RRelative Entropy of Random States and Black Holes
Jonah Kudler-Flam ∗ Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, IL 60637, USA (Dated: February 11, 2021)We study the subsystem relative entropy of highly excited quantum states. First, we indepen-dently draw the two density matrices from the Wishart ensemble and develop a large-N diagrammatictechnique to compute the relative entropy. The solution is monotonically increasing from zero andexactly expressed in terms of hypergeometric functions. We compare the analytic results to small-Nnumerics, finding precise agreement. We then apply this formalism to “fixed-area” states in theAdS/CFT correspondence. In this context, the relative entropy measures the distinguishability be-tween different black hole microstates. We find that black hole microstates are distinguishable evenwhen the observer has arbitrarily small access to the boundary region, though the distinguishabilityis nonperturbatively small in Newton’s constant. Finally, we interpret these results in the context ofthe subsystem Eigenstate Thermalization Hypothesis (ETH), concluding that holographic systemsobey subsystem ETH up to subsystems half the size of the total system.
Introduction.
Random matrices are a unifying sub-ject in quantum physics. From encoding quantum in-formation [1], to characterizing complicated many-bodysystems and quantum chaos [2], to serving as toy mod-els of the black hole information problem [3, 4], randomquantum states have become invaluable across many dis-tinct subfields. Moreover, the mathematical field of ran-dom matrix theory is very mature, enabling analyticalcalculations in random states that would be otherwiseintractable.With these general motivations in mind, we study therelative entropy of random quantum states. The relativeentropy between two density matrices ρ and σ is definedas D ( ρ || σ ) = Tr [ ρ (log ρ − log σ )] . (1)As a distinguishability measure, it obeys various niceproperties, such as positivity with D ( ρ || σ ) = 0 if andonly if ρ = σ . Crucially, the relative entropy is mono-tonic under quantum operations [5] D ( N ( ρ ) ||N ( σ )) ≤ D ( ρ || σ ) , (2)where N is any completely-positive trace-preservingmap. A particularly important quantum channel that wewill come back to is the partial trace operation. Mono-tonicity here means that density matrices become lessdistinguishable as you throw out more information aboutthem, an intuitive notion.Relative entropy is truly the mother of all quantitiesin quantum information theory. While at face value, itjust measures the distinguishability between two densitymatrices, upon further inspection, its fundamental prop-erties underlie many of the deepest universal statementsabout quantum mechanics [6, 7], quantum field theory[8–10], and quantum gravity [11, 12]. ∗ jkudlerfl[email protected] We first introduce random mixed states and their dia-grammatic representation. We then embark on our maincomputation of the relative entropy between indepen-dently sampled random density matrices, finding a closedform solution in terms of hypergeometric functions. Thegeneral formula is exact in the limit of large Hilbert spacedimensions but we find it to be remarkably accurate evenfor small Hilbert space dimensions. Afterwards, we ap-ply our formalism to the AdS/CFT correspondence. Ourcentral conclusion in this analysis is that the relative en-tropy between black hole microstates is finite, thoughnon-perturbatively small in Newton’s constant up untilthe subsystem is half of the total system size. This is anextremely strong version of the eigenstate thermalizationhypothesis [13, 14], a statement that we subsequentlymake precise. We wrap up with speculations regardingfuture directions and the general applicability of our re-sults to chaotic quantum systems.
Random Mixed States.
We begin with a Haar randompure state on a bipartite Hilbert space H A ⊗ H B | Ψ (cid:105) = d A (cid:88) i =1 d B (cid:88) α =1 X iα | i (cid:105) A ⊗ | α (cid:105) B , (3)where the states in the sum are orthonormal bases for thesub-Hilbert spaces of dimensions d A and d B . The X iα ’sare independently distributed complex Gaussian randomvariables with joint probability distribution P ( { X iα } ) = Z − exp (cid:2) − d A d B Tr (cid:0) XX † (cid:1)(cid:3) , (4)where Z is the normalization constant, ensuring theexpression defines a probability. The random inducedstates on H A are then ρ A = XX † Tr( XX † ) . (5)We note that the denominator is a random variable thatis sharply peaked around unity, so we can ignore it in thelimit of large Hilbert space dimension [15] ρ A (cid:39) XX † . (6) a r X i v : . [ h e p - t h ] F e b This defines the Wishart ensemble. We now introduce adiagrammatic representation of the density matrix [16–18] [ | Ψ (cid:105) (cid:104) Ψ | ] iα,jβ = X ∗ iα X jβ = α βi j . (7)The solid and dashed lines correspond to subsystems A and B respectively. Matrix manipulations are done atthe bottom edge of the diagram. For example, the partialtrace over H B is[ ρ A ] i,j = d B (cid:88) α =1 X ∗ iα X jα ≡ α αi j . (8)Ensemble averaging is done at the top of the diagramwith propagators carrying weight ≡ (cid:104) X ∗ iα X jβ (cid:105) = 1 d A d B δ ij δ αβ . (9)Putting these operations together, we can, for example,take the trace of the density matrix (cid:104) Tr ρ A (cid:105) = = 1 , (10)where every closed loop gives a factor of the Hilbert spacedimension. The diagrammatic rules for averaging assertthat we must sum over all possible contractions of thebras and kets. For relative entropy, we need two inde-pendent density matrices, ρ A and σ A . These must beaveraged over the ensemble separately. To make this dis-tinction, we color σ A red.The logarithms in the definition of relative entropymake the quantity significantly more difficult to computeanalytically than simple powers of the density matrices.Happily, a replica trick for the relative entropy has beendeveloped that re-expresses the logarithm as a limit ofappropriate powers [19] D ( ρ || σ ) = lim n → n − (cid:0) log Tr ρ n − log Tr ρσ n − (cid:1) . (11)We will compute these two terms separately. The firstterm is recognized as minus the R´enyi entropy. For n = 2,we have Tr ρ A = . (12)The ensemble average is a sum of the two contractions (cid:104) Tr ρ A (cid:105) = + , (13)immediately giving d − A + d − B . This can be generalizedto arbitrary powers. Because of the sum over all possible contractions, in general, the moments are expressible asa sum over the permutation group (cid:104) Tr ρ nA (cid:105) = 1( d A d B ) n (cid:88) τ ∈ S n d D ( η − ◦ τ ) A d D ( τ ) B , (14)where D ( · ) is the number of cycles in the permutationand η is the cyclic permutation. When the Hilbert spacesare large, only the terms that maximize D ( η − ◦ τ ) + D ( τ ) will contribute to the sum at leading order. Theseare known as the non-crossing permutations and have D ( η − ◦ τ ) + D ( τ ) = n + 1. Much is known about thisspecial subset of permutations including that the numberof such permutations with D ( η − ◦ τ ) = k is given by theNarayana number N n,k = 1 n (cid:18) nk (cid:19)(cid:18) nk − (cid:19) . (15)Thus, the sum becomes (cid:104) Tr ρ nA (cid:105) = 1( d A d B ) n n (cid:88) k =1 N n,k d kA d n +1 − kB = d − nA F (cid:18) − n, − n ; 2; d A d B (cid:19) , (16)where F is a hypergeometric function. This reproducesPage’s famous result [20].Next, we consider the second term of (11). For simplic-ity, we first consider the overlap between the two densitymatrices which, as a diagram, looks likeTr( ρ A σ A ) = . (17)We must ensemble average the black and red lines sepa-rately, so there is only a single term (cid:104) Tr( ρ A σ A ) (cid:105) = , (18)giving d − A . We again generalize this to arbitrary powersby expressing the moments in terms of a sum over thepermutation group (cid:104) Tr( ρ A σ n − A ) (cid:105) = 1( d A d B ) n (cid:88) τ ∈ × S n − d D ( η − ◦ τ ) A d D ( τ ) B . (19)The crucial difference between this expression and (14)is that the sum is only over a subgroup of permutations,namely the ones that stabilize the first element. Thesepermutations still contain many non-crossing permuta-tions that will dominate the sum.The combinatorics of these non-crossing permutationsare encoded within the beautiful formula of Kreweraswhich states that the number of non-crossing permuta-tions of type (1 m m . . . n m n ) is [21, 22] N C n { m i } = n ( n − . . . ( n − b + 2) m ! m ! . . . m n ! , (20)where b ≡ (cid:80) i m i ≥
2. We are able to deduce that thesum of non-crossing permutations can be reorganized as (cid:104) Tr ρ A σ n − A (cid:105) = 1( d A d B ) n n − (cid:88) k =1 C n,k d kA d n +1 − kB ,C n,k ≡ n − k (cid:18) n − k (cid:19)(cid:18) n − k − (cid:19) . (21)Like the R´enyi entropies, this may also be written as ahypergeometric function (cid:104) Tr ρ A σ n − A (cid:105) = d − nA F (cid:18) − n, − n ; 2; d A d B (cid:19) . (22)Combining (16) and (22), we can unambiguously take the n → D ( ρ A || σ A ) = − F (0 , , , (cid:18) − , , , d A d B (cid:19) + F (0 , , , (cid:18) , , , d A d B (cid:19) − F (1 , , , (cid:18) − , , , d A d B (cid:19) + F (1 , , , (cid:18) , , , d A d B (cid:19) . (23)This is our main result. The superscripts denote deriva-tives on the corresponding arguments. This formula iszero when d A /d B →
0. This is to be expected becausedensity matrices become indistiguishable when most ofthe information is “traced away.” The relative entropymonotonically increases with d A /d B , reaching a curiousvalue of 3 / d A = d B . This monotonic behav-ior is just a restatement of the monotonicity of relativeentropy under the partial trace quantum channel. For d A > d B , the density matrices are rank deficient lead-ing to the formula giving a value with a small imaginarypart asymptoting to − π . The real part of the functioncontinues to monotonically increase, diverging linearlyas D ( ρ A || σ A ) → d A d B . This suggests that this may stillbe a meaningful quantity, though we should really onlytrust the solution when it is real. We plot this functionin Fig. 1 and compare to numerics, finding very goodagreement even for the relatively small Hilbert space di-mensions that are accessible on a classical computer. Black Hole Microstates.
Here, we reinterpret (23) inthe context of the AdS/CFT correspondence. In thiscorrespondence, high energy pure states in the bound-ary conformal field theory are dual to black holes mi-crostates in the bulk. Relative entropy then tells us howwell we can distinguish different black hole microstatesof equal energy, a notoriously difficult task that, a priori,one would expect to require knowledge of the full ultra-violet complete quantum gravity theory, such as string In general, the ensemble average and logarithm do not commute,requiring a further replica trick. However, these operations ap-proximately commute for large Hilbert space dimensions. - FIG. 1. Comparison of equation (23) (dashed line) with nu-merics. The blue, red, and green data points are for totalHilbert space sizes of 1024, 6561, and 15625 respectively. Thefluctuations in the relative entropy are clearly suppressed asthe dimension is increased, signaling self averaging. theory [23]. Surprisingly, we show that this is actuallypossible just from semiclassical gravity, which is relatedto the recent surprise that the Page curve can be calcu-lated from semiclassical gravity [24, 25].This is simplest for “fixed-area states” [26, 27], whichwill be enough for our purposes. Fixed area states areholographic states where the areas of gauge-invariant sur-faces in the bulk have been measured. These states haveplayed an important role in the understanding of holo-graphic entanglement entropy and quantum error correc-tion.While many details can be found in the original papersand illuminating follow-ups [28–32], we will only presentwhat is necessary for our analysis. We consider stateswhere two surfaces have been measured. These are thetwo candidate extremal surfaces, γ and γ , that computethe von Neumann entropy [33], depicted in Fig. 2. Thetwo surfaces wrap the black hole horizon in topologicallydistinct manners. In the gravitational replica trick, thecodimension-one region bounded by γ and A is gluedcyclically to the next replica, while the region between γ and B is simply glued to the same replica. These gluingsare determined by the asymptotic boundary conditions.The interesting region between γ and γ is not fixed bythe boundary conditions and can therefore be glued usingany permutation. This can be thought of as a sum over“replica wormholes” [32].To compute the relative entropy between two differentblack hole microstates, we must computeTr( ρ A σ n − A ) = Z ( ρ A σ n − A ) Z ( ρ A ) Z ( σ A ) n − , (24)where Z is the gravitational path integral evaluated on-shell with the boundary conditions dictated by the ar-gument. Due to nice properties of fixed area states, theonly contributions to the on-shell action (after normal-ization) come from the conical deficits that can occur at FIG. 2. Depicted is a black hole geometry with the boundarypartitioned into regions A and B . There are two competingextremal surfaces, γ and γ , that we fix the area of. Whenperforming the replica trick, we compute the path integral on n copies of this geometry. Each bulk region is labeled by thepermutation element that governs how it is glued among thecopies. γ and γ , leading toTr( ρ A σ n − A ) = (cid:88) τ ∈ × S n − e ( D ( η − ◦ τ ) A + D ( τ ) A ) / G N e n ( A + A ) / G N , (25)where A and A are the areas of the fixed surfaces and G N is Newton’s constant. This expression is identical to(19) once we identify d A = e A / G N and d B = e A / G N .A similar conclusion is made for Tr( ρ nA ). Therefore,the relative entropy between black hole microstates isgiven by (23), which is UV finite because while theareas are themselves divergent, their difference is reg-ulator independent. It is important to note that forsmall A , i.e. small boundary subregion A , the rela-tive entropy is nonzero, meaning the two black hole mi-crostates are distinguishable! The catch is that the dis-tinguishability is non-perturbatively small in Newton’sconstant, O ( e − /G N ). However, as A approaches A ,the relative entropy becomes O (1). The transition from O ( e − /G N ) to O (1) occurs in an extremely tiny windowwhen ( A − A ) / G N (cid:46) log 2, roughly meaning that re-gion A contains one less qubit of information than region B .In passing, we note that these results also apply to therelative entropy of two states in the Jackiw-Teitelboimgravity plus end-of-the-world brane model of black holeevaporation from Ref. [32] in the case that the black holeis in the microcanonical ensemble. Subsystem Eigenstate Thermalization.
SubsystemETH is a generalization of the standard local ETH storyand is significantly stronger. While ETH is a statementabout local operators, subsystem ETH is a statementthat finite subregions appear thermal. Precisely, subsys-tem ETH holds when sufficiently highly excited eigen-states have reduced density matrices that are exponen-tially close in trace distance to some universal density matrices, such as the microcanonical ensemble [34] T ( ρ ψ , ρ univ ) ≡ | ρ ψ − ρ univ | ≤ O ( e − S ( E ) / ) , (26)where e S ( E ) is the density of states of the full system. Inthe context of holography, the entropy scales as O ( G − N ),so subsystem ETH just means that the trace distance isnonperturbatively small in Newton’s constant.We now invoke the quantum Pinsker inequality D ( ρ || σ ) ≥ T ( ρ, σ ) . (27)We previously found D ( ρ A || σ A ) to scale as O ( e − /G N )on average for any two black hole microstates with fixedarea. This implies that the trace distance is, at most, O ( e − /G N ). The trace distance defines a metric on thespace of density matrices, so if a typical state is close toa measure one set of all other states, then the universaldensity matrix should sit within this ball. We thereforeclaim that fixed area states in all dimensions obey subsys-tem ETH for subsystems less than half the total systemsize. The violation of subsystem ETH only occurs when( A − A ) / G N (cid:46) log 2. Discussion.
There are various interesting directionsthat lie outside the scope of this paper: (i) We havecomputed the average relative entropy between typicalrandom mixed states. However, we have not fixed thecomplete distribution. It would be interesting to charac-terize the fluctuations in relative entropy. The numericalresults suggest that these are suppressed in the Hilbertspace dimension. Higher moments of the relative entropycan be computed using the same technology we have de-veloped in this paper. (ii) In our applications to hologra-phy, we focused on fixed-area states. More generic statesare superpositions of fixed-area states with a sharplypeaked Gaussian distributions of width O ( √ G N ) [29]. Itis important to study the relative entropy of these moregeneric states to verify that the relative entropy is qual-itatively the same. We can invoke convexity of the tracedistance T (cid:32)(cid:88) i p i ρ i , (cid:88) i q i σ i (cid:33) ≤ T ( p i , q i ) + (cid:88) i p i T ( ρ i , σ i ) , (28)where the ρ i and σ i ’s are fixed area states and T ( p i , q i )is the classical trace distance between probability distri-butions. We have already shown that the second term onthe right hand side is O ( e − /G N ). If we assume that theprobability distributions are Gaussian with equal widthsbut centered at fixed areas a distance at most O ( e − /G N )apart i.e. within the same microcanonical window, then itis a straightforward exercise to confirm that the first termis also O ( e − /G N ), confirming subsystem ETH. However,if the widths of the Gaussian distributions are different,even by an amount polynomial in G N , the bound will nolonger be tight. It would be fascinating if these correc-tions led to violations of eigenstate thermalization. (iii)One of our motivations to study random states is thatthey should be representative of generic excited states innonintegrable quantum systems. It is clearly interestingto look into how accurately our results characterize realHamiltonian systems (beyond holography). We providepreliminary numerical results for the Sachdev-Ye-Kitaev(SYK) model and for spin chains in the supplemental ma-terial. While the SYK eigenstates mimic random matrixtheory, we find that chaotic spin chain eigenstates haveclose to, but larger, relative entropy than random statesand integrable eigenstates have even larger relative en- tropy and much larger fluctuations. We hope to report amore systematic study in the future. Acknowledgements.
I am grateful to Chris Akers,Hong Liu, Pratik Rath, Shinsei Ryu, Hassan Shapourian,and Shreya Vardhan for helpful discussions and com-ments. I am supported through a Simons Investiga-tor Award to Shinsei Ryu from the Simons Foundation(Award Number: 566166).
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We provide a numerical study of relative entropy between mid-spectrum eigenstates of integrable and chaotic spinchains of length N with Hamiltonian H = − N (cid:88) i =1 ( Z i Z i +1 + h x X i + h z Z i ) , (29)where X and Z are Pauli spin operators. We take h x = 1, h z = 0 for the integrable limit and h x = − . h z = 0 . H = N (cid:88) j