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Renyi Entropies from Random Quenches in Atomic Hubbard and Spin Models

Andreas Elben

with B. Vermersch, M. Dalmonte, I. Cirac and P. Zoller

arXiv: 1709.05060

University of Innsbruck/IQOQI

UQUAM ERC Synergy Grant

UNIVERSITY OF INNSBRUCK Outline

1.Why Renyi entropies and Entanglement? Which setups?

A B

2. Measurement of Renyi entropies in atomic Hubbard and Spin models

3. Examples - Area Law and Many-Body Localization Renyi entropies as an entanglement measure

Two subsystems A and B are bipartite entangled iff = A B | ABi6 | Ai⌦| Bi

Sufﬁcient condition for bipartite entanglement Reduced density matrix ⇢ =Tr A B| ABih AB| ⇢AB

2 | 2{z } Purity of subsystem Tr ⇢A < Tr ⇢AB Purity of full system ⇥ ⇤ ⇥ ⇤

Entanglement entropies

S = TrA [⇢A log ⇢A] von-Neumann A

(n) 1 n S = log TrA [⇢ ] SA Renyi A 1 n A  Why Entanglement?

Entanglement as a Entanglement as a characterising property resource 5

a. - that its scope of application is even broader, allowing char- 0 -0 b.0.5 2 acterisation of many-body entangled states and their dynam- 0.2 Quantum computing2 N Quantum many body systems ics even in systems without ﬁnite-range interactions. Since 4 0 0.4 1 2 3 4 5 6 7 8 9 10111213 no prior knowledge of the state in the laboratory is required, Spin pair Time (1/J) 6 Time (ms) Scaling with (sub-)sytem size - MPS tomography provides a practical and ecient approach 0.6 0.5 8 3 to obtaining a reliable state estimate and should therefore be a 0.8 N Spreading of quantum correlations, 10 powerful addition to the toolbox for verifying and benchmark- Trapped 2 4 6 8 ion10 quantum12 14 0 computer 0 1 Spin 1 2 3 4 5 6 7 8 9 101112 Topological phases, … ing engineered quantum systems. c. d. e. Spin triplet

Acknowledgments. Work in Innsbruck was supported by 2 0.6 DMRG the Austrian Science Fund (FWF) under the grant number 4 6 0.4 Area law / @ A P25354-N20, by the European Commission via the inte- Spin 8 grated project SIQS, by the Institut für Quanteninformation 10 (2) 0.2 12 S GmbH and by the U.S. Army Research Oce through grant Lab ZZ Lab XY Lab YY 0.0 W911NF-14-1-0103. All statements of fact, opinion or con- 14 2 4 6 8 clusions contained herein are those of the authors and should 2 4 @A not be construed as representing the o cial views or poli- 6 Dynamics - Thermalisation vs. cies of ARO, the ODNI, or the U.S. Government. Work in Spin 8 Ulm was supported by an Alexander von Humboldt Professor- 10 Many-Body Localization 12 MPS ZZ MPS XY MPS YY ship, the ERC Synergy grant BioQ, the EU projects QUCHIP 14 and EQUAM, the US-Army Research Oce Grant No. W91- 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 0.6 Spin Spin Spin -0.5 0.5 U/J =1, /J = 10 1NF-14-1-0133 and the BMBF Verbundproject QuOReP. Nu- U/J =0, /J = 10 0.5 merical computations have been supported by the state of FIG. 4: MPS tomography results for a 14 spin quench. a.-b. Re- (2) Lanyon et al., arXiv:1612.08000z Baden-Württemberg through bwHPC and the German Re- sults from local measurements. a. Spin magnetisation (1 + (t) )/2 S h i i 0.4 search Foundation (DFG) through grant no INST 40/467-1 with two approximate light-like cones (Supp. Mat.). b. Entangle- FUGG. I.D. acknowledges support from the Alexander von ment in local reductions at t = 4 ms, from density matrices recon- 0.3 0 1 2 Humboldt Foundation. M. H. acknowledges contributions structed via QST. Upper: between neighbouring spin pairs (negativ- 10 10 10 ity). Lower: spin triplets (tripartite negativity), from QST of corre- Jt from Daniel Suess to jointly developed code used for data sponding density matrices. c.-e. Comparison of correlation matrices analysis. Work at Strathclyde is supported by the European directly measured in lab and from MPS estimates at t = 4 ms, show- Union Horizon 2020 collaborative project QuProCS (grant ing observable A(t) B(t) A(t) B(t) , for spins i and j. A, B h i jih iih ji agreement 641277), and by AFOSR grant FA9550-12-1-0057 as labelled. Not all correlations were measured in the lab (hatched M.C. acknowledges: the ERC grant QFTCMPS and SIQS, squares). the cluster of excellence EXC201 Quantum Engineering and Space-Time Research, and the DFG SFB 1227 (DQ-mat). T. [4] Flammia, S. T. & Liu, Y.-K. Direct Fidelity Estimation from B. acknowledges: EPSRC (EP/K04057X/2) and the UK Na- Few Pauli Measurements. Phys. Rev. Lett. 106, 230501 (2011). tional Quantum Technologies Programme (EP/M01326X/1). [5] da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practi- Author contributions. B.P.L, C.F.R, M.B.P. and M.C. de- cal Characterization of Quantum Devices without Tomography. veloped and supervised the project; C.M., C.H., B.P.L., P.J., Phys. Rev. Lett. 107, 210404 (2011). [6] Baumgratz, T., Gross, D., Cramer, M. & Plenio, M. B. Scal- R.B. and C.F.R. performed and contributed to the experi- able Reconstruction of Density Matrices. Phys. Rev. Lett. 111, ments; B.P.L., M.H., T.B., C.M, C.F.R., I.D., A.B. and A.D. 020401 (2013). performed data analysis and modelling; B.P.L. wrote the [7] Baumgratz, T., Nüßeler, A., Cramer, M. & Plenio, M. B. A manuscript, with contributions from all authors. scalable maximum likelihood method for quantum state tomography. New J. Phys. 15, 125004 (2013). [8] Tóth, G. et al. Permutationally invariant quantum tomography. Phys. Rev. Lett. 105, 250403 (2010). [9] Knap, M. et al. Probing real-space and time-resolved correlation functions with many-body ramsey interferometry. Phys. Rev. Lett. 111, 147205 (2013). [10] Senko, C. et al. Coherent imaging spectroscopy of a quantum ⇤ Electronic address: [email protected] many-body spin system. Science 345, 430–433 (2014). † These authors contributed equally to this work. [11] Jurcevic, P. et al. Spectroscopy of interacting quasiparticles in [1] Vogel, K. & Risken, H. Determination of quasiprobability dis- trapped ions. Phys. Rev. Lett. 115, 100501 (2015). tributions in terms of probability distributions for the rotated [12] Shabani, A. et al. Ecient measurement of quantum dynamics quadrature phase. Phys. Rev. A 40, 2847–2849 (1989). via compressive sensing. Phys. Rev. Lett. 106, 100401 (2011). [2] Cramer, M. et al. Ecient quantum state tomography. Nat. [13] Ste↵ens, A. et al. Towards experimental quantum-ﬁeld tomog- Commun. 1, 149 (2010). raphy with ultracold atoms. Nat. Commun. 6, 7663 (2015). [3] Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J. [14] Schollwöck, U. The density-matrix renormalization group in Quantum state tomography via compressed sensing. Phys. Rev. the age of matrix product states. Ann. Phys. 326, 96–192 Lett. 105, 150401 (2010). (2011). Scaling of entanglement

B Volume law - Extensive scaling

S (⇢ ) A A ⇠ | | • ‘generic’ (random) quantum states • thermal states

e.g. Berges et al., arXiv:1707.05338

Area Law S (⇢ ) @A A ⇠ | | • ground states of typical (local, gapped) A @A Hamiltonians ⇢A =TrB [⇢]

Holographic principle and Topological entanglement Complexity of Simulations Black holes entropy - MPS, PEPS

Bekenstein- Hawking: (Logarithmic) corrections Success of DMRG in 1D horizon area to area laws systems S = BH 4 Srednicki: Massless scalar ﬁelds Eisert et al., Rev. Mod. Phys. 82, 277 (2010) Area Law in a Heisenberg model

Isotropic Heisenberg model (here: 2D)

x x y y z z Hh = J i l + i l + i l il Xh i A Ground state ⇢GS @A S(2)(A)= log Tr ⇢2 A A 8x8 sites with ⇢A =TrS A [⇥⇢GS⇤] \

0.6 DMRG How to verify/measure this in an experiment? 0.4

/ @ A • Physical realization?

(2) 0.2 • Measurement of entanglement?

S Area law 0.0 Quantum simulation 2 4 6 8 @A Examples of atomic quantum simulators

Ultracold atoms in optical lattices Rydberg Atoms in optical tweezers (MPQ, HD, CUA, JQI, …) (MPQ, CUA, IOGS,…)

Endres et al., Science (2016) Barredo et al., Science (2016) Choi et al., Science (2016) Kaufman et al., Science (2016) bosonic/fermionic Hubbard models Spin (Ising) models,… Heisenberg model,…

Trapped Ions (IBK, JQI, Oxford,…)

How to measure Renyi entropies demonstrating entanglement? R. Blatt, Innsbruck

Spin (Ising) models, … ARTICLE RESEARCH

To initialize two independent and identical copies of a state with a 680 nm b Mott J fixed particle number N, we start with a low-entropy two-dimensional y 28 Mott insulator with unity filling in the atomic limit and determin- 1 istically retain a plaquette of 2 × N atoms while removing all others 0.8 (Supplementary Information). This is illustrated in Fig. 3a. The plaquette of 2 × N atoms contains two copies (along the y direction) 0.6 50/50/50 beam splitterspli (1,1) of an N-atom one-dimensional system (along the x direction), with 0.4 N = 4 in this figure. The desired quantum state is prepared by manip- ulating the depth of the optical lattice along xMeasurement, varying the parameter of Renyi0.2 entropies U/Jx, where Jx is the tunnelling rate along x. A box potential created by J 0 the spatial light modulator is superimposed onto this optical lattice to y 0 1/8 1/4 3/8 1/2 y constrain the dynamics to the sites within each copy. During the state Jyt preparation, a deep lattice barrier separates the two copies and makes x Jx them independent of each other. The beam splitter operation required for the many-body interference c Twin state Many-body QuantumSite-resolved State3 Tomography is realized in a double-well potential along y. The dynamics of atoms Initialization interference parity readout in the double well is likewisecorrelations described is evident by [11,the 26,Bose–Hubbard 33]. a. b. Odd Even Hamiltonian, equation (4). A singleMPS atom, tomography initially is appliedlocalized to in quench one well, dynamics, starting 0.5 Gross et al. PRL 105, coherently oscillates betweenfrom the the wells initial with antiferromagnetic a Rabi frequency Néel-ordered of J = Jy product state 0.25 (0) = , , , ,... . This highly excited initial state (N/2 150401 (2010) (oscillation frequency in the| amplitude).i |" # "At# discretei times during this 0 − excitations) leads to the emergence of locally-correlated en- ↑↑ B. P. Lanyon et al., Nat.Phys., ()n 21n = AB evolution, t ==tBS , withtangled n states 1, 2, involving ..., the atom all N particlesis delocalized and evolves in a sub- ↑↓ 8Jy 1 (2017) ↓↑ space whose size, contrary to those of low-excitation sub- ↓↓ equally over the two wells with a fixed phase relationship. Each of these 0.5 ↓↓ ↓↑ spaces [33], grows exponentially with N. After preparing ↑↑ ↑↓ times realizes a beam splitter operation, for which the same two wells A even Pure If available (0) with a spatially-steerable laser, focused()n on a single ion, serve as the input ports at time| ti = 0 and output ports at time t = t . c. spin interactions are abruptly turned on (aBS quench) andMott then insulator A+B even Pure Two indistinguishable atoms with negligible interaction strength 0.5 0 ms 3 ms 5 ms

o↵ after a desired evolution time t, freezing the generated state 2 ≪ Quench dynamics, N InterferenceA and B product stat ofe copies (U/Jy 1) in this double welland will allowing interfere for as spin they measurement. tunnel. The Thedynam ideal- model state is 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 ics of two atoms in the double (wellt) = isexp( demonstratediH t) (0) in. Fig. Through 3b in repeated terms stated prepara- Spin pair Spin pair Spin pair | Adiabatici XY | preparation,i d. ofInitial the joint probability state P(1, 1)tion of and finding measurement, them in estimatesseparate ofwells the expectationversus values for 0.5 0 ms 3 ms 5 ms

3 Islam et al., Nature 528, the normalized time Jyt. Thelocal jointk-spin probabili observablesty P…(1, are 1) obtained. oscillates For at example, a to esti- N 0 frequency of 772(16) Hzmate = 4J eachy, with of the a Ncontrast2 local reductions of 95(3)%. of neighbouring At k = 3 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 77–83 (2015) ()n k Spin Triplet Spin Triplet Spin Triplet the beam splitter times, t =sitest , (spin P(1, triplets), 1) ≈ 0. measurementsThe first beam in 3splitter= 27 di↵erent bases AB BS FIG. 2: Local measurement results for an 8 spin system. a. Sin- ()1 1 are carried out. The results are input into a combination of two Daley et al . PRL, 109(2), time, t ≡ t = is used for all the following experiments, with gle site magnetisation: Probability of finding a spin up at each site, BS BS 8J y e cient MPS tomography algorithms [2, 7], which output an during quench dynamics. The interaction range ↵ 1.6. Lefthand 20505 (2012) ⇢ ⇢ ⇡ P(1, 1) = 0.05(2). This is a initialsignature MPS of estimate bosonic for theinterference simulator state of two⇢lab. Finally, a cer- time axisA is renormalised by the average nearest-neighbour J cou- k A odd or even mixed ⇢I37tified,38 MPS state estimate c is found. The lower bound on !plings. Two light-like cones are shown, exemplifying an estimate for indistinguishable particles , akin to the photonic| iHOM interfer- A+B even pure Pichler et al. PRX, 6(4), 41033 ence36. This high interferencethe contrast fidelity of indicates this state the with near-perfect the actual state sup in- the laboratorySuperfuidthe maximum speed at which correlations spread (see Supp. Mat.). is given by Fk, i.e. k ⇢ k Fk (see Supp. Mat.). b. Density matrix (absolute value) of spinsA 3and & 4B atentangled time of 3 ms, (2016) pression of classical noise and fluctuationsc andh c includes| lab | ci an expectedc Measurement of The largest application of full QST was for an 8 qubit W- reconstructed via QST. The state is entangled, with a bipartite neg- 0.6% reduction due to finite interaction strength (U/Jy ≈ 0.3). The Figure 3 | Many-bodyativity of interferenceN2 = 0.31 0 to.01 probe and a fidelityentanglement with an ideal in optical theoretical state [34], for which measurements were made in 6561 dif- ± lattices.Renyi a, A high-resolution modelentropy of over 0 .microscope99. c.-d. Entanglement is used to directly in all neighbouring image the spin results from this interferenceferent can basesbe interpreted taken over as a a period measurement of ten hours of [37]. We be- the quantum purity of the initial Fock state as measured from the aver- number parity pairsof ultracold (c.) and spinbosonic triplets atoms (d.) aton three each evolution lattice site times, (in as the labelled: raw gin experiments with 8 spin (qubit) quench dynamics,ofimages, a and (sub-)system re-green representsvalues calculated odd and from black measured represents density even). matrices Two (e.g. adjacent panel b. ). age parity (equation (3)), 〈 Pi〉construct = 1 − 2 × 8-spin P(1, entangled 1) = 0.90(4), states where via MPS i = tomography,1, 2 using one-dimensionalThe lattices entanglement are created measure by is combining bipartiteRandom negativity an opticalN2 (tripartite latticemeasurements neg- on single are the two copies. measurements in only 27 bases taken over a periodand of aroundpotentials ativitycreatedN 3nby) for a spatial spin pairs light (triplets). modulator.N3 is theWe geometric initialize mean two of all ten minutes. Local measurements are performedidentical to recon- many-bodyTrthree [⇢ bipartite statesA] negativityby filling splittings. the potentialscopies from ain low-entropy a QC Entanglement in the groundstruct all statek-local reductions of individual spins (k = 1), neigh- two-dimensional⇠ Mott insulator. The tunnelling rates Jx and Jy can be The Bose–Hubbard model providesbouring spin an interesting pairs (k = 2) system and spin in triplets which ( kto= 3),tuned at various independentlysistent withby changing the time the at which depth the of the information potential. wavefronts b, The are investigate entanglement. Insimulator optical lattice evolution systems, times. Thea lower results bound of these of measurementsatomic beam splitterexpected operation to reach is next-nearest-neighbours realized in a tunnel-coupled (light-like cones, van Enk, Beenaker, PRL the spatial entanglement hasdirectly been previously reveal important estimated properties. from time-of- Single-site ‘magnetisa-double-well potential.Figure 2a), An allowing atom, initially for correlations localized beyond in one 3 of sites the to wells, develop. Realization in a atomic quantum many-body = flight measurements39 and entanglementtion’ shows how dynamics spin excitations in spin disperse degrees and of thendelocalizes partially withMeasurements equal probability on k into3 sitesboth reveal the wells an MPS by this description beam splitter. with 108, 110503 (2012) refocus (Figure 2a). In the first few ms, strong entanglement more than 0.8 fidelity up to t = 3 ms, before rapidly dropping freedom has beensystems investigated with existing partial state reconstruction today40. Here,in theHere, welab?! show the atomic analogue of the HOM interference of two states. is seen to develop in all neighbouring spin pairs and triplets, to 0 at 6P ms. This is consistent with the model and the entan- we directly measure entanglement in real space occupational particle The joint probability (1, 1) measures the probabilityS. Boixo, ofet coincidenceal., arXiv:1608.00263 thenHubbard/Spin later reducing, first in pairs thenmodels in triplets,detection consistent of theglement atoms in properties separate measured wells as a directly function in of the normalized local reductions number in a site-resolved way.with In correlations the strongly spreading interacting out across atomic larger limit numbers tunnel of spins time J t,(Figure with the 2b-c): single At particlet = 3 ms tunnelling entanglement J = in 193(4) spin triplets Hz. max- ≫ y y of U/Jx 1, the ground state inis a the Mott system insulator (Figure corresponding 2c-d). to a Fock At the beam splitterimises, duration before reducing (Jyt = 1/8) to bosonic almost zero interference at 6ms as leads correlations k state of one atom at each latticeFidelity site. The lower quantum bounds F statec from has MPS no tomographyspatial to during a nearly the vanishinghave spread P(1, 1), out corresponding to include more to distant an even spins. parity In thisin the case, 3- entanglement with respect to8-spin any partitioning quench are shown in this in Figphase—it 3a. The is results in a closelyoutput match states. Thissite localcan be reductions interpreted are as not a sumeasurementcient to uniquely of the purity distinguish product state of the Fock states.an idealised As the interaction model: MPS strength tomography is reduced applied to theof exactthe initial lo- Fockthe globalstate, here state. measured Note, even to if beF k0.90(4).= 0, the The MPS data estimates shown k c | ci cal reductions of the ideal states (t) . The di↵erenceshere are be- averagedcan over still betwo an independent accurate description double ofwells. the lab The state blue (F curvek are only adiabatically, atoms begin to tunnel across the lattice sites, and| ultimatelyi c the Mott insulator melts into tweena superfluid model andwith data a fixed are largelyatom number. due imperfect The knowledgeis a maximum-likelihood of lower bounds). fit to the data, and the error bars reflect 1σ delocalization of atoms createslocal entanglement reduction due between to the finite spatial number subsystems. of measurements statistical used error. cThe, When data two in copies Figure of 3a a clearlyproduct reveal state, thesuch generation as the Mott and This entanglement originatesin41 experiments,42 from correlated (Projection fluctuations noise, see Supp. in the Mat.). insulator Measure- in thespreading-out atomic limit, of are entanglement interfered on during the beam simulator splitter, evolution the up output states contain even particle numbers globally (full system) as well number of particles betweenments the subsystems on k=1 sites due at tot =the0 super-selection provides a certified MPS state to 3 - 4 ms, and are consistent with this behaviour continu- reconstruction 1 , with F1 = 0.98 0.01 and as1 locally(0) 2 = (subsystem),ing beyond indicating this time. pure To states confirm in this,both. it d would, On the be other necessary rule that the total particle number in the full| ci system cis fixed, ±as well as |h c| i| 0.98, proving that the system is initially well describedhand, byfor atwo copiesto measure of an on entangled increasingly state, large such numbers as a superfluid of sites, demandingstate, the coherence between various configurationspure product Néel without state (Figureany such 3a). fluctuation. The fidelity loweroutput bounds states containmeasurements even particle that grow numbers exponentially globally in (purek. That state) the but amount a To probe the emergence ofbased entanglement, on single-site we measurements first prepare the rapidly ground degrade mixture as the sim- of oddof and entanglement even outcomes in the locally simulator (mixed is growingstate). This in timedirectly can be state of equation (4) in both copiesulator evolves,by adiabatically falling to lowering 0 by t = 2 the ms. optical Nevertheless, demonstrates an ac- entanglement.seen from the inset in figure 3a: the half-chain entropies of curate pure-state description is still achieved by measuring on the certified MPSs 3 are seen to grow as expected for a | ci larger (k = 2) and larger (k = 3) reduced sites (Figure 3a). The sudden quench, closely following that in ideal model states model fidelity bounds F3 begin to drop after t = 2 ms, con- (t) . For all times at which F3 > 0 (except t = 0), the c | i3 DECEMBER 2015 | VOLc 528 | NATURE | 79 © 2015 Macmillan Publishers Limited. All rights reserved Outline

1.Why Renyi entropies and Entanglement? Which systems?

A B

2. Measurement of Renyi entropies in atomic Hubbard and Spin models • Mini-Review: Entanglement from random measurements van Enk, Beenaker (PRL 2012) • Physical Realization

3. Examples - Area Law and MBL Random measurements & Quantum information

Protocol for a chain of qubits: A van Enk, Beenaker (PRL 2012) ⇢A ` Random measurement random unitary UA by random gates P (s )=Tr U ⇢AU † s s A A A A | Aih A| ` h i Average over the Circular Unitary Ensemble (CUE) Measurement of qubit states

2 (0, 1 , 0)=sA 1 2 1+Tr ⇢A P (sA) = P (sA) = h i N A h i N (N + 1) H HA H⇥A ⇤ Hilbertspace dimension of A Random measurements & Quantum information

Random unitaries from the Circular Unitary Ensemble (CUE)

Unitaries distributed according the Haar measure on the unitary group

N : Hilbert space 1 HA UA CUE(N A ) (Uij), (Uij) 0, dimension of subsystem 2 H < = ⇠ N N ✓ HA ◆ up to unitary constraints

Projector Application to the protocol: describing measurement Measurement f ⇢A † ⇢A = U⇢AU † P(sA)=TrUA⇢AUA sA with outcome P sA h i

Virtual copies 2 2 2 Tr [⇢A] + Tr ⇢A P(sA) = Tr1 2 ...UA⇢AUA† UA⇢AUA† ... = h i h ⌦ ⌦ i N A (N A + 1) h i H H ⇥ ⇤ klmn + knml CUE (2-design) : U U ⇤U U ⇤ = h ik il im ini N (N + 1) HA HA ~ Gaussian Random measurements & Quantum information

Protocol for a chain of qubits: A van Enk, Beenaker (PRL 2012) ⇢A ` Random measurement random unitary UA by random gates P (s )=Tr U ⇢ U † s s A A A A | Aih A| ` h i Average over the Circular Unitary Ensemble (CUE) Measurement of qubit states

2 (0, 1 , 0)=sA 1 2 1+Tr ⇢A P (sA) = P (sA) = h i N A h i N (N + 1) H HA H⇥A ⇤ Hilbertspace dimension of A Realization in a Hubbard or Spin model:

How to generate random How many measurements unitaries? per unitary and how many unitaries? Outline

1.Why entanglement? How to quantify? A B

2. Measurement of Renyi entropies in atomic Hubbard and Spin models • Mini-Review: Entanglement from random measurements van Enk, Beenaker (PRL 2012) • Physical Realization

3. Examples - Area Law and MBL Measurement protocol for Hubbard and Spin models

Quench dynamics, Adiabatic preparation,…

⇢ ⇢ Time ⇢I ! A Measurement of Renyi entropy of a (sub-)system Tr [⇢n ] ⇠ A Measurement protocol for Hubbard and Spin models

Random unitary as time evolution operator under random quenches iH T iH T U = e ⌘ e 1 A ··· Quench dynamics, 1 ⌘ Adiabatic preparation,…

Time ⇢I ⇢ ⇢ UA⇢AU † ! A A

Disorder pattern " #

potential offsets

See also: M Ohliger, V Nesme, J Eisert - NJP 2013 Generation of random unitaries

Idea: Random unitary as time evolution operator resulting from a series of random quenches

Heisenberg model (e.g. strong interaction limit of FH) H = J h i · l il A h Xi2

Disorder patterns j z Hj = Hh + i i i A 2 j X i from gaussian distribution with standard deviation

potential offsets

iH⌘ T iH1T Vary disorder in discrete steps in time UA = e e ··· Random unitary? Generation of random unitaries

iH T iH T Question: Is U = e ⌘ e 1 a random unitary? A ··· Apply the protocol to a known input state and compare estimated to true purity to test the ensemble

Heisenberg model (here: 1D, 8 sites)

j z Hj = J i l + 1 AF · i i 1010 il A i A PS h Xi2 X2 e j

) Rand i from gaussian distribution 2 with standard deviation = J p AF + PS ( 101.00 J = =1/T exponential 0.5 AF: convergence 0 16 32 |"#"#"#"#i 0 2 4 PS: JT⌘tot/L |""""####i Rand: random pure state Random unitaries using ✓ generic interactions ✓ engineered disorder Scaling with system size 101 | 1 | 2 10 AF 2 2 8 p p 0 100 PS 10 4 10 − − Rand 6 e Heisenberg model with L sites e 1 1 ) ) 10− j z 10− 2 2 Hj = J i l + i AFi + PS

· p iH T iH T p ⌘ 1 2 il A i A 2 ( UA = e e ( 10j h Xi2 X2 − | | 10− ··· i from gaussian distribution with standard deviation = J 0 2 4 0 2 4 6 8 JTtot/L JTtot/L 1D N = 500 2D 101 U 1 NU = 500

| 10 1 | 1 | | 2 10 2 2 8 10 2 2 2 3 3 AF 2 JT =16 × × tot p p p

0 p 0 10 4 10 0 3 2 5 2 JT =8 10 PS 10 × × tot − −

− 4 2 6 L = L L − Rand × x y e e 1

e ⇥ 1 1 e ) ) 10− ) 10− 10− ) 1 2 2

AF + PS 2 2 10− p p p 2 2 p ( ( ( 10 2 ( − | | 10− | 10− | Zeno MBL 2 1 0 1 0 2 4 0 2 4 6 8 0 2 4 6 8 10 10− 10− 10 10 JTtot/L JT⌘/Ltot/L JT⌘tot/L/L statistical error threshold Number of necessary JT 1 1 1 10 due to ﬁnite number1 10

| 10 | |

random quenches |

10 2 L =4 2 2 2 3 3 2 JT =16 × ×(500) random unitaries tot Ions 2 Rydberg L =4 p 0 p p 0 3 2 5 2 ⌘ L 10 L =6 p 0 10 × × JTtot =8 ⇠ 10 atoms L =6 − − − 4 2 L =8 − Efﬁcient generation of random1 L =8

× e 1 e e e 10− 1 ) 10− ) ) ) 10− 1 unitaries for purity measurements2 2 2 2 10− 2 p 2 Random quantum 2 p 2 p p

P (s ) Tr ⇢( A A10− ( 10− ( | ( 2 h i⇠ circuits | | 10− | Zeno MBL ⇥ ⇤ 2 1 0 1 0 1 2 3 0 1 2 3 0 2 4 6 8 10 10− 10− 10 10 JT /L JT /L JTtot/L JT tot tot 101 1

| 10 |

2 L =4 Ions 2 Rydberg L =4 p 0

p 0 10 L =6 10 atoms L =6 −

L =8 − 1 L =8 e 1 10− e ) 10− ) 2 2 p 2 p 2 ( 10− ( 10− | | 0 1 2 3 0 1 2 3 JTtot/L JTtot/L Ising and Hubbard models

Hubbard models Quantum Ising models (Bosons/Fermions) (Rydberg atoms / Ions)

C6 r6 2 J nP3/2 4 U 3 Rb 1 ⌦1 87 ⌦ ⌦1 ⌦1 1 5S1/2 2 ⌦1 ⌦1 ⌦1 ⌦1

U j x j z C6 z z Hj = J a† ai +h.c. + ni(ni 1) + ni Hj = ⌦ + + i+1 2 i i i i r r 6 i j i A i A i A i i i

Fock states 1 e

e 10 1 ) ) 10 2 2

Random unitariesp p ( ( |

| created with existing tools 3 10 ✓ generic interactions 1 2 3 0 2 ⌘/L ⌘/L ✓ engineered disorder Measurement protocol for Hubbard and Spin models

Random unitary as time evolution operator under random quenches iH T iH T U = e ⌘ e 1 A ··· Quench dynamics, 1 ⌘ Adiabatic preparation,…

Time ⇢I ⇢ ⇢ UA⇢AU † ! A A

Disorder pattern " #

potential offsets Measurement protocol for Hubbard and Spin models

Random unitary Measurement State Quench dynamics, preparation Adiabatic preparation,…

Time ⇢I ⇢ ⇢ U ⇢ U † ! A A A A

(n , n ) Disorder " # patterns Quantum Gas " Microscope #

Potential barriers

Repeat scheme

For the same random unitary probabilities P( n , n ) for all measurement outcomes ( n , n ) " # " # 2 2 For many random unitaries correlations P(n , n ) Tr ⇢A h " # i⇠ ⇥ ⇤ Scaling of statistical errors

NM : number of measurements per unitary NU : number of unitaries N : Hilbert space dimension of A HA Error for estimated purity (averaged over all outcomes)

N = 256 NU = 1000 HA 100 | | N A 2 2 NM = p A 64 0 N H p 10 p A 128 N 1 1 10 256

e 10 1 e ) ) 2 2 1 2 p N A 10 p 2 ( H

( 10 |

⇠ pN | U ⇠ pNM 1 2 3 10 10 10 100 101 102 NU NNMM//pN AA NH 1 N A Analytics: (p2)e p2 1+ H p | | ⇠ NU N NM HA ✓ ◆ p ﬁniteNU ﬁniteNM Scaling of statistical errors

NM : number of measurements per unitary NU : number of unitaries N : Hilbert space dimension of A HA Error for estimated purity (averaged over all outcomes)

N = 256 NU = 1000 HA 100 | | N A 2 2 NM = p A 64 0 N H p 10 p A 128 N 1 1 10 256

e 10 1 e ) ) 2 2 1 2 p N A 10 p 2 ( H

( 10 |

⇠ pN | U ⇠ pNM 1 2 3 10 10 10 100 101 102 NU NNMM//pN AA NH p 2 2 Number of measurements to determineasfp2 Tr ⇢ = NH (NH + 1) Prob(s ) 1 ⌘ A h i 1/ N up to error sdsdfgss⇠ U ⇥ ⇤ NM N A p ⇠ H p Outline

1.Why Renyi entropies and Entanglement? Which systems?

2. Measurement of Renyi entropies in atomic Hubbard and Spin models • Mini-Review: Entanglement from random measurements van Enk, Beenaker (PRL 2012) • Physical Realization

3. Examples - Area Law and MBL Area Law in a Heisenberg model

Isotropic Heisenberg model (here: 2D)

x x y y z z Hh = J i l + i l + i l il Xh i A Ground state ⇢GS @A S(2)(A)= log Tr ⇢2 A A 8x8 sites with ⇢A =TrS A [⇥⇢GS⇤] \

0.6 DMRG 0.4 / @ A / @ A (2) (2) ⌘ 0.2 1 4 S S 0.0 2 8

2 4 6 8 NU = 100 @A NM = 100 Entanglement dynamics: Many-Body Localization

Many Body Localization A quantum phase characterized by the interplay of disorder (localization) and interactions Only known generic exception of thermalisation (ETH)

D.M. Basko, I.L. Aleiner, B.L. Altshuler Annals of Physics 321, 1126 (2006) Rahul Nandkishore, David A. HuseAnnual Review of Condensed Matter Physics, Vol. 6: 15-38 (2015)

Hallmark of MBL: slow (logarithmic, but nonzero) growth of entanglement week ending PRL 109, 017202 (2012) PHYSICAL REVIEW LETTERS 6 JULY 2012

subsystem boundary, even though there is no conserved currentMBL of entanglement. The ﬁrst question we seek to answer is whether there is any qualitatively different behavior of physical quantities Von Neumann whenAnderson a small interaction

Entropy Localization z z Hint Jz Si Si 1 (2) Bardarson, et al., PRL 2012 ¼ þ Xi is added. With Heisenberg couplings between the spins (Jz NoJ ),direct the model observation is believed to have a dynamical tran- ¼ ? sition asof a functionentanglement of the dimensionless disorder strength =J [4,5,7]. This transition is present in generic eigenstates z growth so far! Time of the system and hence exists at infinite temperature at some nonzero . The spin conductivity, or equivalently particle conductivity after the Jordan-Wigner transforma- tion, is zero in the many-body localized phase and nonzero for small enough =Jz. However, with exact diagonalization the system size is so limited that it has not been possible to estimate the location in the thermodynamic limit of the transition of eigenstates or conductivities. We find that entanglement growth shows a qualitative change in behavior at infinitesimal Jz. Instead of the expected behavior that a small interaction strength leads to a small delay in saturation and a small increase in final entanglement, we find that the increase of entanglement continues to times orders of magnitude larger than the initial localization time in the Jz 0 case (Fig. 1). This slow growth of entanglement is consistent¼ with prior observations for shorter times and FIG. 1 (color online). (a) Entanglement growth after a quench larger interactions Jz 0:5J and Jz J [12,13], starting from a site factorized Sz eigenstate for different inter- although the saturation behavior¼ ? was unclear.¼ Note? that ob- action strengths Jz (we consider a bipartition into two half chains serving a sudden effect of turning on interactions requires of equal size). All data are for 5 and L 10, except for large systems, as a small change in the Hamiltonian applied J 0:1 where L 20 is shown¼ for comparison.¼ The inset z ¼ ¼ to the same initial state will take a long time to affect the shows the same data but with a rescaled time axis and subtracted behavior significantly. We next explain briefly the methods J 0 values. (b) Saturation values of the entanglement entropy z ¼ enabling large systems to be studied. as a function of L for different interaction strengths Jz. The inset shows the approach to saturation. Numerical methodology.—To simulate the quench, we use the time evolving block decimation (TEBD) [15,16] method which provides an efficient method to perform a subregions A and B. But the total amount of entanglement time evolution of quantum states, c t U t c 0 , in entropy generated remains finite as t (Fig. 1), and the one-dimensional systems. The TEBDj algorithmð Þi ¼ ð canÞj beð Þ seeni !1 fluctuations of particle number eventually saturate as well as a descendant of the density matrix renormalization group (see below). The entanglement entropy for the pure state [17] method and is based on a matrix product state (MPS) of the whole system is defined as the von Neumann entropy representation [18,19] of the wave functions. We use a S trA logA trB logB of the reduced density second-order Trotter decomposition of the short time propa- ¼À ¼À matrix of either subsystem. We always form the two biparti- gator U Át exp iÁtH into a product of term which tions by dividing the system at the center bond. acts onlyð onÞ¼ two nearest-neighborðÀ Þ sites (two-site gates). After The type of evolution considered here can be viewed as a each application, the dimension of the MPS increases. To ‘‘global quench’’ in the language of Calabrese and Cardy avoid an uncontrolled growth of the matrix dimensions, [14] as the initial state is the ground state of an artificial the MPS is truncated by keeping only the states which have Hamiltonian with local fields. Evolution from an initial the largest weight in a Schmidt decomposition. product state with zero entanglement can be studied effi- In order to control the error, we check that the neglected 6 ciently via time-dependent matrix product state methods weight after each step is small (< 10À ). Algorithms of until a time where the entanglement becomes too large for this type are efficient because they exploit the fact that the a fixed matrix dimension. Since entanglement cannot ground-state wave functions are only slightly entangled increase purely by local operations within each subsystem, which allows for an efficient truncation. Generally the its growth results only from propagation across the entanglement grows linearly as a function of time which

017202-2 Entropy growth in the many-body localised phase

Bose Hubbard model (1D)

U H = J a†a +h.c. + n (n 1) + n BH i i 2 i i i i J i ⇣ ⌘ i i U U X X X H = J a†a +h.c. + n (n 1) + n BH i i 2 i i i i i i i X ⇣ ⌘ X X 2 Disorder: [ , ] i 2 L = 10,N =5 Initial state: 1010101010 | i Quench dynamics of entanglement

101 ‘Thermalization’

100 power laws (2)

S 1 10 U = J, =0 10 2 1 0 1 2 10 10 10 10 Jt Entropy growth in the many-body localised phase

Bose Hubbard model (1D)

101 ‘Thermalization’

100 ‘Localization’ (2)

S 1 10 U = J = U = J, =0 10 2 1 0 1 2 10 10 10 10 Jt Entropy growth in the many-body localised phase

Bose Hubbard model (1D)

101 ‘Thermalization’

‘Localization’ (2) 1 U/J =1, /J = 10 S 10 U/J =0, /J = 10 U/J =1, /J =0

1 0 1 2 10 10 10 10 Jt Entropy growth in the many-body localised phase

Bose Hubbard model (1D)

101 ‘Thermalization’ 0.6 U/J =1, /J = 10 MBL U/J =0, /J = 10 ‘Localization’ 0.5 (2) (2) 1 U/J =1, /J = 10 S 10 S U/J =0, /J = 10 0.4 U/J =1, /J =0 Anderson Localization 0.3 1 0 1 2 0 1 2 10 10 10 10 10 10 10 Jt Jt EntropyEntropy growth growth in the - Thermalizationmany-body localised vs MBL phase

Bose Hubbard model (1D)

101 ‘Thermalization’ 0.6 U/J =1, /J = 10 MBL ‘Localization’ 0.5 U/J =0, /J = 10 (2) (2) 1 U/J =1, /J = 10 S 10 S U/J =0, /J = 10 0.4 U/J =1, /J =0 Anderson localised 0.3 1 0 1 2 0 1 2 10 10 10 10 10 10 10 Jt Jt

NM = 100 NU = 100 Conclusion

Measurement of Renyi entropies on single copies Ion Traps in atomic Hubbard and Spin models

• Renyi entropies can be inferred from random measurements Rydberg atoms • Uses only existing tools: ✓ Generic interactions ✓ Engineered disorder ✓ Quantum Gas microscope Hubbard models • Application to systems of moderate size

• Variant: Protocol based on local unitaries

Thank you very much for your attention!

arXiv: 1709.05060