It is shown that the Hartle-Hawking state of a scalar field is a maximum of
entanglement entropy in the space of pure quantum states satisfying the
condition that backreaction is finite. In other words, the Hartle-Hawking state
is a curved-space analogue of the EPR state, which is also a maximum of
entanglement entropy.Comment: Latex, 4 pages, Some comments are added on the "small backreaction
  condition

Bruyn, Arnoud

Campos, J.

Goessens, Wil

Groot, Ronald

Hermans, Peter

Sluijter, Marcel

Smith, Arnold

English

arXiv

Shinji Mukohyama

L. Bombelli

M. Srednicki

J. D. Bekenstein

S. W. Hawking

S. Mukohyama

J. B. Hartle

G. ’t Hooft

F. Belgiorno

S. Liberati

W. Israel

Crossref

Hartle-Hawking state is a maximum of entanglement entropy

Mukohyama, S

Kyoto University Research Information Repository

Title Hartle-Hawking state is a maximum of entanglement entropyAuthor(s)Mukohyama, SCitationPHYSICAL REVIEW D (2000), 61(6)Issue Date2000-03-15URL http://hdl.handle.net/2433/50456RightCopyright 2000 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityHartle-Hawking state is a maximum of entanglement entropyShinji MukohyamaYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japanand Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada V8W 3P6~Received 2 April 1999; published 24 February 2000!It is shown that the Hartle-Hawking state of a scalar field is a maximum of entanglement entropy in thespace of pure quantum states satisfying the condition that the back reaction be finite. In other words, theHartle-Hawking state is a curved-space analogue of the EPR state, which is also a maximum of entanglemententropy.PACS number~s!: 04.70.DyEntanglement entropy @1,2# is one of the strongest candi-dates for the origin of black hole entropy @3,4#. It originatesfrom a direct-sum structure of a Hilbert space of a quantumsystem: for an element uc& of the Hilbert space F of the formF5FI ^¯ FII , ~1!the entanglement entropy Sent is defined bySent52TrI@r I ln r I# ,r I5TrIIuc&^cu. ~2!Here ^¯ denotes a tensor product followed by a suitablecompletion and TrI ,II denotes a partial trace over FI ,II , re-spectively.In Ref. @5#, a new interpretation of the entanglement en-tropy was proposed based on its relation to the so-calledconditional entropy and a well-known meaning of the latter.It was conjectured that the entanglement entropy for a purestate is an amount of information, which can be transmittedthrough FII and FI from a system interacting with FII toanother system interacting with FI by using quantum en-tanglement. Thus, it is important to seek quantum states hav-ing maximal entanglement entropy and to investigate prop-erties of the states. In fact, it was shown that a state havingmaximal entanglement entropy plays an important role inquantum teleportation @5#.In this paper, we show that the Hartle-Hawking state @6#of a scalar field is a maximum of entanglement entropy in thespace of pure states satisfying a consistency condition.For simplicity, we consider a minimally coupled, real sca-lar field described by the actionS5212E d4xA2g@gmn]mf]nf1m2f2# , ~3!in the spherically symmetric, static black-hole spacetimeds252 f ~r !dt21 dr2f ~r ! 1r2dV2. ~4!We denote the area radius of the horizon by r0 and the sur-face gravity by k0 (Þ0):f ~r0!50,k0512 f 8~r0!. ~5!We quantize the system of the scalar field with respect to theKilling time t in a Kruskal-like extension of the black holespacetime. The corresponding ground state is called theBoulware state and its energy density is known to divergenear the horizon. Although we shall only consider states withbounded energy density, it is convenient to express thesestates as excited states above the Boulware ground state fortechnical reasons. Hence, we would like to introduce an ul-traviolet cutoff a with dimension of length to control thedivergence. Off course, we shall take the limit a→0 in theend. The cutoff parameter a is implemented so that we onlyconsider two regions satisfying r.r1 ~shaded regions I andII in Fig. 1!, where r1 (.r0) is determined bya5Er0r1 drAf ~r ! . ~6!~Strictly speaking, we also have to introduce outer bound-aries, say at r5L(@r0), to control the infinite volume of theconstant-t surface. However, even if there are outer bound-aries, the following arguments still hold and we can take thelimit L→‘ in the end.! Evidently, the limit a→0 corre-sponds to the limit r1→r0. Thus, in this limit, the wholeregion in which ]/]t is timelike is considered.In this situation, there is a natural choice for division ofthe system of the scalar field: let HI be the space of modefunctions with supports in the region I and HII be the spaceof mode functions with supports in the region II. Thence, theFIG. 1. The Kruskal-like extension of the static, sphericallysymmetric black-hole spacetime. We consider only the regions sat-isfying r.r1 ~the shaded regions I and II!.PHYSICAL REVIEW D, VOLUME 61, 0640150556-2821/2000/61~6!/064015~4!/$15.00 ©2000 The American Physical Society61 064015-1space F of all states are of the form ~1!, where FI and FII aredefined as symmetric Fock spaces constructed from HI andHII , respectively:FI[C % HI % ~HI ^¯ HI!sym %  ,FII[C % HII % ~HII ^¯ HII!sym %  . ~7!Here ()sym denotes the symmetrization.Let us investigate what kind of condition should be im-posed for our arguments to be self-consistent. A clear condi-tion is that the back reaction of the scalar field to the back-ground geometry should be finite. Thus, the contribution DMof the subsystem FI to the mass of the background geometryshould be bounded in the limit a→0. Although this conditionis not a sufficient condition in order to make the back reac-tion finite, what is surprising is that we can obtain the Hartle-Hawking state without using any other conditions strongerthan this. In this sense, our strategy of maximizing entangle-ment entropy seems as strong as the maximum entropy prin-ciple in the statistical thermodynamics.As shown in Ref. @7#, DM is given byDM[2ExPITtt4pr2dr5HI , ~8!where HI is the Hamiltonian of the subsystem FI with re-spect to the Killing time t. Hence, the expectation value ofDM with respect to a state uc& of the scalar field is decom-posed into the contribution of excitations and the contribu-tion from the zero-point energy:^cuDM uc&5Eent1DM B , ~9!where Eent is the entanglement energy1 defined byEent[^cu:HI :uc&, ~10!and DM B is the zero-point energy of the Boulware state.Here, the colons denote the usual normal ordering.Since the Boulware energy DM B diverges as DM B.2cATHa22 in the limit a→0, we should impose the condi-tionEent.uDM Bu ~11!in the lowest order in a, where A54pr02 is the area of thehorizon, TH5k0/2p is the Hawking temperature and c is apositive constant of order unity. We call this condition thesmall back reaction condition (SBC).2Note that the right hand side of SBC ~11! is independentof the state uc&. Thus, SBC implies that the entanglementenergy should be fixed when we maximize the entanglemententropy. In statistical thermodynamics, it is well known inwhat situation we should fix the average energy: we have tofix it when we know the observed value of the energy. How-ever, the corresponding situation seems not to have beenessentially known for ~quantum or thermal! excitations offields on a black hole background. In fact, for the brick wallmodel, in which thermal excitations are considered, Ref. @7#is the first which pointed out that the ground state for thebrick wall model is the Boulware state and that the Boulware~negative, divergent! energy should be added to thermal en-ergy. ~See Refs. @10–12# for the complete confusion reigningin this subject.! For our system in this paper, it is the Boul-ware ~negative, divergent! energy that makes us fix the en-tanglement energy.Now, we shall show that the Hartle-Hawking state is amaximum of the entanglement entropy in the space of purequantum states satisfying SBC. For this purpose, we prove amore general statement for a quantum system with a statespace of the form ~1!: a state of the formuc&5N(ne2En/2Tun& I ^ un& II ~12!is a maximum of the entanglement entropy in the space ofpure states with fixed expectation value of the operator EIdefined byEI5S (nEnun& I I^nu D ^ S (mum& II II^mu D , ~13!provided that the real constant T is determined so that theexpectation value of EI is actually a fixed value. Here, FIand FII are assumed to be isomorphic to each other, $un& I%and $un& II% (n51,2, . . . ) are bases of the subspaces FI andFII , respectively, and En are assumed to be real and posi-tive. Note that this statement looks almost the same as thefollowing statement in statistical thermodynamics: a canoni-cal state is a maximum of statistical entropy in the space ofstates with fixed energy, provided that the temperature of thecanonical state is determined so that the energy is actuallythe fixed value. Thus, it might be expected that the abovegeneral statement might be directly derived from the stan-dard Jaynes method @13# as this statement in statistical ther-modynamics can be derived. However, the Jaynes methodcannot be applied directly to our system since the Jaynesmethod treats entropy of not a subsystem but a total system.Thus, in the proof given below, we seek a correspondencebetween our variational principle and the standard variationalprinciple in statistical thermodynamics. @See Eq. ~23!.#Note that the expectation value of EI is equal to the en-tanglement energy ~10!, provided that un& I and En are aneigenstate and an eigenvalue of the normal-ordered Hamil-tonian :HI : of the subsystem FI . Hence, for the system ofthe scalar field, the above general statement insists that thestate ~12! is a maximum of the entanglement entropy in thespace of pure states satisfying the SBC, which corresponds1This definition corresponds to Eent(I8) in Ref. @8# and ^:H2 :& in Ref.@9#.2Off course, finite energy can be added to the entanglement en-ergy without generating a divergent back reaction. However, thefinite energy is higher order in the a expansion. Thus, effects of thefinite energy are higher order in a.SHINJI MUKOHYAMA PHYSICAL REVIEW D 61 064015064015-2to fixing the entanglement energy. Of course, in this case, theconstant T should be determined so that SBC ~11! is satis-fied.Returning to the subject, let us prove the general state-ment. ~The following proof is almost the same as that givenin the Appendix of Ref. @5# for a slightly different statement.However, for completeness, we shall give the proof.!First, we decompose an element uc& of F asuc&5(n ,mCnmun& I ^ um& II , ~14!where the coefficients Cnm (n ,m51,2, . . . ) are complexnumbers satisfying (n ,muCnmu251 and can be considered asmatrix elements of a matrix C. Since C†C is a non-negativeHermitian matrix, it can be diagonalized asC†C5V†PV , ~15!where P is a diagonal matrix with diagonal elements pn(>0) and V is a unitary matrix. For this decomposition anddiagonalization, the entanglement entropy and the expecta-tion value of the operator EI are written as follows:Sent52(npn ln pn , ~16!Eent5(n ,mEnpmuVnmu2, ~17!where Vnm is matrix elements of V. The constraints(n ,muCnmu251 and V†V51 are equivalent to(npn51,(lVln* Vlm5dnm . ~18!Next, we shall show that these expressions are equivalentto those appearing in statistical mechanics in FI . Let us con-sider a density operator r¯ on FI :r¯5(n ,mP˜ nmun& I I^mu, ~19!where (P˜ nm) is a non-negative Hermitian matrix with unittrace. By diagonalizing the matrix P˜ asP˜ 5V¯ †P¯ V¯ , ~20!we obtain the following expressions for entropy S and anexpectation value E of the operator E¯ I[(nEnun& I I^nu:S52(np¯ n ln p¯ n ,E5(n ,mEnp¯ muVnmu2, ~21!where p¯ n is the diagonal elements of P¯ . The constraintsTr r¯51 and V¯ †V¯ 51 are restated as(np¯ n51,(lV¯ ln* V¯ lm5dnm . ~22!From these and those expressions, the following corre-spondence is easily seen:Sent↔S ,Eent↔E ,C†C↔P˜ . ~23!Hence, a maximum of S in the space of statistical states witha fixed value of E gives a set of maxima of Sent in a space ofquantum states with a fixed value of Eent . ~All of them arerelated by unitary transformations in the subspace FII .!Thus, since the thermal state P˜ nm5e2En /Tdnm is a maximumof S in a space of statistical states with a fixed value ofE , Cnm5e2En/2Tdnm is a maximum of Sent in a space ofpure quantum states with a fixed value of Eent . Here thetemperature T should be determined so that E ~or Eent! has afixed value. This completes the proof of the general state-ment.Therefore, for the system of the scalar field, a state of theform ~12! is a maximum of the entanglement entropy in aspace of pure quantum states satisfying the SBC, providedthat the constant T is determined so that the SBC is satisfied.The value of T is easily determined as T5TH by using thewell-known fact that the negative divergence in the Boul-ware energy density can be canceled by thermal excitationsif and only if temperature with respect to the time t is equalto the Hawking temperature.3Finally, by taking the limit a→0, we obtain the statementthat the Hartle-Hawking state is a maximum of entanglemententropy in a space of pure quantum states satisfying theSBC.4 In other words, the Hartle-Hawking state is a curved-space analogue of the Einstein-Podolsky-Rosen ~EPR! state,which is also a maximum of entanglement entropy @5#.From this result we can say that the brick wall model of ’tHooft @10# seeks the maximal value of entanglement entropy@15#. Thus, the maximal entanglement entropy is equal to theblack hole entropy if the cutoff length a is set to be of theorder of the Planck length. Although in our arguments we3See e.g., Ref. @7#.4The Hartle-Hawking state is actually of the form ~12! with T5TH @14#, provided that a→0.HARTLE-HAWKING STATE IS A MAXIMUM OF . . . PHYSICAL REVIEW D 61 064015064015-3have taken the limit a→0, it will be valuable to investigatepossibilities that quantum fluctuations of geometries mayprevent a from being zero and that the fluctuations of thehorizon may be effectively represented as a Planck-ordervalue of a. Note that the effect of a non-zero value of a onour arguments should be small enough if the mass of a back-ground black hole is sufficiently large in Planck units.Our arguments suggests a strong connection among threekinds of thermodynamics: black hole thermodynamics, sta-tistical thermodynamics, and entanglement thermodynamics@5,8,9,16#. Moreover, from the interpretation of entangle-ment entropy proposed in Ref. @5#, it is expected that theHartle-Hawking state may play an important role in transmit-ting information by using quantum entanglement to restoretemporarily missing information. It will be interesting to in-vestigate such a possibility in detail.The author would like to thank Professor H. Kodama andProfessor W. Israel for their continuing encouragement. Thiswork was supported partially by a Grant-in-Aid for ScientificResearch Fund ~No. 9809228!.@1# L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev.D 34, 373 ~1986!.@2# M. Srednicki, Phys. Rev. Lett. 71, 666 ~1993!.@3# J. D. Bekenstein, Phys. Rev. D 7, 949 ~1973!.@4# S. W. Hawking, Commun. Math. Phys. 43, 199 ~1975!.@5# S. Mukohyama, Phys. Rev. D 58, 104023 ~1998!.@6# J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188~1976!.@7# S. Mukohyama and W. Israel, Phys. Rev. D 58, 104005~1998!.@8# S. Mukohyama, M. Seriu, and H. Kodama, Phys. Rev. D 58,064001 ~1998!.@9# S. Mukohyama, Ph. D. thesis, gr-qc/9812079.@10# G. ’t Hooft, Nucl. Phys. B256, 727 ~1985!.@11# F. Belgiorno and S. Liberati, Phys. Rev. D 53, 3172 ~1996!; S.Liberati, Nuovo Cimento Soc. Ital. Fis., B 112, 405 ~1997!.@12# F. Belgiorno and M. Martellini, Phys. Rev. D 53, 7073 ~1996!.@13# E. T. Jaynes: Papers on Probability, Statistics and StatisticalPhysics, edited by R. D. Rosenkrantz ~Reidel, Boston, 1983!.@14# W. Israel, Phys. Lett. 57A, 107 ~1976!.@15# S. Mukohyama, in the proceedings of the the 8th CanadianConference on General Relativity and Relativistic Astrophys-ics, gr-qc/9907090.@16# S. Mukohyama, M. Seriu, and H. Kodama, Phys. Rev. D 55,7666 ~1997!.SHINJI MUKOHYAMA PHYSICAL REVIEW D 61 064015064015-4

We previously demonstrated that trimethoprim (Tmp) resistance in
          Haemophilus influenzae is mediated by chromosomally encoded dihydrofolate
          reductase (DHFR) with a modified primary structure and distinct kinetic
          properties. To gain insight into the relationship of the DHFR structure
          and the level of Tmp resistance that it confers on the host bacterium, we
          cloned and characterized the folH genes of one Tmp-susceptible and two
          Tmp-resistant H. influenzae strains. Differences were observed between
          Tmp-susceptible and Tmp-resistant isolates both in the promoter region and
          in the coding sequences. The effect of differences between H. influenzae
          folH genes on Tmp susceptibility was investigated in Escherichia coli.
          Various folH gene hybrids were constructed, and their influence on Tmp
          susceptibility was determined. Resistance in E. coli mediated by folH from
          H. influenzae strain R1047 was associated with alterations in the promoter
          and the central part of folH. In contrast, the E. coli Tmp resistance
          phenotype associated with the folH gene of H. influenzae R1042 was
          characterized by alterations in one or more of three amino acid residues
          at the C-terminal part of the protein. These data indicate that Tmp
          resistance is not only related to alterations in the promoter region of
          the folH gene and the Tmp binding domains at the N-terminal and central
          part of DHFR. Alterations in the C-terminal part may also cause Tmp
          resistance, probably as a result of a change in secondary structure and
          the subsequent loss of Tmp binding affinity

Hartle-Hawking state is a maximum of entanglement entropy

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