Lattice QCD study of static quark and antiquark correlations via entanglement entropies
LLattice QCD study of static quark and antiquark correlations via entanglemententropies
Toru T. Takahashi
National Institute of Technology, Gunma college, Maebashi, Gunma 371-8530, Japan
Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Sakyo, Kyoto 606-8502, Japan (Dated: October 4, 2019)We study the color correlation between static quark and antiquark ( q ¯ q ) in the confined phase viareduced density matrices ρ defined in color space. We adopt the standard Wilson gauge action andperform quenched calculations with the Coulomb gauge condition for reduced density matrices. Thespatial volumes are L = 8 , 16 , 32 and 48 , with the gauge couplings β = 5 .
7, 5.8 and 6.0. Eachelement of the reduced density matrix in the sub space of quarks’ color degrees of freedom of the q ¯ q pair is calculated from staples defined by link variables. As a result, we find that ρ is well written bya linear combination of the strongly correlated q ¯ q pair state with the color-singlet component andthe uncorrelated q ¯ q pair state with random color configurations. We compute the Renyi entropies S Renyi from ρ to investigate the q ¯ q distance dependence of the color correlation of the q ¯ q pair andfind that the color correlation is quenched as the distance increases. I. INTRODUCTION
Color confinement is one of the nonperturbative fea-tures of Quantum ChromoDynamics (QCD), the fun-damental theory of the strong interaction. The staticinterquark potential ( q ¯ q potential) in the confinementphase exhibits a linearly rising potential in the large-separation limit giving the diverging energy, and quarkscannot be isolated. Such confining features have beenstudied and confirmed in several approaches [1].The color confinement may be illustrated by the fluxtube formation between quark and antiquark. A colorflux tube which has a constant energy per length isformed between (color singlet) q ¯ q pair and this tube givesthe linearly rising q ¯ q potential [2, 3]. Note that QCDis nonabelian gauge theory and hence such gluon fluxeshave colors. In other words, the color charge first asso-ciated with a color-singlet q ¯ q pair flows into interquarkflux tube as the q ¯ q separation is enlarged keeping the to-tal system color singlet [4, 5]. If the color charge of the q ¯ q part and that of the gluon part are separately consid-ered, this color transfer can be regarded as a color chargeleak from q ¯ q part to the gluon part in association withthe screening effect. This color leak should depends onthe q ¯ q distance and would be observed as the distancedependence of the color correlation between quark andantiquark.Such color correlation of the q ¯ q pair may be detectedby entanglement entropy (EE) defined by the reduceddensity matrix. EE quantifies an entanglement betweendegrees of freedom in purely quantum systems, and havebeen utilized in variety of physical systems [6–17]. If the q ¯ q pair’s correlation is strong, the q ¯ q part is well decou-pled from the gluon part and there is no entanglementbetween the q ¯ q and gluon parts. In other words, the colorleak from q ¯ q part can be measured by EE. In this paper,we define the reduced density matrix ρ for a static q ¯ q pair in terms of color degrees of freedom. The density matrixis reduced into subspace of q ¯ q color configurations byintegrating out the gluons’ degrees of freedom, which issimply done by averaging the density matrix componentsover gauge configurations, and compute entanglement en-tropy S with the reduced density matrix. Constructinga simple ansatz for the reduced density matrix ρ , we in-vestigate the dependence of S on the interquark distance R .In Sec. II, we give the formalism to compute the re-duced density matrix ρ of q ¯ q system and the entangle-ment entropy S of it. The details of numerical calcula-tions and ansatz for ρ are also shown in Sec. II. Resultsare presented in Sec. III. Sec. IV is devoted to the sum-mary and concluding remarks. II. FORMALISMA. reduced 2-body density matrix and q ¯ q correlation The entanglement between two subsystems A and B can be quantified with entanglement entropy (EE). Fromthe density matrix ρ AB for a whole system A + B , the re-duced density matrix ρ A is obtained as ρ A = Tr B ( ρ AB ).Here, Tr B is taken over the degrees of freedom of thesubsystem B . The entanglement entropy S EE A of the sub-system A is then defined as S EE A = − Tr A ( ρ A log ρ A ) inthe functional form of the von Neumann entropy. Thedensity matrix ρ A defined for the reduced space (the sub-system A) can give a non-zero value of EE because a partof information is lost from the ρ AB for the full space bytracing out degree of freedom (Dof) of the subsystem B .The EE is zero only in the case of ρ A = ρ A when thesubsystems A and B are completely decoupled from eachother (not entangled).Since our interest is being focused on the static q ¯ q a r X i v : . [ h e p - l a t ] O c t pair’s color correlations, we divide the whole color-singlet system into ( possibly colored ) two subsystems, static(anti)quarks ( Q ) and “others”( G ), and consider colorDoF of the subsystems ( Q = A and G = B ). OtherDoF contains all the gluon’s DoF including the vacuumpolarization by the sea quark’s loop.In the actual calculations, we compute the reducedtwo-body density matrix ρ in the subsystem Q by takinginto account static quark’s color configuration only. Thusdefined density matrix is nothing but the reduced densitymatrix ρ Q that is obtained integrating out the other DoF G in the full density matrix ρ QG ; ρ Q = Tr G ( ρ QG ).The reduced two-body density operator ˆ ρ ( R ) in a q ¯ q system with the interquark distance R is defined asˆ ρ ( R ) = | ¯ q (0) q ( R ) (cid:105)(cid:104) ¯ q (0) q ( R ) | . (1)Here | ¯ q (0) q ( R ) (cid:105) represents a quantum state in which theantiquark is located at the origin and the other quarklies at x = R . The reduced density matrix components ρ ( R ) ij,kl , where i ( j ) are quark’s (antiquark’s) color in-dices, are expressed as ρ ( R ) ij,kl = (cid:104) q i (0)¯ q j ( R ) | ˆ ρ ( R ) | q k (0)¯ q l ( R ) (cid:105) . (2) ρ ( R ) is a m × m square matrix with the dimension m = N c . Note again that ρ is defined using only quark’s DoFand gluon’s wavefunction is not considered and then thusdefined ρ can be regarded as a reduced density matrixwhere gluon’s DoF are integrated out.The von Neumann entanglement entropy S VN ( R ) for q ¯ q pair at a distance of R can be computed with thereduced density matrix ρ ( R ) as S VN ( R ) ≡ − Tr ρ ( R ) log ρ ( R ) = − (cid:88) ij [ ρ ( R ) log ρ ( R )] ij , (3)which can be regarded as an entanglement entropy rep-resenting the correlation between static-quark pair (sub-system Q ) and other DoF (subsystem G ).In the actual computation of S VN , one needs to di-agonalize ρ or approximate the logarithmic function. Inorder to avoid such numerically demanding processes, weadopt Renyi entropy [18] for EE for detailed analysis.Renyi entanglement entropy S Renyi − α of order α ( α > α (cid:54) = 1) is given as S Renyi − α = 11 − α log Tr ( ρ α ) , (4)with a reduced density matrix ρ . Note that in thelimit when α →
1, it goes to von Neumann entropy as S Renyi − α → S VN . Renyi entanglement entropy is a kindof generalized entropies that quantify uncertainty or ran-domness, and used to measure entanglement in quantuminformation theory. Since entanglement entropy is in-variant under unitary transformations, it enables repre-sentation independent analysis. We use the second orderRenyi entanglement entropy by taking α = 2, which issimply given by the squared ρ ( R ) as S Renyi − = − log Tr (cid:0) ρ (cid:1) . (5) We here comment on the relationship between q ¯ q corre-lation and the entanglement entropy. Our main interestis the q ¯ q pair’s color correlation defined in the subsystem Q . The whole pure state in Q + G system can be writtenas (cid:88) α | α (cid:105) Q ⊗ | α (cid:105) G . (6)Here, α denotes all the possible color states of the q ¯ q pair, and total system is kept in a color singlet state.When quark and antiquark’s colors are strongly corre-lated forming a color singlet combination | (cid:105) Q with nocolor charge leak from Q to G , the subsystems Q and G are well decoupled in the color space and therefore thewhole state can be expressed in a simple product of Q and G parts as (cid:88) α = | α (cid:105) Q ⊗ | α (cid:105) G = | (cid:105) Q ⊗ | (cid:105) G . (7)In this strongly correlated case, the entanglement entropy S EE goes to zero, since two subsystems Q and G decou-ple and the entanglement between subsystems Q and G vanishes.On the other hand, when q ¯ q pair’s color charge leaksinto inbetween gluons and the color correlation betweenthem decreases, the whole state cannot be written in aseparable form, and S would take a positive finite valueas S > B. Ansatz for reduced density matrix ρ ij,kl ( R ) Let us consider a possible functional form of the re-duced density matrix ρ ij,kl ( R ) based on the simple ansatzthat the contamination mixed to the correlated color sin-glet component is the random color component withoutany color correlation between quark and antiquark of the q ¯ q pair. We first define the density operator ˆ ρ s , s for quarkand antiquark in a color singlet state | s (cid:105) = (cid:80) N c i | ¯ q i q i (cid:105) inthe Coulomb gauge asˆ ρ s , s = | s (cid:105)(cid:104) s | . (8)In color SU( N c ) QCD, the density operator ˆ ρ a i , a i ( i =1 , , ..., N c −
1) for q ¯ q in an adjoint state | a i (cid:105) ( i =1 , , ..., N c −
1) is expressed asˆ ρ a i , a i = | a i (cid:105)(cid:104) a i | ( i = 1 , , ..., N c − . (9)In the limit R →
0, quark and antiquark are consideredto form a color-singlet state ( | s (cid:105) ) corresponding to thestrong correlation limit, and its density operator will bewritten as ˆ ρ = ˆ ρ s , s = diag(1 , , ..., α − rep . (10)Here, “ α − rep . ” means that the matrix is expressed interms of q ¯ q ’s color representation with the vector set of { s , a , ... a } . As R increases, it is expected that adjointcomponents mix into the singlet component due to theQCD interaction. We assume that contamination mixedinto the pure singlet (correlated) state is the uncorrelatedstate with random color configurations where N c compo-nents mix with equal weights. The density operator forsuch the random state is given asˆ ρ rand = 1 N c ˆ ρ s , s + 1 N c ˆ ρ a , a + 1 N c ˆ ρ a , a + ... = 1 N c ˆ I = 1 N c diag(1 , , ..., α − rep . (11) Letting the fraction of the original (maximally corre-lated) singlet state being F ( R ) and that of the mixing(random) components being (1 − F ( R )), the density op-erator in this ansatz is written asˆ ρ ansatz ( R ) = F ( R )ˆ ρ + (1 − F ( R ))ˆ ρ rand . (12)The matrix elements of ˆ ρ ansatz ( R ) in the α -representationare explicitly written asˆ ρ ansatz ( R ) = F ( R )ˆ ρ + (1 − F ( R ))ˆ ρ rand (13)= diag (cid:18) F ( R ) + 1 N c (1 − F ( R )) , N c (1 − F ( R )) , ..., N c (1 − F ( R )) (cid:19) α − rep . (14)= F ( R ) + N c (1 − F ( R )) 0 · · · N c (1 − F ( R )) ...... . . . ...... 00 · · · N c (1 − F ( R )) α − rep . (15)When N c = 3, (cid:40) ρ ( R ) , = ρ ( R ) , = ... = ρ ( R ) , ≡ ρ ( R ) , ρ ( R ) α,β = 0 (for α (cid:54) = β ) (16)would be satisfied at any R in this ansatz. The firstrelation should be satisfied due to the color SU(3) sym-metry. The second, which means that the off-diagonalcomponents are all zero, comes from the ansatz of therandom state. The normalization condition Tr ρ = 1 istrivially satisfied in this ansatz as ρ ( R ) , + ( N c − ρ ( R ) , = 1 . (17)In the strong correlation limit when q ¯ q pair’s colorforms | (cid:105) , F ( R ) = 1. On the other hand, in the ran-dom limit when quarks’ colors are screened, F ( R ) = 0. C. Lattice QCD formalism
Let the site on the lattice r = ( x, y, z, t ) = x e x + y e y + z e z + t e t and µ -direction ( µ = x, y, z, t ) link variablesbeing U µ ( r ). With a lower staple S L ( R, T ) representing q ¯ q pair creation and propagation and an upper staple S U ( R, T ) for q ¯ q pair annihiration that are defined as S Lij ( R, T ) ≡ (cid:32) − T (cid:89) t = − U † t ( t e t ) R − (cid:89) x =0 U x ( x e x − T e t ) × − (cid:89) t = − T U t ( R e x + t e t ) (cid:33) ij , (18) S Uij ( R, T ) ≡ (cid:32) T − (cid:89) t =0 U t ( t e t ) R − (cid:89) x =0 U x ( x e x + T e t ) × (cid:89) t = T − U † t ( R e x + t e t ) (cid:33) ij , (19)we define L ij ( R, T ) as L ij,kl ( R, T ) ≡ S Uij ( R, T ) S L † kl ( R, T ) . (20)When the euclidean time separation T is largeenough and excited state contributions can be ignored, (cid:104) L ij,kl ( R, T ) (cid:105) is expressed as (cid:104) L ij,kl ( R, T ) (cid:105) = C (cid:104) q (0)¯ q ( R ) | e − ˆ HT | q i (0)¯ q j ( R ) (cid:105)× (cid:104) ¯ q k (0) q l ( R ) | e − ˆ HT | q (0)¯ q ( R ) (cid:105) = Ce − E T (cid:104) q (0)¯ q ( R ) | q i (0)¯ q j ( R ) (cid:105)(cid:104) ¯ q k (0) q l ( R ) | q (0)¯ q ( R ) (cid:105) = Ce − E T ρ ( R ) ij,kl , (21)where E is the ground-state energy. Normalizing (cid:104) L ( R, T ) (cid:105) so that Tr (cid:104) L ( R, T ) (cid:105) = (cid:80) ij (cid:104) L ij,ij ( R, T ) (cid:105) = 1,we obtain ρ ( R ) whose trace is unity (Tr ρ ( R ) = 1).Once we obtain ρ ( R ), Renyi entropy of order α as afunction of R is obtained as S Renyi − α ( R ) = 11 − α log Tr ( ρ ( R ) α ) . (22) D. Lattice QCD parameters
We adopt the standard Wilson gauge action and per-form quenched calculations for reduced density matri-ces of static quark and antiquark ( q ¯ q ) systems. Thegauge configurations are generated on the spatial vol-umes L = 8 , 16 , 32 and 48 , with the gauge cou-plings β = 5 .
7, 5.8 and 6.0. All the gauge configurationsare gauge-fixed with the Coulomb gauge condition. Theparameters adopted in the present work are summarizedin Table. 7. β a [fm] L L [fm ]5.7 0.18 8 TABLE I: Lattice QCD parameters. Coupling β , lattice spac-ing a , spatial volume L in the lattice unit and the physicalunit. III. LATTICE QCD RESULTSA. Ground-state dominance
In order to confirm the ground-state dominance, weinvestigate the static quark and antiquark potential. InFigs. 1, 2 and 3, we show the effective energy plots forstatic q ¯ q systems with several interquark distances R as afunction of the Euclidean time separation T . For all theinterquark distances R , effective energies show plateauxagainst T at T ≥ T ≥
2. Hereafter, we adopt nor-malized reduced density matrix ρ ( R,
2) measured with T = 2 for ρ ij,kl ( R ); ρ ij,kl ( R ) ≡ ρ ij,kl ( R, / Tr ρ ( R, ✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁☎ ✥ ✄ ✆ ✝ ✞ ✂✥❊✟✟✠✡☛☞✌✠✍✎✏✏✎☛❜❂✑✒✓✔✕✎☛☛☞✡✠✖✗☞☛ ✘✙✚✛✜✢✣✤✦ ✧✜★✣ ✩✛✤✧✧✜✚✣ ✙✦✜✧✪✫✬✭✮✫✬✭✯✫✬✭✰✫✬✭✱✫✬✭✲✫✬✭✳✫✬✭✴✫✬✭✵✫✬✭✶✫✬✮✭✫✬✮✮✫✬✮✯ FIG. 1: Effective energy plot as a function of the Euclideantime separation at β = 5 . R denotes the interquark distancein lattice unit. ✥✥(cid:0)✁✂✂(cid:0)✁✄ ✥ ✄ ☎ ✆ ✝ ✂✥❊✞✞✟✠✡☛☞✟✌✍✎✎✍✡❜❂✏✑✒✓✔✍✡✡☛✠✟✕✖☛✡ ✗✘✙✚✛✜✢✣✤ ✦✛✧✢ ★✚✣✦✦✛✙✢ ✘✤✛✦✩✪✫✬✭✪✫✬✮✪✫✬✯✪✫✬✰✪✫✬✱✪✫✬✲✪✫✬✳✪✫✬✴✪✫✬✵✪✫✭✬✪✫✭✭✪✫✭✮ FIG. 2: Effective energy plot as a function of the Euclideantime separation at β = 5 . R denotes the interquark distancein lattice unit. B. Reduced density matrix elements
In this subsection, we take a detailed look at thereduced density matrix elements obtained with latticeQCD. In order to see the validity of the first condition inEq.(16), we define the average ρ ( R ) , = 1 N c − (cid:88) i ρ ( R ) i , i (23)and the deviation D i ( R ) = ( ρ ( R ) i , i − ρ ( R ) , ) . (24)In Fig. 4, D i ( R ) (1 ≤ i ≤
8) are plotted as a functionof the interquark distance. All the values are consistentwith zero and it is confirmed that the first condition issatisfied for all the R and i within statistical errors. Here-after, the octet components of ρ ( R ) is represented by theaveraged value ρ ( R ) , . ✥✥(cid:0)✁✥(cid:0)✂✥(cid:0)✄✥(cid:0)☎✆✆(cid:0)✁✆(cid:0)✂ ✥ ✁ ✂ ✄ ☎ ✆✥❊✝✝✞✟✠✡☛✞☞✌✍✍✌✠❜❂✎✏✑✒✓✌✠✠✡✟✞✔✕✡✠ ✖✗✘✙✚✛✜✢✣ ✤✚✦✜ ✧✙✢✤✤✚✘✜ ✗✣✚✤★✩✪✫✬✩✪✫✭✩✪✫✮✩✪✫✯✩✪✫✰✩✪✫✱✩✪✫✲✩✪✫✳✩✪✫✴✩✪✬✫✩✪✬✬✩✪✬✭ FIG. 3: Effective energy plot as a function of the Euclideantime separation at β = 6 . R denotes the interquark distancein lattice unit. ✲(cid:0)✁(cid:0)✂✲(cid:0)✁(cid:0)✄(cid:0)(cid:0)✁(cid:0)✄(cid:0)✁(cid:0)✂ (cid:0) (cid:0)✁✄ (cid:0)✁✂ (cid:0)✁✥ (cid:0)✁☎ ✆ ✆✁✄ ✆✁✂ ✆✁✥r✭✝✞ ✽ ✐✽ ✐✟r✭✝✞ ✽✽ q✠q ✡☛☞✌✍✎✏✑ ✒ ✓✔✕✖✗✘✙ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘★ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘✩ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘✣ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘❂ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘✢ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘✧ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧✗✘✛ ✚✘✛✜✢✣ ✤✦ ❜✘❂✜✧ FIG. 4: The deviation of each component ρ ( R ) i , i from theaveraged value ρ ( R ) , = N c − (cid:80) i ρ ( R ) i , i is plotted as afunction of the interquark distance. They are evaluated at β = 5 . L = 48. All the values are consistent with zerowithin the errors. The second condition in Eq.(16) is the assumption inthe ansatz. To see to what extent this assumption isvalid in the actual reduced density matrices, we definefollowing two independent components. ρ ( R ) , = − √ ρ ( R ) , + ρ ( R ) , + ρ ( R ) , ) , (25) ρ ( R ) , = ρ ( R ) , . (26) ρ ( R ) , and ρ ( R ) , are plotted in Fig. 5. We findthat they are consistent with zero and we conjecture thatthe off-diagonal components of the reduced density ma-trix ρ ( R ) are considerably small. From these analyses,we can conclude that the reduced density matrix ρ ( R )obtained with lattice calculations in the static q ¯ q systemis expressed by the ansatz with high accuracy. Indeed, ✲(cid:0)✁(cid:0)(cid:0)✂✲(cid:0)✁(cid:0)(cid:0)✄(cid:0)(cid:0)✁(cid:0)(cid:0)✄(cid:0)✁(cid:0)(cid:0)✂ (cid:0) (cid:0)✁✄ (cid:0)✁✂ (cid:0)✁✥ (cid:0)✁☎ ✆ ✆✁✄ ✆✁✂ ✆✁✥r✭✝✞ ✶✟ ✠✡r✭✝✞ ✟ ✸✟ ✹ q☛q ☞✌✍✎✏✑✒✓ ✔ ✕✖✗✘✙✚✛✜✢ ✣✤ ✦✧★✩✪✫ ✬✮ ❜✧❂✩✯✙✚✛✜✣✰ ✣✱ ✦✧★✩✪✫ ✬✮ ❜✧❂✩✯ FIG. 5: In order to see the magnitudes of the off-diagonal components, two independent off-diagonal compo-nents ρ ( R ) , and ρ ( R ) , are plotted as a function of theinterquark distance. They are consistent with zero and weconjecture that the off-diagonal components of the reduceddensity matrix ρ ( R ) are consideably small. even when we replace the octet components and the off-diagonal components of ρ ( R ) with the average ρ ( R ) , and with zero by hand, all the results remain almost un-changed. C. R dependence of F ( R ) Taking into account the normalization condition ρ ( R ) , + ( N c − ρ ( R ) , = 1 , (27)the independent quantity at a given R is only ρ , , andwe can compute the fraction F ( R ) of the remaining cor-related q ¯ q component as, F ( R ) = ρ ( R ) , − ρ ( R ) , = 1 − N c ρ ( R ) , . (28)When the q ¯ q system forms a random state with no colorcorrelation between q and ¯ q , the calculated ρ ( R ) equalsto ˆ ρ rand and gives F ( R ) = 0. In the upper panel in Fig. 6, F ( R ) is plotted as a function of the interquark distance R . F ( R ) linearly decreases at small R , and exponentiallyapproaches zero at large R , which can be also seen in thelower panel (logarithmic plot of F ( R )).The exponential decay of the q ¯ q correlation indicatesthe color screening effects due to inbetween gluons. Wefit F ( R ) with an exponential function as F ( R ) = A exp( − BR ) (29)and extract the “screening mass” B . In Fig. 7, the fit-ted parameters A and B are plotted as functions of thespatial lattice size L . The plot includes all the data ob-tained at β =5.7, 5.8 and 6.0 so that one can see the β (lattice spacing) dependence. While a tiny deviation is ✥✥(cid:0)✁✥(cid:0)✂✥(cid:0)✄✥(cid:0)☎✆ ✥ ✥(cid:0)✝ ✆ ✆(cid:0)✝ ✁❋✞✟✠ q✲q ✡☛☞✌✍✎✏✑ ✒ ✓✔✕✖▲✗✘✙✚✛ ✜✢ ❜✗❂✙✣❆ ✤✦✧★✩✪✫✬✥(cid:0)✁✁ ✥ ✥(cid:0)✂ ✁ ✁(cid:0)✂ ✄❋☎✆✝ q✲q ✞✟✠✡☛☞✌✍ ✎ ✏✑✒✓▲✔✕✖✗✘ ✙✚ ❜✔❂✖✛❆ ✜✢✣✤✦✧★✩ FIG. 6: F ( R ) is plotted as a function of the interquarkdistance R in the upper panel. F ( R ) monotonously decreasesand approaches zero. The lower panel shows the log plot for F ( R ). The fit function, F ( R ) = A exp( − BR ) with A = 1 . B = 1 .
347 fm − , is shown as a solid line. found among three β ’s, all the data seem lie in a monoto-neous line, which means the systematic errors for A and B mainly arise from the lattice size L . For L > A and B are deteremined as A = 1 . B = 1 . − = 265(7) MeV (31)from F ( R ) obtained in the largest volume.In Fig. 8, the singlet component ρ ( R ) , and the av-eraged octet component ρ ( R ) , are plotted as a func-tion of the interquark distance R . One finds that bothcomponents approach ρ ( R ) , = ρ ( R ) , = N c = atlarge R , which ensures that the reduced density matrixat large interquark separation R is governed by the ran-dom component ˆ ρ rand and the original correlated stateˆ ρ vanishes. D. Finite volume effects
Within the present numerical accuracy, the only in-dependent quantity in the reduced density matrix ρ ( R ) ✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁ ✥ ✂ ✄ ☎ ✆ ✁ ✝ ✞ ✟ ✠❆ ▲✡☛☛☞✌✍ ✎☞✏✍ ▲ ✑✒✓✔ ❜❂✕✖✗❜❂✕✖✘❜❂✙✖✚✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁ ✥ ✂ ✄ ☎ ✆ ✁ ✝ ✞ ✟ ✠❇✡☛☞✲✌ ❪ ▲✍✎✎✏✑✒ ✓✏✔✒ ▲ ✕✖✗✘ ❜❂✙✚✛❜❂✙✚✜❜❂✢✚✣ FIG. 7: The fitted parameters A and B plotted as functionsof the spatial lattice size L . is ρ , , and all the finite volume effects are reflected in F ( R ) = 1 − N c ρ ( R ) , .In Fig. 9, F ( R ) for several L (lattice size) and β (lat-tice spacing) are plotted as a function of the interquarkdistance R . At L > . F ( R ) shows almost no vol-ume dependence and ρ ( R ) is safe from the finite volumeeffects at this L range. When the lattice size L is small, F ( R ) rapidly decreases with increasing R . On the otherhand, β dependence seems smaller than the finite volumeeffect. The systematic errors mainly comes from the fi-nite size effect.This finite volume effect would be due to the periodicboundary condition, with which identical q ¯ q -systems ex-ist with the period L . Quark and antiquark ( q (0)¯ q ( R ))separated by R in a system can also form color singletpairs with quarks that are separated with the distance L − R , which additionally enters in ρ ( R ) as a randommixture decreasing F ( R ). E. Entanglement entropy
In the following, we consider α = 2 case for the eval-uation of the EE. (We will go back to S VN in the latterpart of this section.) The S Renyi − ( R ) is correctly cal- ✥✥(cid:0)✁✥(cid:0)✂✥(cid:0)✄✥(cid:0)☎✆ ✥ ✥(cid:0)✝ ✆ ✆(cid:0)✝ ✁ ✁(cid:0)✝❙✞✟✠✡☛☞✌✟✍✎✏☞☛☞✏✑✒✓✑✟☛✟☞✔ q✲q ✕✖✗✘✙✚✛✜ ✢ ✣✤✦✧★✩✪✫✬✭✮ ✯✰✱✳✴✵ ✶✷ ❜✰❂✳✸❖✹✮✭✮ ✯✰✱✳✴✵ ✶✷ ❜✰❂✳✸✺✻ ✼❝✽ ✰ ✺✻✾ FIG. 8: The singlet component ρ ( R ) , and the averagedoctet component ρ ( R ) , are plotted as a function of the in-terquark distance R . Both are approaching 1 /Nc = 1 / R . ✥✥(cid:0)✁✥(cid:0)✂✥(cid:0)✄✥(cid:0)☎✆ ✥ ✥(cid:0)✁ ✥(cid:0)✂ ✥(cid:0)✄ ✥(cid:0)☎ ✆ ✆(cid:0)✁ ✆(cid:0)✂ ✆(cid:0)✄❋✝✞✟ q✲q ✠✡☛☞✌✍✎✏ ✑ ✒✓✔✕▲✖✗✘✙✗ ✚✛ ❜✖❂✘✗▲✖✜✘❂✗ ✚✛ ❜✖❂✘✗▲✖✢✘✢✣ ✚✛ ❜✖✤✘✙▲✖✢✘✙✙ ✚✛ ❜✖✤✘✦▲✖✧✘✢✗ ✚✛ ❜✖❂✘✗▲✖✣✘✣✙ ✚✛ ❜✖✤✘✙▲✖✣✘✙✗ ✚✛ ❜✖❂✘✗▲✖✤✘✦❂ ✚✛ ❜✖✤✘✦▲✖❂✘✦✢ ✚✛ ❜✖✤✘✙▲✖✙✘❂✣ ✚✛ ❜✖✤✘✦ FIG. 9: F ( R ) for different L (lattice size) are plotted as afunction of the interquark distance R . At L > . F ( R )shows almost no volume dependence and ρ ( R ) is safe fromthe finite volume effects at this L range. culated from the trace of the squared reduced densitymatrix ρ ( R ) as S Renyi − = − log Tr( ρ ( R ) ) . (32)Taking into account that Tr( ρ ( R )) = 1, the maximum of S Renyi − is obtained when all the N c diagonal elementsare equal to 1 /N c in the diagonal representation of ρ ( R ).From the representation invariance of S , the maximumvalue of S is proved to bemax (cid:2) S Renyi − ( R ) (cid:3) = 2 log N c . (33)In Fig. 10, S Renyi − ( R ) calculated with the ρ ( R )obtained on the lattice are plotted as S Renyi − ( R ). S Renyi − ( R ) approaches 2 log N c as R increases, whichindicates that ρ ( R ) is described by the random compo-nent ˆ ρ rand in the large R limit. In the ansatz, the density matrix ρ ansatz ( R ) is a diag-onal matrix and Tr( ρ ansatz ( R ) ) is given by F ( R ) asTr( ρ ansatz ( R ) ) = F ( R ) + 1 N c − F ( R ) N c . (34)Then S Renyi − ( R ), the Renyi entropy evaluated using theansatz, is expressed as S Renyi − ( R ) = − log Tr( ρ ansatz ( R ) )= − log (cid:18) F ( R ) + 1 N c − F ( R ) N c (cid:19) . (35) ✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁☎ ✥ ✥(cid:0)✁ ✂ ✂(cid:0)✁ ✄❙ ❘✆✝✞✟✭✠✡ q✲q ☛☞✌✍✎✏✑✒ ✓ ✔✕✖✗❡✘✙✚✛✜✢ ✣✤✦✙✙✧★❡✩ ✪✫✬✮✯✰ ✱✳ ❜✫❂✮✴❡✘✙✚✛✜✢ ✣✦✘✵✦✙✶✩ ✪✫✬✮✯✰ ✱✳ ❜✫❂✮✴✷ ✤✛✸ ✹❝ ✫ ✷✮✺✻✴ FIG. 10: S Renyi − ( R ) obtained from the original reduceddensity matrix ρ ( R ) and S Renyi − ( R ) obtained using theansatz are plotted as a function of the interquark distance R . Fig. 10 shows S Renyi − ( R ) obtained using the ansatzplotted as a function of the interquark distance R . S Renyi − ( R ) approaches 2 log N c at large R , which againconfirms that F ( R ) goes to zero and ρ ansatz ( R ) is ex-pressed by the random elements ˆ ρ rand in the R → ∞ limit. The remarkable fact is that S Renyi − ( R ) and S Renyi − ( R ) are almost identical for all R , which indi-cates that the reduced density matrix ρ ( R ) can be verywell expressed by the ansatz. S Renyi − and S Renyi − for different lattice sizes L areplotted as a function of R in Fig. 11. As expected, when L < . S Renyi − and S Renyi − are both affected. On the otherhand, for all the L , S Renyi − (cid:39) S Renyi − is found and theansatz is valid with a good accuracy even when the finitevolume effects are large.It is well known that any averaging leads to the growthof the entropy. The reduced density-matrix componentsare averaged in the ansatz and one may think S Renyi − >S Renyi − should be observed. Although such tendencycan be sometimes seen in figures, statistical errors aremuch larger and both data are consistent with each otherwithin the present statistics. ✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁☎ ✥ ✥(cid:0)✁ ✂ ✂(cid:0)✁ ✄❙❘✆✝✞✟ ✭✠✡ q✲q ☛☞✌✍✎✏✑✒ ✓ ✔✕✖✗❡✘✙✚✛✜✢ ✣✤✦✙✙✧★❡✩ ✪✫✬✮✯✬ ✰✱ ❜✫❂✮✬❡✘✙✚✛✜✢ ✣✦✘✳✦✙✴✩ ✪✫✬✮✯✬ ✰✱ ❜✫❂✮✬❡✘✙✚✛✜✢ ✣✤✦✙✙✧★❡✩ ✪✫✵✮❂✬ ✰✱ ❜✫❂✮✬❡✘✙✚✛✜✢ ✣✦✘✳✦✙✴✩ ✪✫✵✮❂✬ ✰✱ ❜✫❂✮✬❡✘✙✚✛✜✢ ✣✤✦✙✙✧★❡✩ ✪✫✶✮✯✯ ✰✱ ❜✫✷✮✸❡✘✙✚✛✜✢ ✣✦✘✳✦✙✴✩ ✪✫✶✮✯✯ ✰✱ ❜✫✷✮✸❡✘✙✚✛✜✢ ✣✤✦✙✙✧★❡✩ ✪✫✯✮❂✹ ✰✱ ❜✫✷✮✸❡✘✙✚✛✜✢ ✣✦✘✳✦✙✴✩ ✪✫✯✮❂✹ ✰✱ ❜✫✷✮✸ FIG. 11: S Renyi − and S Renyi − , which are obtained fromthe original reduced density matrix ρ ( R ) and that obtainedusing the ansatz, are plotted as a function of R . Finally, we show the von Neumann entropy S VN basedon the ansatz. The direct calculation of S VN from thereduced density matrix on the lattice is numerically de-manding. Instead of such a straightforward approach, wetake an alternative way to calculate S VN with an approx-imation using ρ ansatz from the ansatz as S VNansatz = − Tr ( ρ ansatz log ρ ansatz ) . (36) ρ evaluated on the lattice coincides with ρ ansatz with highaccuracy as shown above, and S VNansatz is expected to bea good approximation of S VN . Now the reduced den-sity matrix in the α -representation has been found to bediagonal and then S VN ( R ) is easily computed as S VNansatz ( R ) = − (cid:18) F ( R ) + 1 N c (1 − F ( R )) (cid:19) log (cid:18) F ( R ) + 1 N c (1 − F ( R )) (cid:19) − ( N c − (cid:18) N c (1 − F ( R )) (cid:19) log (cid:18) N c (1 − F ( R )) (cid:19) . (37)Fig. 12 shows S VNansatz ( R ) as a function of R , and S Renyi − ( R ) and S Renyi − ( R ) are also plotted for refer-ence. S VNansatz ( R ) increases towards 2 log N c faster than S Renyi − ( R ) as the VN EE is a more sensitive measureof the entanglement than the Renyi-2 EE in general. As R increases, the reduced density matrix ˆ ρ is dominatedby the random contribution ˆ ρ rand , and all the matrix el-ements are equipartitioned in this limit giving the maxi-mum value of entropy.In order to see the finite volume effects, we plot S VNansatz as a function of R in Fig. 13. The tendency that S isincreased by the finite size effects remains unchanged. IV. SUMMARY AND CONCLUDINGREMARKS
We have studied the color correlation of static quarkand antiquark ( q ¯ q ) systems in the confined phase fromthe viewpoint of the entanglement entropy (EE) definedby reduced density matrices ρ in color space. We haveadopted the standard Wilson gauge action and performedquenched calculations for density matrices. The gaugecouplings are β = 5 .
7, 5.8 and 6.0, and the spatial vol-umes are L = 8 , 16 , 32 and 48 . In order to eval-uate each component of ρ ij,kl , all the gauge configura-tions are Coulomb-gauge fixed. We have also proposedan ansatz for the reduced density matrix ρ , in which ρ ✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁☎ ✥ ✥(cid:0)✁ ✂ ✂(cid:0)✁ ✄❙❱✆ ✭✝✞✟❙❘✠✡☛☞ ✭✝✞ q✲q ✌✍✎✏✑✒✓✔ ✕ ✖✗✘✙✚✛✜✢✣✤✦ ✛✜✧★✩✪✢ ✫✬✮✧✧✣✯✛✰ ✱✳✴✵✶✷ ✸✹ ❜✳❂✵✺✚✛✜✢✣✤✦ ✛✜✧★✩✪✢ ✫✮✜✻✮✧✼✰ ✱✳✴✵✶✷ ✸✹ ❜✳❂✵✺✽✾ ✛✜✧★✩✪✢ ✫✮✜✻✮✧✼✰ ✱✳✴✵✶✷ ✸✹ ❜✳❂✵✺✦ ✬✩✿ ✾❝ ✳ ✦✵❀❁✺ FIG. 12: S VNansatz , S Renyi − and S Renyi − , which are obtainedfrom the original reduced density matrix ρ ( R ) and that ob-tained using the ansatz, are plotted as a function of R . is written by a sum of the color-singlet (correlated) state | (cid:105)(cid:104) | and random (uncorrlated) elements | (cid:105)(cid:104) | , | i (cid:105)(cid:104) i | ( i = 1 , .., N c −
1) induced by the QCD interaction.We have quantitatively evaluated the q ¯ q correlation bymeans of the entanglement entropy constructed from thereduced density matrix ρ . We have adopted the von Neu-mann entropy S VN and the Renyi entropy of the order α S Renyi − α for the evaluation of EE. Especially when ✥✥(cid:0)✁✂✂(cid:0)✁✄✄(cid:0)✁☎ ✥ ✥(cid:0)✁ ✂ ✂(cid:0)✁ ✄❙❱✆ ✭✝✞ q✲q ✟✠✡☛☞✌✍✎ ✏ ✑✒✓✔✕✖ ✗✘✙✚✛✜✢ ✣✤✘✦✤✙✧★ ✩✪✫✬✮✫ ✯✰ ❜✪❂✬✫✕✖ ✗✘✙✚✛✜✢ ✣✤✘✦✤✙✧★ ✩✪✱✬❂✫ ✯✰ ❜✪❂✬✫✕✖ ✗✘✙✚✛✜✢ ✣✤✘✦✤✙✧★ ✩✪✳✬✮✮ ✯✰ ❜✪✴✬✵✕✖ ✗✘✙✚✛✜✢ ✣✤✘✦✤✙✧★ ✩✪✮✬❂✶ ✯✰ ❜✪✴✬✵ FIG. 13: S VNansatz obtained using the ansatz are plotted as afunction of R for different lattice size L . α is an integer, S Renyi − α can be computed easily fromthe density matrix product ρ α , and we need no diago-nalization of ρ . Note that color indices in EEs are allcontracted, and color-correlation measurement by meansof EEs can be performed in a gauge (representation) in-dependent way.As a result, we have found that the reduced densitymatrix ρ can be reproduced well with the ansatz: Thereduced density matrix ρ consists of the color-singlet(correlated) state | (cid:105)(cid:104) | when q ¯ q distance is small, andrandom (uncorrlated) diagonal elements | (cid:105)(cid:104) | , | i (cid:105)(cid:104) i | ( i = 1 , .., N c −
1) are equally mixed as q ¯ q distance is in-creased. The q ¯ q color correlations have been found to bewell quantified by entanglement entropies, and we con-clude that entanglement entropy can be a gauge indepen-dent measure for color correlations. Appendix A: α -representation and ij -representation In SU(3) QCD, q ¯ q state in α -representation, | (cid:105) and | i (cid:105) ( i = 1 , , .., N c − ij -representation, | ¯ q i q j (cid:105) , as following. | (cid:105) = 1 √ (cid:32)(cid:88) i | ¯ q i q i (cid:105) (cid:33) , | (cid:105) = −| ¯ q q (cid:105) , | (cid:105) = − √ | ¯ q q (cid:105) − | ¯ q q (cid:105) ) , | (cid:105) = | ¯ q q (cid:105) , | (cid:105) = | ¯ q q (cid:105) , | (cid:105) = −| ¯ q q (cid:105) , | (cid:105) = | ¯ q q (cid:105) , | (cid:105) = | ¯ q q (cid:105) , | (cid:105) = 1 √ | ¯ q q (cid:105) + | ¯ q q (cid:105) − | ¯ q q (cid:105) ) . Then, the elements of ˆ ρ in α -represenation can be relatedwith those in ij -representation asˆ ρ , = | (cid:105)(cid:104) | = 13 (cid:32)(cid:88) i | ¯ q i q i (cid:105) (cid:33) (cid:32)(cid:88) i (cid:104) ¯ q i q i | (cid:33) = 13 (cid:88) ij ˆ ρ ii,jj , ˆ ρ , = | (cid:105)(cid:104) | = | ¯ q q (cid:105)(cid:104) ¯ q q | = ˆ ρ , , ˆ ρ , = | (cid:105)(cid:104) | = 12 (ˆ ρ , + ˆ ρ , − ˆ ρ , − ˆ ρ , ) , ˆ ρ , = | (cid:105)(cid:104) | = | ¯ q q (cid:105)(cid:104) ¯ q q | = ˆ ρ , , ˆ ρ , = | (cid:105)(cid:104) | = | ¯ q q (cid:105)(cid:104) ¯ q q | = ˆ ρ , , ˆ ρ , = | (cid:105)(cid:104) | = | ¯ q q (cid:105)(cid:104) ¯ q q | = ˆ ρ , , ˆ ρ , = | (cid:105)(cid:104) | = | ¯ q q (cid:105)(cid:104) ¯ q q | = ˆ ρ , , ˆ ρ , = | (cid:105)(cid:104) | = | ¯ q q (cid:105)(cid:104) ¯ q q | = ˆ ρ , , ˆ ρ , = | (cid:105)(cid:104) | = 16 (ˆ ρ , + ˆ ρ , + ˆ ρ , + ˆ ρ , − ρ , − ρ , − ρ , − ρ , + 4ˆ ρ , ) . [1] J. Greensite, “An Introduction to the ConfinementProblem,” Lecture Notes in Physics (Springer, 2011)doi:10.1007/978-3-642-14382-3, and references therein.[2] G. S. Bali, K. Schilling and C. Schlichter, Phys. Rev.D , 5165 (1995) doi:10.1103/PhysRevD.51.5165 [hep-lat/9409005].[3] V. G. Bornyakov et al. [DIK Collaboration], Phys. Rev.D , 054506 (2004) doi:10.1103/PhysRevD.70.054506[hep-lat/0401026].[4] G. Tiktopoulos, Phys. Lett. , 271 (1977).doi:10.1016/0370-2693(77)90878-4[5] J. Greensite and C. B. Thorn, JHEP , 014 (2002)doi:10.1088/1126-6708/2002/02/014 [hep-ph/0112326].[6] E. Itou, K. Nagata, Y. Nakagawa, A. Nakamura and V. I. Zakharov, PTEP , no. 6, 061B01 (2016)doi:10.1093/ptep/ptw050 [arXiv:1512.01334 [hep-th]].[7] S. Aoki, T. Iritani, M. Nozaki, T. Numasawa,N. Shiba and H. Tasaki, JHEP , 187 (2015)doi:10.1007/JHEP06(2015)187 [arXiv:1502.04267 [hep-th]].[8] Y. Kanada-En’yo, Phys. Rev. C , no. 3, 034303 (2015)doi:10.1103/PhysRevC.91.034303 [arXiv:1501.06231[nucl-th]].[9] T. Takayanagi, Class. Quant. Grav. , 153001 (2012)doi:10.1088/0264-9381/29/15/153001 [arXiv:1204.2450[gr-qc]].[10] C. H. Bennett, H. J. Bernstein, S. Popescu andB. Schumacher, Phys. Rev. A , 2046 (1996) doi:10.1103/PhysRevA.53.2046 [quant-ph/9511030].[11] P. Calabrese and J. L. Cardy, J. Stat. Mech. ,P06002 (2004) doi:10.1088/1742-5468/2004/06/P06002[hep-th/0405152].[12] G. Vidal, J. I. Latorre, E. Rico and A. Ki-taev, Phys. Rev. Lett. , 227902 (2003)doi:10.1103/PhysRevLett.90.227902 [quant-ph/0211074].[13] L. Amico, R. Fazio, A. Osterloh and V. Ve-dral, Rev. Mod. Phys. , 517 (2008)doi:10.1103/RevModPhys.80.517 [quant-ph/0703044[QUANT-PH]].[14] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. , 865 (2009)doi:10.1103/RevModPhys.81.865 [quant-ph/0702225].[15] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin andW. K. Wootters, Phys. Rev. A , 3824 (1996)doi:10.1103/PhysRevA.54.3824 [quant-ph/9604024].[16] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998)doi:10.1103/PhysRevLett.80.2245 [quant-ph/9709029].[17] G. Vidal and R. F. Werner, Phys. Rev. A65