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