformation
Just like in quantum optics!
flat
flat
Spacetime-----Cristal
particular spacetimes with asymptotically flat regions
Squeezed states
Bogolubov
THE CHALLENGE
flat spacetime
inertial
observer
non-relativistic obs-independent
conserved massive
entangled
massive/ massless
obs-independent
varies (interactions) interaction
flat spacetime
accelerated
observer
obs-dependent (Unruh effect)
varies (interactions)
massive/ massless
BH radiation
curved spacetime generic observer
Killing observer
ill-defined ill-defined
massive/ massless
obs-dependent varies (also free field)
identify regions
??? ???
Theory Particles Particle number Mathematics Effects
Particle creation
PART TWO
FLAT SPACETIME
Alice Bob
k k’
Rob
k’
acce
lera
tio
n r
Entanglement • observer-dependent • degrades with acceleration , vanishes for ∞ acceleration
Fuentes-Schuller, Mann PRL 2005 Adesso, Fuentes-S, Ericsson PRA 2007
RESULT 1 — Alice and Rob
more realistic states:
RESULT 2 — Ralice and Rob
entanglement
fixed detection frequencies k , k’ fixed acceleration & squeezing
entanglement
Entanglement vanishes at
finite acceleration
TWO ACCELERATED OBSERVERS (same direction, same acceleration)
Entanglement very fragile (gravity in the lab!) but high frequencies help
Adesso, Fuentes-S, Ericsson PARA 2007
RESULT 3 — Fermionic Surprise BELL STATE OF FERMIONIC FIELD
Bob Rob Alice
k k’ k’
acce
lera
tio
n r
tanh ( acceleration )
Fermionic entanglement more robust under acceleration.
entanglement
Alsing, Fuentes-S, Mann, Tessier PRA 2006
RESULT 4 — Entanglement sharing Where did the lost entanglement between Alice and Bob go?
Alice Rob
k’ k k’
Again important differences between fermions and bosons.
Alsing, Fuentes-S, Mann, Tessier PRA 2006 Adesso, Fuentes-S , Ericsson PRA 2007
multipartite entanglement!
Rob Alice
k’ k k’
Bosonic field Fermionic field
Bipartite entanglement between Alice and mode in region II
Mode in
region II Mode in
region II
PART THREE
BLACK HOLES &
COSMOLOGY
RESULT 1 — Alice falls into a BH
BH
horizon
BH
horizon
“3+1” 1+1 part of Rindler space
Rob Alice
Entanglement Classical correlations
degraded for escaping observers
Lost entanglement multipartite entanglement between modes inside and outside the BH
Fuentes-S, Mann PRL 2005 Adesso & Fuentes-S 2007
RESULT 2 — Entanglement cosmology
no particle interpretation
unentangled state
“History of the universe encoded in entanglement”
toy model expansion rate
expansion factor
• calculate entanglement
asymptotic past
asymptotic future
• excitingly, can solve for
Ball, Fuentes-S, Schuller PLA 2006
RESULT 3 — Fermionic entanglement cosmology
Fuentes, Mann, Martin-Martinez, Moradi PRD 2010
“The Universe entangles less fermionic fields”
bosons
fermions
bosons
fermions
enta
ngl
emen
t
enta
ngl
emen
t
expansion factor expansion rate
Fermionic fields in 3+1 dimensions: more realistic model
PART FOUR
•Entanglement between localized systems
•cavities
•detectors
•localized wave-packets
•gravity effects on quantum properties
•Time to get real: earth-based and space-based experiments
inertial cavity
field equation
solutions: plane waves+ boundary
creation and annihilation operators
Minkowski coordinates
uniformly accelerated cavity
Klein-Gordon takes the same form
Rindler coordinates
Bogoliubov transformations
entangling moving cavities in non-inertial frames
Downes, Fuentes & Ralph PRL 2011
ability to entangle: depends on acceleration
entanglement preserved: inertial or uniform acceleration motion
entangle two cavities: one inertial and one accelerated
idea
results
non-uniform motion Bruschi, Fuentes & Louko PRD (R) 2011
Bogoliubov transformations
acceleration length
computable transformations
Friis and Fuentes JMO (invited) 2012
general symplectic matrix
general trajectories
entanglement: negativity
Friis, Bruschi, Louko & Fuentes PRD 2012 Friis and Fuentes invited at JMO 2012 Bruschi, Louko, Faccio & Fuentes 2012
entanglement generated
initial separable squeezed state
general trajectories continuous motion including circular acceleration
motion and gravity create entanglement
non-uniform motion creates entanglement
gravity creates entanglement
results
single cavity
Friis, Bruschi, Louko, Fuentes PRD (R) 2012
entanglement resonances Bruschi, Lee, Dragan, Fuentes, Louko arXiv:1201.0663
Bruschi, Louko, Faccio & Fuentes 2012
Entanglement vs. number of oscillations and period of oscillation
amount of entanglement
entanglement resonance
total segment time
trajectory details
Also: entanglement resonance without particle creation observable by mechanical means
quantum gates
the relativistic motion of quantum systems can be used to produce quantum gates
two-mode squeezer beam splitter multi-qubit gates: Dicke states Multi-mode squeezer
Friis, Huber, Fuentes, Bruschi PRD 2012 Bruschi, Lee, Dragan, Fuentes, Louko arXiv:1201.0663
Bruschi, Louko, Faccio & Fuentes 2012
multipartite case
Entanglement: genuine multipartite entanglement created
dates creating Dicke states
Friis, Huber, Fuentes, Bruschi PRD 2012
resonance:
teleporation with an accelerated partner
the fidelity of teleportation is effected by motion it is possible to correct by local rotations and trip planning
Friis, Lee, Truong, Sabin, Solano, Johansson & Fuentes PRL 2003
new directions: experiments
simulate field inside a cavity which travels in a spaceship using superconducting circuits
To infinity and beyond…….
Art by Philip Krantz (Chalmers)
Friis, Lee, Truong, Sabin, Solano, Johansson & Fuentes 2012
Analog Relativistic Quantum Information
test our results in an analog spacetime using a Bose-Einstein condensates
Thank you