Quantum Fisher informa0on as efficient entanglement witness in quantum many-body systems

Transcription

1 Quantum Fisher informa0on as efficient entanglement witness in quantum many-body systems Philipp Hauke (IQOQI, University of Innsbruck) With M. Heyl, L. Tagliacozzo, P. Zoller J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. Hess, D. Huse, C. Monroe M. GärWner, A. Safavi-Naini, M. Wall, and A. M. Rey Heidelberg,

2 Entanglement: Resource of Quantum Simula0on nodd od Theory Experiment c U/J = 5.16(7) K/J = me 4Jt Too much entanglement for classical computer Trotzky et al., Nat. Phys. (2012)

3 Entanglement: Fingerprint of Quantum Phases Example: Disordered systems and many-body localiza0on S vn Many-body localized Anderson localized J z /J =0 J z /J =0.1 J z /J =0.05 J z /J =0.01 log(j t) Znidaric, Prosen, Prelovsek, PRB 2008 Bardarson, Pollmann, Moore, PRL 2012 Vosk and Altman, PRL 2013 Serbyn, Papic, Abanin, PRL 2013

4 The problem with quan0fying many-body entanglement Entanglement measures S Rényi2 = log Tr( 2 A) S vn = Tr( A log A ) E geom =1 max 0 sep Tr r! q q 0 sep 0 sep Problem 1: Which is the relevant one? Problem 2: Non-linear func0ons of density matrix E = f nl ( ) =Tr( Â) =hâi Exponen0al number of measurements! Entanglement entropy Theory: Daley, Pichler, Schachenmayer, Zoller, PRL 2013 Experiments: Greiner group, Nature 2015, Science 2016

5 Entanglement is really hard to measure in experiment entangliometer

6 Workaround: entanglement witnesses Toth, Gühne, Cramer, Brukner, Lewenstein... entangliometer witness

7 Workaround: entanglement witnesses Toth, Gühne, Cramer, Brukner, Lewenstein... The art is to find witnesses that are easy to measure but also tell us something relevant Here: Quantum Fisher Informa0on + genuine mul0par0teness + efficient at T=0 and T>0 + interes0ng many-body sejngs witness

8 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

9 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

10 Quantum Fisher Information F Q bounds parameter estimation Braunstein, Caves PRL 1994 Quantum Cramer-Rao bound ( ) 2 1 F Q #meas. How sensi0ve is a state to change in parameter? = dis0nguishability of from 0 =e io e io X i x i h x squeezed state entangled but not squeezed See experiments groups Oberthaler, Klempt, Bollinger entanglement e.g., Ma et al., Phys. Rep. 2011

11 It witnesses mul0par0te entanglement Pezzé and Smerzi PRL 2009, Hyllus et al., PRA 2012, Toth PRA 2012 Precision limit of es0ma0on ( ) 2 1 F Q #meas. N N 1 F Q /N Need entanglement for F Q >N Need k +1-body entanglement F Q /N > k 2 1 0

12 Calculation of the QFI Braunstein, Caves PRL 1994 F Q = F Q [ ( ), O] ( ) $ ( + ) =e io ( )e io Pure states Mixed states = X p ih F Q =4 ho 2 i hoi 2 F Q =2 X, 0 (p p 0) 2 p + p 0 h O 0 i 2 How to measure this?

13 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

14 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

15 Quantum Fisher Information F Q bounds parameter estimation Braunstein, Caves PRL 1994 Quantum Cramer-Rao bound ( ) 2 1 F Q #meas. How sensi0ve is a state to change in parameter? = dis0nguishability of from 0 =e io e io X i x i h x squeezed state entangled but not squeezed See experiments groups Oberthaler, Klempt, Bollinger entanglement e.g., Ma et al., Phys. Rep. 2011

16 In condensed mawer: Sensi0vity of state = suscep0bility Sta0c suscep0bility: H = H 0 + ho M = Dynamic suscep0bility: (!,T)=i Z 1 00 (!,T)==( (!,T)) 0 H = H 0 + h cos(!t)o dt e i!t h[o(t), O]i T Response func0on rou0nely measured in neutron scawering, Bragg spectroscopy,... Sengstock group Nat. Phys. 2010

17 Quantum Fisher Informa0on and dynamic suscep0bility are the same (in thermal states) F Q (T )= 4 Z 1 0 d! tanh! 2T 00 (!,T) PH, Heyl, Tagliacozzo, Zoller, Nat. Phys. (2016) + Makes QFI directly measurable + Connects no0ons of sensi0vity from two fields + Allows efficient calcula0ons and deriva0on of scaling laws See also Kolodrubetz, Mehta, Polkovnikov, arxiv: Greschner, Kolezhuk, and Vekua, PRB 2013

18 QFI = suscep0bility a short proof Have (!,T)=i Z 1 0 dt e i!t h[o(t), O]i T 00 (!,T)==( (!,T)) Lehmann representa0on (energy eigenbasis) = X p ih, p =e E /Z 00 (!) = X, 0 (p p 0) h O 0 i 2 (! + E 0 E ) Use 2 Z 1 0 d! tanh! 2T E 0 (! + E 0 E ) = tanh 2T E = p p 0 p + p 0 F Q (T )= 4 Z 1 0 d! tanh! 2T 00 (!,T) Independent of microscopic details ( only thermal states, Kubo linear response ) Want F Q =2 X, 0 (p p 0) 2 p + p 0 h O 0 i 2

19 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

20 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

21 Example Ising chain in transverse field H = J X x l x l+1 + h X z l ferromagnet paramagnet quantum phase transi0on

22 Divergence of many-body entanglement at cri0cal point f Q For N=64 at least 13-body entanglement Entangled region at T>0 see also Tóth, and Wu et al., PRA 2005 Test scaling transverse field Entanglement peak at cri0cality

23 Strong scaling in Ising chain Finite-size f Q L 3/4 Q 1 (LT ) Finite-temperature f Q T 3/4 Q 2 (LT ) ln(fq T 3/4 ) f Q L 3/4 f Q = ct 3/4 L = (!,T) from known scaling of Sachdev, Int. J. Mod. Phys. B 1997 F Q (T )= 4 Z 1 0 d! tanh! 2T 00 (!,T) ln(lt) Works at T=0: Zheng et al., Commun. Theor. Phys. 2015, Ma and Wang, PRA 2009, Liu, Ma, Wang, J. Phys. A 2013, Wang et al., NJP 2014 see also works by Zanardi, Campos-Venu0, Gu, and others for metric tensor/fidelity suscep0bility

24 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

25 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

26 QFI as fingerprint for exo0c quantum behavior: many-body localiza0on (MBL) Smith, Lee, Richerme, Neyenhuis, Hess, PH, Heyl, Huse, Monroe, Nat. Phys entanglement entropy QFI N =12 J z /J =0 0.1 J z /J = J z /J =0.05 J z /J = S vn f Q non-interac0ng substracted S vn S vn (0) f Q f Q (0) Entanglement growth and MBL, e.g.: Znidaric, Prosen, Prelovsek, PRB 2008 Bardarson, Pollmann, Moore, PRL 2012 Vosk and Altman, PRL 2013 Serbyn, Papic, Abanin, PRL 2013 J z /J =0 J z /J =0.1 J z /J =0.05 J z /J = J t J z t 0.025

27 Indica0ons for log-growth in trapped-ion experiment Smith, Lee, Richerme, Neyenhuis, Hess, PH, Heyl, Huse, Monroe, Nat. Phys Variance [= QFI if state pure] Quantum Fisher Info (a) No Disorder J max t 1.10 W=6J max (b) W=8J max J max t

28 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

29 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

30 Resource theore0c defini0ons of coherence 1. Define what is an incoherent state 2. Define what are free opera0ons (do not generate coherence) 3. Define measure for coherence: does not increase under free opera0ons

31 Resource theore0c defini0ons of coherence Baumgratz, Cramer, Plenio, PRL 2014 Coherence with respect to a basis e.g., S z i inc. = """i ""#i ###i """i ""#i ###i Marvian and Spekkens, several works Coherence with respect to a Hamiltonian e.g., H = i S z i Resource for various tasks: reference-frame alignment, thermodynamic tasks, quantum metrology """i ""#i "##i ###i """i ""#i "##i ###i F Q is a strict measure for coherence

32 Coherence in NMR: mul0-quantum coherences Collabora0on M. G ärwner, A. Safavi-Naini, M. Wall, and A. M. Rey Different magne0za0on sectors S z M z i = M z M z i NX = (m) (m) = X Mz,M z m M z ihm z m M z m= N """i ""#i "##i ###i """i ""#i "##i mul0-quantum coherences I m =tr( ( m) (m) ) ###i See, e.g., experiments of Alvarez, Kaiser, and Suter, Ann. Phys. 2013, Science 2015

33 Mul0-quantum coherence and quantum Fisher informa0on Mar0n GaerWner et al., in prepara0on 2 NX m= N X I m m 2 apple F Q [,S z ] apple 4dq Im m 2 F Q is a strict measure for coherence (in the Marvian-Spekkens sense) I m are actually only witnesses for coherence I m are also witnesses for entanglement

34 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

35 Content Background Quantum Fisher Informa0on metrological defini0on witness for mul0par0te entanglement Measurability via response func0ons QFI in quantum many-body systems Divergent entanglement at quantum phase transi0on Fingerprint of many-body localiza0on Measure of coherence Conclusions

36 Conclusion Connected Quantum Fisher Info and dynamic suscep0bility measure QFI efficiently, also at T>0 independent of microscopic details F Q (T )= 4 Scaling theory for QFI (predic0ons from known cri0cal exponents) Z 1 0! d! tanh 2T 00 (!,T) QFI: from metrology to quantum many-body systems divergent entanglement at quantum phase transi0on fingerprint of many-body localiza0on measure of coherence as resource EU IP SIQS, SFB FoQuS (FWF Project No. F4016-N23), ERC synergy grant UQUAM