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Universality in Volume-Law Entanglement of Scrambled Pure Quantum States

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Universality in Volume-Law Entanglement of Scrambled Pure Quantum States ARTICLE DOI: 10.1038/s41467-018-03883-9 OPEN Universality in volume-law entanglement of scrambled pure quantum states Yuya O. Nakagawa 1,2, Masataka Watanabe 2,3, Hiroyuki Fujita 1,2 & Sho Sugiura 1 A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. The thermodynamic entropy can also be recovered as the entanglement entropy of small subsystems. When the size of the subsystem increases, however, quantum 1234567890():,; correlations break the correspondence and mandate a correction to this simple volume law. The elucidation of the size dependence of the entanglement entropy is thus essentially important in linking quantum physics with thermodynamics. Here we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing equilibrium. We numerically find that our formula applies universally to any sufficiently scrambled pure state representing thermal equilibrium, i.e., energy eigenstates of non-integrable models and states after quantum quenches. Our formula is exploited as diagnostics for chaotic systems; it can distinguish integrable models from non-integrable models and many-body localization phases from chaotic phases. 1 Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan. 2 Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 133-0022, Japan. 3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan. Correspondence and requests for materials should be addressed to Y.O.N. (email: [email protected]) or to S.S. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:1635 | DOI: 10.1038/s41467-018-03883-9 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03883-9 s a measure of the quantum correlations in many-body (cTPQ) states, which are pure states representing thermal − systems, the entanglement entropy (EE) has become an equilibrium at a temperature β 117,18. In particular, the Page A fi indispensable tool in modern physics. The EE quanti es curve of the 2REE is controlled only by two parameters, an the amount of non-local correlation between a subsystem and its effective dimension ln a and an offset ln K, for any Hamiltonian compliment. The ability of the EE to expose the non-local fea- and at any temperature. We then conjecture and numerically tures of a system offers a way to characterize topological orders1,2, verify that this feature of the Page curve universally appears in solve the black hole information paradox3, and quantify any sufficiently scrambled pure states representing equilibrium information scrambling in a thermalization process under unitary states; that is, our function fits the 2REE of the energy eigenstates – time evolution4 6. of a non-integrable system and the states after quantum quenches Recently, the EE was measured in quantum many-body sys- including the state realized in the above-mentioned experiment8. tems for the first time7,8. Specifically, the second Rényi EE By contrast, in the case of the energy eigenstates of an integrable (2REE), one of the variants of the EE, of a pure quantum state was system, which are not scrambled at all, we find that their Page measured in quantum quench experiments using ultra-cold curves deviate from our function. Since our function enables us to atoms. For such pure quantum states, it is believed that the EE estimate the slope of the volume law from small systems with of any small subsystem increases in proportion to the size of the high accuracy and precision, our result is also numerically subsystem just like the thermodynamic entropy9. This is called effective in detecting the ETH-MBL transition15,16,19,20 and the volume law of the EE. However, when the size of a subsystem improves the estimation of the critical exponent. becomes comparable to that of its complement, it is observed experimentally that the EE starts deviating from the volume law, Results and eventually decreases (Fig. 1). The curved structure of the size Derivation of the Page curve in cTPQ states. Let us consider a dependence of the EE, which first increases linearly and then lattice Σ containing L × M sites (Fig. 1), equipped with a decreases, universally appears in various excited pure states, for translation-invariant and local Hamiltonian H. We divide Σ into example, energy eigenstates10 and states after quantum quen- two parts, A and B, each containing ‘ ´ M and ðL À ‘Þ ´ M sites. ches11,12. We call this curved structure a Page curve, after D. The n-th Rényi EE of a pure quantum state |ψ〉 is defined as Page13, and it is of fundamental importance in explaining these ÀÁ ‘ 1 ρn ; examples to reveal a universal behavior of the Page curve. Snð Þ¼ ln tr A ð1Þ Despite their ubiquitous appearance, the theoretical under- 1 À n standing of the Page curves is limited to the case of a random where ρ ≡ tr |ψ〉〈ψ|. We call S ð‘Þ as a function of ‘ the n-th pure state13,14, which is a state at infinite temperature and can be A B n Rényi Page curve (nRPC). We note that we use the term defined only in a finite-dimensional Hilbert space. By contrast, differently from how it is used in the context of quantum gravity, the cold-atom experiments address finite temperature systems where it denotes the temporal dependence of the entanglement and an infinite-dimensional Hilbert space; therefore, it is during the formation of a black hole. To simplify the calculation, important to develop a theory of the Page curve applicable to we assume ‘; L 1. these situations. Additionally, there are practical needs for the To derive the behaviors of nRPCs for any Hamiltonian, we estimation of the slope of the volume law. The slope is often utilize cTPQ states, which are proposed along with studies of employed, for example, to calculate the corresponding thermo- – typicality in quantum statistical mechanics21 27.The cTPQ state dynamic entropy8,10 and to detect a transition between the energy at the inverse temperature β is defined as eigenstate thermalization hypothesis (ETH) phase and the 15,16 X many-body localized (MBL) phase . However, since the ψ p1ffiffiffiffiffiffi ÀβH=2 ; ji¼ zje jij ð2Þ experimentally or numerically accessible sizes of the systems are Zψ j small, the estimation of the volume-law slope is deteriorated by P the curved structure of the Page curve. Ã ÀβH 〉 where Zψ i;j zi zjhije jji is a normalization constant, {|j }j In this work, we show that Page curves in broad classes of is an arbitrary complete orthonormal basis of the Hilbert space excited pure states exhibit universal behaviors. We first derive the fi HΣ, and thep coefffiffiffi cients {zj} are random complex numbers function of the Page curve for canonical thermal pure quantum = zj ðxj þ iyjÞ 2, with xj and yj obeying the standard normal distribution Nð0; 1Þ. For any local observable, the cTPQ states at β reproduce their averages in thermal equilibrium at the same L – inverse temperature18. As a starting point, we here derive the − following exact formula of the 2RPC at a temperature β 1 for any Hamiltonian (see the Methods section for the calculations and A B results for the nRPCs), "# ÀβH 2 ÀβH 2 S trAðtrBe Þ þ trBðtrAe Þ 2 S2ð‘Þ¼ À ln : ð3Þ ðtreÀβHÞ2 Volume law In(2) We also give several simplifications of Eq. (3). The first step is = + + to decompose the Hamiltonian H as H HA HB Hint, where O L⏐2 L HA,B are the Hamiltonians of the corresponding subregion and Fig. 1 A schematic picture of our setup. The second Rényi Page curve for Hint describes the interactions between them. Since the range of interaction H is much smaller than ‘ and L À ‘ due to the pure states, S ð‘Þ, follows the volume law when ‘ is small, but gradually int 2 locality of H, we obtain the simplified expression deviates from it as ‘ grows. At the middle, ‘¼L=2, the maximal value is ! obtained, where the deviation from the volume law is ln 2 (see the Results Z ð2βÞ Z ð2βÞ ‘ L= ‘ L S ð‘Þ¼ À ln A þ B þlnRðβÞ; ð4Þ section). Past the middle ¼ 2, it decreases toward ¼ and becomes 2 β 2 β 2 symmetric under ‘ $ L À ‘ ZAð Þ ZBð Þ 2 NATURE COMMUNICATIONS | (2018) 9:1635 | DOI: 10.1038/s41467-018-03883-9 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-03883-9 ARTICLE β where R( ), coming from Hint,isanO(1) constant dependent 3.0 β L=14 0.25 β β À HA;B only on and ZA;Bð ÞtrA;Bðe Þ. L=16 0.20 0.15 Further simplification occurs through the extensiveness of the 2.5 L=18 In(α) L=20 0.10 S (L 2) (L 2) − β 2 free energy, ln ZA,B/ , which is approximately valid when L=22 0.05 ΔS Δ in [1,5] ‘; ‘ 0.00 2 L À 1. In the region in question, ln ZA,B is proportional to 2.0 L=50 02468 the volume of the corresponding subregion, and thus, we L=100 1 L [× 10–2] 2 2 À‘ β β β β β β 2 replace ZA(2 )/ZA( ) and ZB(2 )/ZB( ) with Qð Það Þ and 1.5 ‘ S QðβÞaðβÞÀðLÀ Þ, respectively. Here, a(β) and Q(β) are O(1) constants dependent only on β, and 1 < a(β) holds because of 1.0 the concavity and monotonicity of the free energy.
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