Entanglement and Correlation of Quantum Fields in an Expanding Universe Yasusada Nambu (Nagoya University, Japan) International Workshop on Strings, Black Holes and Quantum Information @ Tohoku Univ 2015/9/9
Contents Introduction Bipartite entanglement 1-dim lattice model of quantum field in a FRW universe Gaussian quantum discord Entanglement in k-space and quantum estimation in a FRW universe 2
Introduction Inflation provides us a mechanism to generate primordial quantum fluctuations that lead to the large scale structures in our present universe. As the present universe is classical, initial quantum fluctuations must lose its quantum nature in course of its evolution (quantum to classical transition). scale wavelength Hubble horizon inflation (de Sitter) 3 time radiation dominant
What are conditions for quantum fluctuations become classical? loss of wave property (freeze out) loss of quantum superposition (decoherence) loss of quantum correlation (dis-entanglement) Entanglement is purely quantum mechanical non-local correlation We want to understand the meaning of classicalization in terms of entanglement of quantum field Evolution of spatial entanglement in an expanding universe two comoving observer in desitter spacetime A B 4
Entanglement (two party) bipartite entanglement A B pure state A, B are separable ja; Bi D jaijbi A, B are entangled ja; Bi D ja 1 ijb 1 i C ja 2 ijb 2 i C.q A ; p A /.q B ; p B / correlation peculiar to quantum mechanics mixed state A, B are separable O AB D X j w j O j A Oj B ; X w j D 1; w j 0 j If the state cannot represented as this form, A, B are entangled 5
Separability: necessary and sufficient condition 1 X 1 Gaussian state (R.Simon 2000, L.Duan et al. 2000) A B D J 0 0 J J D 0 1 1 0 covariance matrix h O Ai D TrΠO O A positivity O D O V C i 2 0 for arbitrary partial transpose p B! p B V! Q V A,B are separable for M X N system A,B is separable 6 Q V C i 2 0 Q V C i 2 0 2X2 2X3 1XN
Symplectic eigenvalue SVS T D diag. C ; C; ; / C > 0 symplectic transformation S 2 Sp.4; R/ S S T D positivity separability Q 1 2 1 2 uncertainty relation Logarithmic negativity If these conditions are satisfied, A, B are separable no entanglement E N D min Œlog 2.2Q /; 0 E N > 0 E N D 0 entangled separable 7
Entanglement of Quantum Field in a FRW Universe 1-dim lattice model (periodic BC) massless scalar D 0 EOM for q D a discretize space q 00 a 00 q 00 j a q r 2 q D 0 a 00 q 1 q j q j 1 q j C1 a q j C 2q j.q j C1 C q j 1 / D 0 D 1 x YN, PRD80(2009)124031 scale factor a. / conformal time D Z dt a q N 1.m x/2 2 x quantization Oq j D p 1 NX 1 N f 00 k C kd0!2 k f k Oa k C f k Oa N a 00 a k e i kj k D 2 k N f k D 0! 2 k D 2.1 cos k/ 8
block variables q A D 1 p n X covariance matrix V D A C C A j 2A q B D 1 p n X j 2B q j q j A D a 1 a 3 a 3 a 2 q A q B n n C D c 1 c 3 c 3 c 2 x a 1 D h Oq 2 A i a 2 D h Op 2 A i a 3 D 1 2 h Oq A Op A C Op A Oq A i c 1 D 1 2 h Oq A Oq B C Oq B Oq A i c 2 D 1 2 h Op A Op B C Op B Op A i c 3 D 1 2 h Oq A Op B C Op B Oq A i these variables are time dependent numerically calculate symplectic eigenvalue obtain logarithmic negativity and judge separability 9 N=100
Minkowski vacuum q A q B 0.5 0.4 n d n x 0.3 d D 0 EN 0.2 0.1 0.0 d D 1 5 10 15 20 block size n Minkowski vacuum is entangled EN<>0 for d=0, EN=0 for d>1 Value of entanglement depends on the definition of spatial regions Negativity is constant in time 10
De Sitter (Bunch-Davies vacuum) a.t/ D e Ht scale wavelength q A q B n n x H 1 Hubble horizon time Evolution of negativity block size dependence 0.4 n=1 0.4 n=1 EN 0.3 0.2 0.1 0.0 0.1 0.0 0.1 0.5 1 5 10 a c Initial entangled state evolves to separable state Separable time c EN 0.3 0.2 n=2 n=3 n=4 0.1 0.5 1 5 10 a a depends on size of block 11
block size and separable time 1/(ac H Δx) 20 15 10 5 n D 1=.a c H x/ 0 0 5 10 15 20 n separable time c n x D c D 1 a c H ) a c n x D H 1 block size Hubble length When the block size is equal to the Hubble horizon scale, entanglement between blocks is lost. initial entangled state A H 1 H 1 B x x 12 (initial vacuum state) de Sitter expansion separable state quantum correlation is lost generation of classical fluctuation quantumness?
Information and Correlation of Scalar Field Correlation of a bipartite system A B Mutual information lack of information entropy I.A W B/ D S.A/ C S.A/ S.X/ D X x p x log p x S.AB/ Shannon entropy for a classical variable S.X/ D tr. X log X / von Neumann entropy for a quantum state For classical variables, by Bayes rule I.A W B/ D S.A/ S.AjB/ J.A W B/ mutual information in terms of conditional entropy conditional entropy S.AjB/ D X b p b S.Ajb/ 13
For quantum case, using a POVM measurement on B, X J.AjB/ D S.A/ p b S. Ajb / and in general, I.A W B/ J.AjB/ Maximal correlation obtained via measurement is defined by J.AjB/ D S.A/ b min f b g X Difference between I and J quantifies quantumness of correlation: b p b S. Ajb / measurement op. of b f b g; X D.AjB/ I.A W B/ J.AjB/ quantum discord b b D 1 p b D tr. b AB b / I D J C D total correlation classical correlation quantum discord Henderson and Vedral 2001 J. Phys. A 34, 6899 Ollivier and Zurek 2002 PRL 88, 01790 We can judge quantumness of quantum fluctuation using quantum discord. 14
Gaussian Quantum Discord For 2-mode Gaussian state with a covariance matrix A C V D C B We consider a Gaussian measurementz B. / D D B. / M D é B. /; 1 D B. / D e bé b State of A after measurement of B is V 0 A D A C.B C V M / 1 C T General form of Gaussian discord: f.x/ D x C 1 log x C 1 2 2 15 d 2 B. / D 1 x coherent state POVM D D f. p p B/ f. / f. C / C max f. V M 1 log 2 x b D 1 2 Adesso & Datta 2010 Giorda & Paris 2010 r! 2 x B C i p 2! p B VA 0 /
De Sitter (Bunch-Davies vacuum) q A q B x n n 0.35 0.30 0.25 negativity scale wavelength EN 0.20 0.15 0.10 H 1 Hubble horizon 0.05 0.00 0.5 1 5 10 50 100 time a D D I J Discord 0.14 0.12 0.10 0.08 0.06 0.04 discord Information 2.6 2.4 2.2 2.0 I J 0.02 1.8 0.00 0.5 1 5 10 50 100 a Asymptotically approaches to zero discord state appearance of classical correlation 16 0.5 1 5 10 50 100 a
De Sitter radiation dominant a / e Ht a /.t C t 0 / 1=2 q A q B n n x EN 0.35 0.30 0.25 0.20 negativity RD scale Hubble horizon wavelength 0.15 0.10 H 1 0.05 0.00 0.5 1 5 10 50 100 a time zero discord state in radiation dominant stage Discord 0.14 0.12 0.10 0.08 0.06 discord RD Information 2.8 2.6 2.4 2.2 I J RD 0.04 2.0 0.02 1.8 0.00 0.5 1 5 10 50 100 1.6 0.5 1 5 10 50 100 a a 17
Minkowski cosmic expansion Minkowski negativity discord q A q B n n x 0.5 Δ=5 EN 0.4 0.3 0.2 0.1 Discord 0.15 0.10 0.05 a. / D 1 C a.1 C tanh. // 0.0-10 -5 0 5 10 15 20 η 0.00-10 -5 0 5 10 15 20 η 2 0.5 0.15 0.4 Δ=1 EN 0.3 0.2 Discord 0.10 large change of negativity 0.1 0.05 0.0-10 -5 0 5 10 15 20 η 0.00-10 -5 0 5 10 15 20 η 0.5 0.4 0.15 small value of final discord Δ=0.5 EN 0.3 0.2 Discord 0.10 0.1 0.05 0.0-10 -5 0 5 10 15 20 η 0.00-10 -5 0 5 10 15 20 η 18
Evolution of entanglement and quantum correlation in De Sitter radiation dominant universe (lattice model) Quantum fluctuations generated in de Sitter phase lose spatial entanglement when their wavelength exceed the Hubble horizon. Entanglement between adjacent spatial regions remains zero after the universe enters the phase of decelerated expansion. Quantum discord has non-zero value even after the entanglement becomes zero. Asymptotically, quantum discord approaches zero and the zero discord state is attained. scale Hubble horizon zero discord state wavelength total correlation=classical correlation 19 H 1 time
Quantum Estimation in Cosmology Estimation of model parameters expansion law, mass of fields, coupling of fields,... Classical theory: measurement result (probability) Fisher information Quantum theory: measurement result (probability) depends on POVM optimize about possible POVM quantum Fisher information We want to understand relation between Fisher information and entanglement in cosmological situations 20
Fisher information P. j / : probability to obtain measurement result respect to POVM f g with : a parameter to be estimated Fisher information F. / D X P. j / @ log P. j / @ 2 Unbiased error for satisfies (Cramer-Rao inequality). / 2 1 F. / Larger Fisher information reduces the lower bound of error 21
Quantum Fisher information F Q. / D max f g F. / defined by optimization wrt all POVM This quantity can be represented by symmetric logarithm derivative: F Q. / D tr. L 2 /; @ 1 2 f ; L g For a state with form D X n nj n ih n j F Q. / D X m.@ m / 2 m C 2 X m n. m n / 2 m C n jh m j@ n ij 2 22
Massive scalar field in a FRW universe L D Z d 3 x p g 1 2 gab @ a @ b C m2 2 2 ' D a ' k D 1 p 2!k b k C b k é p k D i r!k 2 b k é b k Hamiltonian H D Z d 3 k bk. / b é k. / D apple!k 2 uk v k b k é b k C b k b k é C i a0 v k u k bk. 0 / b é k. 0 / Bogoluibov coefficients 23! 2 k D k2 C m 2 a 2 a b k é b k é b k b k generates entanglement between k,-k u k. 0 / D 1; v k. 0 / D 0
Initial vacuum state evolves to many particle state: j0; i S D 1 apple vk exp a é é ju k j1=2 k a k j0; 0 id 1 ju k j 2u k 2 mode squeezed state: entangled 1X nd0 vk u k n jn k ;n k i Reduced density matrix D 1X nd0 njn k ihn k j; n D n ju k j 2 ; ˇ v k u k ˇ 2 Entanglement entropy S E D Quantum Fisher information F Q D 1X n log nd0 1X nd0 24 n n.@ log n / 2
previous works model universe with expansion law a. / D 1 C.1 C tanh / (k,-k) entanglement of a bosonic and a fermionic field I. Fuentes et al. PRD82, 045030(2010) estimation of expansion law using a fermionic field J. Wang et al. Nucl. Phys. B892(2015)390 We consider a massive scalar field in de Sitter universe. 25
Entanglement entropy 10 1 m2=0.0001 no specific feature due to mass S 0.100 m2=0.1 0.010 m2=0.5 m2=1 0.001 10-4 0.001 0.010 0.100 1 10 100 k Quantum Fisher information: estimation parameter is mass F Q 10 10-4 10-9 10-14 10-4 0.001 0.010 0.100 1 10 100 k m2=0.01 m2=0.1 m2=0.2 m2=0.3 m2=0.4 oscillation due to mass appears in super horizon scale QFI is sensitive to behavior of entanglement physical meaning? 26
Summary Classicality (quantumness) of quantum field in a FRW universe Entanglement and quantum discord We confirm the quantum to classical transition of the scalar field using 1-dim lattice model. The considering system approaches the zero discord state Cause of this behavior? : particle creation due to cosmic expansion increase of k-space entanglement particle creation decrease of x-space entanglement What is the relation between (k,-k)-entanglement and x-space entanglement? Entanglement, classical correlation, strength of energy(density) fluctuation 27
F P hf. Oq A ; Op A ; Oq B ; Op B /i D Z d 2 qd 2 p P.q A ; p A ; q B ; p B /F.q A ; p A ; q B ; p B / X Gaussian state Z separable O AB D d 2 d 2ˇ P. ; ˇ/ j ; ˇih ; ˇj (R.Simon 2000, L.Duan et al. 2000) P-function h WF. Oq; Op/W i D Z P 0 j ; ˇi D j ijˇi A, B coherent state d 2 qd 2 p P.q; p/f.q; p/ 28
standard form P-func separability 29
Wigner function D.q A ; p A ; q B ; p B / T Z hff. Oq; Op/g sym i D d 2 qd 2 p W.q; p/f.q; p/ Wigner func. : Z h WF. Oq; Op/W i D d 2 qd 2 p P.q; p/f.q; p/ P-func. : separable separable hff. Oq; Op/g sym i h WF. Oq; Op/W i hf. Oq; Op/i Oq; Op P.q; p/ W.q; p/ (nambu, 2008) hf. Oq; Op/i Z d 2 qd 2 p W.q; p/f.q; p/ 30
lattice model Ν 3.0 2.5 2.0 1.5 1.0 0.5 2.0 1.5 1.0 0.5 Η symplectic eigenvalue 31
A B A, B 32
Summary 2 entanglement scale entangled separable classical wavelength time horizon scale separable horizon separable one Hubble time 33
mass (Compton wavelength) inflation model entanglement geometry N-party entanglement 34