week ending 21 MARCH 2014

PHYSICAL REVIEW LETTERS

PRL 112, 111602 (2014)

Quantum Entanglement of Local Operators in Conformal Field Theories 1

Masahiro Nozaki,1 Tokiro Numasawa,1 and Tadashi Takayanagi1,2 Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8582, Japan (Received 16 January 2014; published 21 March 2014)

2

We introduce a series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement. It is defined by the increased amount of (Rényi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in two, four and six dimensions. We find that these results are interpreted in terms of quantum entanglement of a finite number of states, including Einstein-Podolsky-Rosen states. They agree with a heuristic picture of propagations of entangled particles. DOI: 10.1103/PhysRevLett.112.111602

PACS numbers: 11.25.Hf, 04.62.+v, 03.65.Ud

Recently, entanglement entropy has become a center of wide interest in a broad array of theoretical physics research. It is defined as the von Neumann entropy SA ¼ −Tr½ρA log ρA  of the reduced density matrix ρA for a subsystem A. It has been used as a useful quantity which characterizes quantum properties of ground states in condensed matter physics (see, e.g., [1,2]). Moreover, it is intriguing to apply entanglement entropy to quantify excited states. For excited states in conformal field theories (CFTs), it was shown that entanglement entropy has an interesting property analogous to the first law of thermodynamics if the size of subsystem A is much smaller than the excitation scale. This property was derived in [3] from the holographic entanglement entropy [4] and later a field theoretic derivation was given in [5]. Refer also to [6] for an earlier related result. Consider a CFT on a sphere times the time axis and pick up an excited state defined by acting a local operator O on the vacuum state j0i. Then, the first law argues that the increased amount of entanglement entropy ΔSA for this excited state is essentially given by the conformal dimension of the operator O if the subsystem size (or equally the excitation energy) is very small. On the other hand, it is natural to ask what will happen if we consider ΔSA in the opposite limit, i.e., the large size limit of subsystem A. One may expect that we get another basic quantity of an operator in CFTs, which can be as fundamental as the conformal dimension. The main aim of this Letter is to take a first step in answering this question. As we will see, this new quantity characterizes the quantum entanglement of an operator itself, together with its Rényi entropic versions. There have been extensive studies on time evolutions of entanglement entropy in certain classes of largely excited states, called quantum quenches. One of them is called a global quench, which is triggered by changing parameters 0031-9007=14=112(11)=111602(5)

homogeneously [7] and is a special example of thermalization. Another class is called a local quench, which occurs by changing the Hamiltonian locally [8,9]. In this Letter we focus on excited states which are defined by acting local operators on the vacuum with a finite and positive conformal dimension in a given CFT. We consider a conformal field theory in the d þ 1 dimensional Euclidean space Rdþ1 , whose coordinates are denoted by ðτ; x1 ; …; xd Þ. The density matrix ρ for the total system is given by ρ ¼ jΨihΨj and we choose the excited state jΨi by acting an operator O as follows jΨi ¼ N Oðxi Þj0i;

(1)

where N is a normalization factor such that hΨjΨi ¼ 1. The constant N becomes finite after a proper regularization as we will explain later. Our state jΨi cannot be treated as a small perturbation from the vacuum state, though it describes an excited state much milder than that in local and global quantum quenches. Define also the ground state density matrix as ρ0 ¼ j0ih0j. To define the entanglement entropy, we choose the subsystem A to be a half of the total space, i.e., x1 > 0. The reduced density matrix ρA is defined by ρA ¼ TrB ρ, tracing out the complement of A, called the subsystem B. ðnÞ The Rényi entanglement entropy SA is defined by ðnÞ

SA ¼ log Tr½ρnA =ð1 − nÞ:

(2)

The limit n → 1 coincides with the entanglement entropy ðnÞ SA . The difference of SA between an excited state and the ðnÞ ground state is defined to be ΔSA . We first calculate the entropies in the Euclidean formulation and, finally, perform an analytical continuation to see the dependence on the real time t. The time evolution of density matrix is described by

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© 2014 American Physical Society

PRL 112, 111602 (2014)

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PHYSICAL REVIEW LETTERS

ρðtÞ ¼ e−iHt e−ϵH Oðxi Þj0ih0jOðxi Þe−ϵH eiHt ¼ Oðτe Þj0ih0jOðτl Þ;

(3)

where we defined τe ¼ −ϵ − it, τl ¼ ϵ − it. An infinitesimal parameter ϵ is an ultraviolet regularization. In general, ΔSA shows a nontrivial time evolution. Our ðnÞ analysis of explicit examples suggests that ΔSA are monotonically increasing with the time t for any local operator O. Moreover, they finally approach to certain ðnÞf finite values ΔSA in the late time limit t → ∞. These ðnÞf values ΔSA depend on the choice of local operator O and are the quantities of our main interest. ðnÞ To calculate SA , we employ the path-integral formalism by extending the replica method analysis in [2] for ground states. We can express TrρnA in terms of correlations functions as TrρnA ¼ Zn =ðZ1 Þn. The two point function on Rdþ1 ð¼ Σ1 Þ, corresponding to h0jOðτl ÞOðτe Þj0i, is written as Z1 , while Zn is the 2n-point function of the operator O on the n-sheeted space Σn (see Fig. 1). It is also useful to define the vacuum partition functions on Σn ðnÞ ΔSA ¼

FIG. 1 (color online). The n-sheeted geometry Σn is constructed by gluing the upper cut along subsystem A on a sheet to the lower cut on the next sheet.

and Rdþ1 by Z0n and Z01 , respectively, so that we have Trρn0A ¼ Z0n =ðZ01 Þn for the ground state. ðnÞ In this way, we find that ΔSA is rewritten as

    1 Zn Z1 1 n n 1 1 ¼ log loghOðrl ;θl ÞOðre ;θe ÞOðrl ;θl ÞOðre ;θe ÞiΣn −nloghOðrl ;θl ÞOðre ;θe ÞiΣ1 : (4) −nlog 1−n Z0n Z01 1−n

The term in the second line is given by a 2n points correlation function of O on Σn . The final term is a two point function of O on Rdþ1 . The values of re;l and θje;l are determined as follows. First, we introduced the polar coordinate as x1 þ iτ ¼ reiθ . The angular coordinate θ takes values 0 ≤ θ < 2nπ on Σn (see Fig. 2). We set x1 ¼ −l < 0 at each location of O and this measures the distance between the excited point and the boundary ∂A of the subsystem A. Since our calculations do not depend on locations in other directions ðx2 ; …; xd Þ, we omit their dependence. By defining re;l eiθe;l ≡ −l þ iτe;l , the 2n locations of the O insertions are given by θje;l ≡ θe;l þ 2πðj − 1Þ, ðj ¼ 1; 2; …; nÞ. ðnÞ To obtain analytical results of ΔSA , we focus on a free massless R scalar field theory defined by the familiar action S ¼ ddþ1 x½ð∂ τ ϕÞ2 þ ð∂ xi ϕÞ2 . We performed explicit calculations for various operators and replica numbers n in 2, 4, and 6 dimensions. We found that the results of late ðnÞf time values ΔSA do not depend on the dimension as long as the dimension is higher than 2 as we summarized in Table I. The results in two dimensions will be discussed later in this Letter. In this Letter we will skip the details of the calculations because they are straightforward (but tedious) computations, employing the Green functions on Σn in [10,11]. Rather, we give a brief summary of our results below. First we describe the results in the four and six dimensional cases. As a series of local operators, we consider the primary operators

O ¼ ∶ϕk ∶ðk ¼ 1; 2;   Þ:

(5)

The time evolutions of the Rényi entropies are all similar. ðnÞ In general, ΔSA are vanishing in the region t < l. They start increasing at t ¼ l and keep increasing in the region t > l. Finally, they approach to certain constant values ðnÞf ΔSA , in the late time limit t → ∞. For example, the n ¼ 2 Rényi entropies for the operator O ¼ ϕ (i.e., k ¼ 1) in four and six dimensions are given as follows when t > l, in the ϵ → 0 limit (see Fig. 3)   2t2 ð2Þ 4D∶ΔSA ¼ log 2 ; (6) t þ l2

FIG. 2 (color online). insertions.

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The Euclidean coordinate and operator

TABLE I.

ðnÞf

ΔSA

ð1Þf

and ΔSfA ð¼ ΔSA Þ for free massless scalar field theories in dimensions higher than 2 (d > 1). k¼1

n ðnÞf

ΔSA

ΔSfA

k¼2



2

log 2

logð8=3Þ



3 .. .

log 2 .. .

ð1=2Þ logð32=5Þ .. .

 .. .

m

log 2

½1=ðm − 1Þ log½22m−1 =ð2m−1 þ 1Þ



1

log 2

ð3=2Þ log 2



ð2Þ 6D∶ΔSA

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PHYSICAL REVIEW LETTERS

PRL 112, 111602 (2014)

  8t6 : ¼ log 6 l − 6t2 l4 þ 9t4 l2 þ 4t6

(7)

matrix ρfA by the following ðk þ 1Þ × ðk þ 1Þ diagonal matrix ρfA ¼ 2−k ðk C0 ; k C1 ; …; k Ck Þ;

ð2Þf

Thus, we find ΔSA ¼ log 2. We can interpret this behavior as follows. First, many entangled pairs are produced at the operator inserted point as in Fig. 4. Each pair of them propagates in the opposite directions at the speed of light. When one of them reaches the boundary ∂A, the entanglement between them starts to contribute to the (Rényi) entanglement entropies between A and B. Thus the minimum time for this event is t ¼ l and generally it takes more than l for the pairs moving in a generic direction. Finally, the entropies approach certain constants as half of them remain to stay in A and the other half in B forever. This also explains the monotonicity of entropy under time evolutions. This argument is an application of the causal or horizon effect based on relativistic quasiparticles, first found in quantum quenches [7,8]. In the above, R we did not take into account the conformal mass term ∝ Σn Rϕ2 of scalar field theory, where R is the scalar curvature. Even though in general this affects results for excited states as noted in [12], our results in the ϵ → 0 limit are not changed by this effect. ðnÞf We calculated the late time values ΔSA for O ¼ ∶ϕk ∶ with various (n, k) and we summarized them in Table I. ðnÞf Interestingly, we can find that ΔSA are equal to the values of Rényi entropies for k þ 1 dimensional Hilbert spaces under a simple rule. Indeed, let us define a reduced density

k¼l P − log ðð1=22l Þ lj¼0 ðl Cj Þ2 Þ P ð−1=2Þ log ðð1=23l Þ lj¼0 ðl Cj Þ3 Þ .. . P ½1=ð1 − mÞ log ½ð1=2ml Þ lj¼0 ðl Cj Þm  P l × log 2 − ð1=2l Þ lj¼0 l Cj log l Cj

(8)

where m Cn ¼ ðm!=n!ðm − nÞ!Þ. Then we can confirm ðnÞf

ΔSA

¼

1 log Tr½ðρfA Þn : 1−n

(9)

ðnÞf

The Rényi entropy ΔSA ðnÞf

ΔSA

can be explicitly written as   X 1 1 k n (10) log nk ¼ ð Cj Þ : 1−n 2 j¼0 k

Taking the limit n → 1 leads to the entanglement entropy ΔSA ¼ k log 2 −

k 1X Cj log k Cj : 2k j¼0 k

(11)

Now we would like to provide an interpretation of the density matrices (8) in terms of the entangled pairs. We decompose the scalar field as ϕ ¼ ϕL þ ϕR , where ϕL and ϕR describe the modes which are moving toward the left (x1 < 0) and right (x1 > 0) direction. The key observation is that the late time entanglement entropy measures the entanglement between the left and right as A is defined by x1 > 0. We can expand our excited states as follows k X j k−j j0i jΨi ¼ N ∶ϕk ∶j0i ¼ N k Cj ðϕL Þ ðϕR Þ j¼0

¼ 2−k=2

k pffiffiffiffiffiffiffi X k Cj jjiL jk − jiR ;

(12)

j¼0

ð2Þ

FIG. 3 (color online). The plots of ΔSA as functions of t in the limit ϵ ¼ 0. We chose l ¼ 10. The red (lowest) and blue (middle) curve correspond to the operator O ¼ ϕ (k ¼ 1) in six and four dimensions, respectively. The green (highest) graph describes the entropy for the operator O ¼ ∶eiαϕ ∶ þ ∶e−iαϕ ∶ in two dimensions.

FIG. 4 (color online). A schematic explanation for the time ðnÞ evolution of ΔSA in terms of entangled pairs. In the left (or right) picture, the border between A and B is outside (inside) of the light cones of influence, respectively.

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PHYSICAL REVIEW LETTERS

PRL 112, 111602 (2014)

where jjiL;R are normalized such that hijjiL;R ¼ δi;j . Indeed, we can confirm that (12) leads to the density matrix (8) after tracing out the right-moving sector. Especially, if we choose k ¼ 1, (12) is equivalent to the maximally entangled two 1=2 spins (i.e., the EPR state). ðnÞf Thus, we find ΔSA ¼ log 2 for any n. It might also be useful to notice that for n ¼ 2 we find the simple formula ð2Þf

ΔSA

¼

k X

log ½2j=ð2j − 1Þ:

(13)

j¼1

Finally, we describe the results for the two dimensional free massless scalar. As opposed to higher dimensions, the operators (5) cannot be regarded as local operators in our sense as their conformal dimensions are vanishing. This motivates us to choose the following primary operators for any real values of α: O1 ¼ ∶eiαϕ ∶ and O2 ¼ ∶eiαϕ ∶ þ ∶e−iαϕ ∶. ðnÞ By explicit calculations, it is easy to show that ΔSA and ΔSA are always vanishing for the operator O1. On the other hand, if we consider the operator O2, we obtain the following result ðnÞ

ΔSA ¼ ΔSA ¼ 0 ðt < lÞ; ðnÞ

ΔSA ¼ ΔSA ¼ log 2

ðt > lÞ:

(14)

Again, we can explain these results in terms of the entangled pairs. In two dimensions, we can exactly decompose the scalar field into left and right-moving modes as ϕ ¼ ϕL ðt þ x1 Þ þ ϕR ðt − x1 Þ. Then it is obvious that the excited state O1 j0i ¼ jeiαϕL iL jeiαϕR iR is a direct product state and should have the vanishing quantum entanglement. On the other hand, pffiffiffi O2 j0i ¼ ½jeiαϕL iL jeiαϕR iR þ je−iαϕL iL je−iαϕR iR = 2; is the EPR state and has the entropy log 2 for any n. Moreover, in two dimensions the lightlike motion of the entangled pair is one dimensional and this is the reason why the entropy instantaneously jumps at t ¼ l as opposed to the results in higher dimensions (see Fig. 3). It is curious to note that the results do not depend on the parameter α, or, equally, the conformal dimension. The results for other local operators can be similarly understood in terms of the entangled pairs. For example, consider operators of the form O ¼ PðzÞQð¯zÞ, where PðzÞ and Qð¯zÞ are arbitrary chiral and antichiral local operators ¯ where ∂ and ∂¯ are the derivatives such as O3 ¼ ∂ϕ · ∂ϕ, with respect to z ¼ x1 þ iτ and z¯ ¼ x1 − iτ. If we act these states on the vacuum j0i, it is obvious that they are all direct product states between the left and right-moving sector. ðnÞ Therefore ΔSA and ΔSA are all vanishing when we take the cut off ϵ to be vanishing. In this Letter, we proposed a series of new quantities ðnÞf ΔSA which characterize local operators in CFTs. In short, they measure the amount of quantum entanglement of an

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operator or, more intuitively, quantum mechanical degrees of freedom included in an operator. They are defined as the increased amount of the nth ðnÞ Rényi entanglement entropy ΔSA at late times considering a time evolution of an excited state obtained by acting an operator on the vacuum. We chose the subsystem A to be ðnÞ half of the total space Rd . We conjectured that ΔSA are monotonically increasing functions of time. We analyzed various explicit examples in free massless scalar field theories in two, four, and six dimensions. They are enough to draw general conclusions for free massless scalar theories in dimensions higher than two, as summarized in Table I. We found that all of our results, including two dimensional ones, can be understood in terms of quantum entanglement in finite dimensional Hilbert spaces like qubits in quantum information theory. ðnÞ The behavior of ΔSA can be understood in terms of relativistic propagations of entangled pairs created by local operators. Indeed, the entropy starts increasing just when one of the entangled pairs reaches the boundary ∂A of subsystem A. The time evolution of entropy becomes step functional in two dimensions, while it gets a smooth function in higher dimensions. This is because there are many directions to propagate and the arrival time at ∂A depends on directions in the latter. Note that taking the subsystem A to be infinitely large is important to obtain a nonzero constant entropy at late time. Our entangled pair interpretation suggests that the late ðnÞf time values ΔSA do not change even if we modify the shape of A continuously. In this sense, they are topological quantities. It will be an interesting future problem to see how our results are changed in interacting CFTs, where our entangled quasiparticle interpretation might be modified. We may think of holographic computations similar to [13]. It will also be intriguing to generalize our arguments to massive quantum field theories. We thank J. Bhattacharya, S. He, T. Nishioka, S. Ryu, N. Shiba, and T. Ugajin for useful discussions. T. T. is supported by JSPS Grant-in-Aid for Scientific Research (B) No. 25287058 and JSPS Grant-in-Aid for Challenging Exploratory Research No. 24654057. T. T. is also supported by the World Premier International Research Center Initiative (WPI Initiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT).

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PHYSICAL REVIEW LETTERS

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Quantum entanglement of local operators in conformal field theories.

We introduce a series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement...
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