ENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE. Ivette Fuentes University of Nottingham

Transcription

1 ENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE Ivette Fuentes University of Nottingham with: Robert B. Mann (U Waterloo) Shahpoor Moradi (U Razi) Eduardo Martin Martinez (CISC Madrid)

2 OUTLINE Motivation PART ONE: theoretical considerations entanglement basics (very short) quantum field theory in curved spacetime basics PART TWO: entanglement in an expanding Universe bosonic fields fermionic fields

3 MOTIVATION Entanglement is a key resource in quantum information theory which has mainly been studied in flat spacetime However spacetime is generally curved Questions: Does the underlying curvature has effects on entanglement? Can the dynamics of spacetime create or destroy entanglement? Entanglement is hard to defined in curved spacetime but what can we learn about it?

4 PART ONE THEORETICAL CONSIDERATIONS

5 ENTANGLEMENT AND ITS USES COMPOSITE SYSTEMS quantum theory rank 1 rank 2 rank 3 classical theory versus one particle two particles entangled qubit two qubits data measurement TELEPORTATION entangled pair 2 classical bits data teleported! manipulation HOW DOES THIS WORK IN A RELATIVISTIC QUANTUM THEORY?

6 QUANTIFYING ENTANGLEMENT BIPARTITE PURE STATES Schmidt basis density matrix ENTANGELMENT between A and B reduced density matrix von Neumann entropy

7 PARTICLES FROM FIELDS Quantum field theory on curved spacetime Quantum field fundamental Particlesderived notion (if at all) QUANTUM FIELD THEORY linear field equation vectorspaceof solutions PARTICLE INTERPRETATION requires classification of modes into positive negative frequency pos. Inner product not positive definite! timelike Killing vector field neg. Killing vector field boson Fock space Spacetime creation annihilation

8 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 KILLING OBSERVERS different timelike Killing vectors K and K different splits of basis in pos/neg Killing vector field K Bogoliubov transformation Spacetime Killing vector field K Squeezed states

9 EXAMPLE: UNRUH EFFECT Minkowski spacetime in 1+1 dimensions (flat spacetime = no gravity!) Rob is causally disconnected from region II k Bob k Rob acceleration r Timelike killing observers (a) inertial observer (b) uniformly accelerated observers trace thermal state Similar effect in black holes: Hawking radiation

10 SPACETIME AS A CRISTAL Curve spacetimes generally do not admit timelike killing vector fields particular spacetimes with asymptotically flat regions flat Spacetime Cristal flat Bogoliubov transformation Squeezed states Just like in quantum optics!

11 THE CHALLENGE Theory Particles Particle number Mathematics Effects curved spacetime generic observer Killing observer ill defined massive/ massless ill defined obs dependent varies (also free field)??? identify regions??? Particle creation flat spacetime accelerated observer massive/ massless obs dependent (Unruh effect) varies (interactions) Bogolubov BH radiation flat spacetime inertial observer massive/ massless obs independent varies (interactions) interaction non relativistic massive obs independent conserved entangled

12 ENTANGLEMENT IN QUANTUM FIELD THEORY RESULTS ON ENTANGLEMENT IN FLAT SPACETIME Entanglement is observer dependent It can be degraded due to horizons Fermionic and Bosonic entanglement is very different IFS, Mann PRL (2005) Adesso, IFS, Ericsson PRA (2007) Alsing, IFS, Mann, Tessier PRA (2006) Black hole accelerated observers bosons Fermions

13 PART TWO ENTANGLEMENT IN AN EXPANDING SPACETIME

14 ROBERTSON WALKER UNIVERSE spacetime with metric EXPANDING UNIVERSE Minkowski slope ~ Minkowski COSMOLOGYCAL PARAMETERS expansion rate expansion factor TIMELIKE KILLING VECTOR FIELD

15 PARTICLE CREATION AND ENTANGLEMENT particle interpretation in the asymptotic past and future Particle creation Entanglement creation no particle interpretation QUANTIFYING ENTANGLEMENT FOR BOSONS AND FERMIONS asymptotic past asymptotic future vacuum excitingly, can solve for

16 FERMIONIC IN AND OUT STATES Duncan PRD 1978 DIRAC EQUATION 1. SOLVE THE EQUATION IN THE GIVEN SPACE TIME Exploiting spatial translational invariance therefore Flat spacetime matrices where

17 2. IN THE ASYMPTOTIC LIMIT IDENTIFY POSITIVE AND NEGATIVE SOLUTIONS therefore the in and out solution are where and F1 are hypergeometric functions

18 3. EXPRESS THE FIELD IN TERMS OF THE IN AND OUT SOLUTIONS WITH THE CORRESPONDING CREATION AND ANNIHILATION OPERATORS where the curved spacetime spinor solutions are flat spacetime spinors with

19 4. USE INNER PRODUCT TO FIND BOGOLIUBOV TRANSFORATIONS BETWEEN THE IN AND OUT SOLUTIONS 5. FIND TRANSFORMATION BETWEEN THE OPERATORS TRANSFORMATION BETWEEN THE OPERATORS particles antiparticles where

20 6. FIND TRANSFORMATION BETWEEN THE STATES where therefore,

21 FERMIONIC ENTANGLEMENT Fuentes, Mann, Moradi, Martin Martinez (in preparation) 7. COMPUTE THE ENTANGLEMENT BETWEEN K AND K MODES in the limit for light particles

22 BOSONIC IN AND OUT STATES KLEIN GORDON EQUATION massive conformally coupled case POSITIVE AND NEGATIVE SOLUTIONS where BOSONIC GAMMA

23 BOSONIC ENTANGLEMENT Ball, IFS, Schuller PLA (2006) HISTORY OF THE UNIVERSE ENCODED IN ENTANGLEMENT in the limit for light particles Entanglement contains information about the underlying spacetime

24 ENTANGLEMENT COSMOLOGY Fuentes, Mann, Moradi, Martin Martinez (in preparation) bosons bosons entanglement entanglement fermions fermions expansion factor expansion rate The Universe entangles less fermionic fields

25 Entanglement vs. frequency Fuentes, Mann, Moradi, Martin Martinez (in preparation) fermions bosons entanglement entanglement Entropy for bosons and fermions as a function of k

26 EXPANSION PARAMETERS FROM ENTANGLEMENT different different Entropy for fermions as a function of k and m

27 EXPANSION PARAMETERS FROM ENTANGLEMENT Entropy for fermions as a function of k and m k m

28 EXPANSION PARAMETERS FROM ENTANGLEMENT optimal k Entropy for fermions as a function of k and m m

29 EXPANSION PARAMETERS FROM ENTANGLEMENT optimal k m Entropy for fermions as a function of k and m

30 CONCLUSIONS Entanglement is created by the expansion of the Universe Fermionic and bosonic entanglement is very different Fermionic entangelment is less sensitive to the underlying spacetime In the fermionic case, we find that entanglement is maximum for a given frequency The entanglement encodes information about the past history of the Universe This information can be better extracted using fermions