Entropy of Berenstein-Maldacena-Nastase Strings

In a previous paper, we proposed a probability interpretation for higher genus amplitudes of BMN (Berenstein-Maldacena-Nastase) strings in a pp-wave background with infinite negative curvature. This provides a natural definition of the entropy of a BMN string as the Shannon entropy of its corresponding probability distribution. We prove a universal upper bound that the entropy grows at most logarithmically in the strong string coupling limit. We also study the entropy by numerical methods and discuss some interesting salient features.


INTRODUCTION
The holographic AdS/CFT correspondence [5,14,18] have been very influential in recent decades, due to its wide-ranging applications in various topics in theoretical physics. A particularly interesting application is the study of the entanglement entropy, which appears in quantum information theory as a measure of information for entangled states. It is also popular in condensed matter physics as a new type of order parameters to understand quantum phases of matters and critical phenomena [3,10,12,16]. In a seminar paper, Ryu and Takayanagi proposed to compute entanglement entropy in conformal field theory holographically in terms of minimal surfaces in AdS space [15]. This provides a novel geometric perspective on entropy and has entailed tremendous research developments. See e.g. the paper [13] for physical arguments that explain the Ryu-Takayanagi conjecture. Nowadays entanglement entropy is a concept with broad appeal in physics.
On the other hand, in our previous works [7][8][9], we have accomplished some spectacular quantitative tests of the AdS/CFT correspondence in a stringy regime. In a pp-wave background with infinite negative curvature, stringy states are identified with the BMN (Berenstein-Maldacena-Nastase) operators [2] in the free N = 4 SU (N ) super-Yang-Mills theory. In this case, string theory turns out to be also extremely simplified. We proposed the string (loop) amplitudes can be computed by cubic diagrams without propagators, and are related to the field theory side calculations by the so-called "factorization principle". The usually notoriously difficult problem of computing higher loop string amplitudes with stringy excited modes, corresponding to non-planar BMN correlators here, becomes a straightforward exercise in our case and can be computed explicitly on both sides of the correspondence.
Let us introduce some notations. The BMN operators * email: minxin@ustc.edu.cn up to two string oscillator modes, orthonormal at planar level, are the followings Here we take the BMN limit J, N → ∞, and identify g := J 2 N as the effective string coupling constant. Some higher genus correlators were first computed by early papers on the subject [4,11]. For example, the free torus (genus one) two-point function is computed by dividing the long strings into 4 segments and Wick contracting according to a "short process" (1234) → (2143). The result for BMN operators with two stringy modes can be written as an integral and evaluated all other cases.
Here as in our previous papers we only consider free gauge theory, and omit the universal spacetime dependent factor in correlators.
In our recent paper [9] we proposed a physical probability interpretation of the free higher genus amplitudes of two single string BMN states. We proposed that the probability of preparing a BMN string O −m,m , then observing another BMN string O −n,n , can be written as with the sum over genus h. The supporting evidences of our proposal are that each term in the formula is always non-negative Ō J −m,m O J −n,n h ≥ 0, and it is simply normalized by the vacuum correlator to sum over all final states to unity ∞ n=−∞ p m,n = 1 for any initial mode m. It is well known that string perturbative series is usually asymptotically double-factorially divergent. However, in our case, since each term in (2) is non-negative, it is apparent that our perturbative series is actually convergent. We think this is probably due to the extremely simple spacetime structure induced by the infinite curvature, and does not necessarily signal an inconsistency by itself. Perhaps this is a rare case of non-trivial "perturbatively complete" string theory, which nevertheless still contains the infinite towers of oscillator modes as in usual critical strings. In any case in this paper we do not consider non-perturbative effects, assuming they are either negligible or non-existent here.

AN UNIVERSAL UPPER BOUND FOR ENTANGLEMENT ENTROPY
Our physical interpretation gives rise to a natural definition of the entanglement entropy S m (g) of a BMN string O −m,m as a function of string coupling constant g. It can be written as the von Neumann entropy of the probability distribution However, an immediate puzzle arises. If a BMN string is a pure quantum state, how can it evolve to a mixed state of different BMN strings? Here we are certainly not claiming a violation of the sacred principle of unitarity in quantum mechanics. To understand the issue, as explained in our previous paper [9], in the comparison with quantum mechanics, it is helpful to think about the BMN strings as different types of particles, instead as different quantum states of the same particle. As usually in collider experiments, we can not prepare or detect a quantum superposition of different types of particles. In [9], we argue that this "on-shell condition" remains valid here and a natural observer in principle can only prepare and measure the BMN states. We note that the spacetime is highly compressed due to the infinite negative curvature. The role of time in usual quantum evolution is somewhat analogously played by the string coupling constant g here. For g = 0, the BMN strings do not change and the probability matrix (2) is simply an identity matrix. The entanglement entropy for any mode m vanishes S m (0) = 0. As we turn on the string coupling g > 0, we argue that a BMN string evolves into an entangled quantum state between the string and a hypothetical observer, or the environment. So here the quantum evolution including the string and the environment may still be unitary, but when we trace out the environment, we get a mixed state of BMN strings with the probability distribution (2). This is very similar to the decoherence process much discussed in quantum computation. Therefore our definition of entanglement entropy (3) should be an intrinsic physical property of the BMN string at finite string coupling, and it is sensible to study some of its salient features. The zero modes represent discretized momenta in one of the 8 traverse directions corresponding to the scalar insertion in BMN operators, while the positive and negative modes represent the left and right moving stringy excited modes. The total positive modes must cancel the negative modes due to the close string level matching condition [2]. The conservation of (discrete) momentum in the traverse directions implies that Ō J 0,0 O J −n,n h = 0 for n = 0 at any genus h, which can be also directly confirmed by integral formula like in (1). We should note a subtlety of the arguments here. As we mentioned in [9], the non-negative condition of our probability interpretation only works for amplitudes with external single string states, while multi-string states should be viewed as some kind of virtual states in the intermediate steps of a quantum process. Likewise, the "conservation of zero mode" may be violated by multi-string states, e.g. in the Summing over all genera in (2), we have p 0,0 = 1 and p 0,n = 0 for n = 0 for any coupling g. The zero mode BMN string is decoupled from the other modes, and the entanglement entropy is simply S 0 (g) = 0. Here the formulas are symmetric for ±m, so without loss fo generality we can just from now on focus on m > 0. For a quantum system with Hilbert space of finite dimension D, the maximal entanglement entropy log(D) is achieved by a mixed state with uniformly distributed probability over an orthogonal basis. Since there are infinitely many BMN strings, it is not immediately clear that our entanglement entropy (3) is even finite. We shall prove an upper bound for the entanglement entropy.
For genus h, the field theory calculations of the twopoint amplitude Ō J −m,m O J −n,n h consist of (4h−1)!! 2h+1 cyclically inequivalent diagrams of dividing the long string into 4h segments [6]. Due to cyclicity we only need to do a one-segment integral for one mode, and 4h integrals for the other mode. See e.g. the case of genus one (torus) in equation (1). For m = n, the absolute value of a segment integration over a stringy oscillator mode is less than 1 π|m−n| . So we have an upper bound for the two-point function We see that at large distance between mode numbers |m − n| ∼ ∞, the strength of BMN string interactions are bounded by an inverse square law. Summing over all genera, we have an estimate of the probability matrix element (2) as where we denote a function which appears in the resulting summation as The function goes like f (g) ∼ g 2 in the large g limit. We can then estimate the entanglement entropy (3). First notice for 0 < p < 1, the function −p log(p) achieved maximum at p = e −1 , and it is monotonic in p ∈ (0, e −1 ). We can choose an integer and evaluate the sum in 3 parts for n ≤ m−n 0 , m−n 0 < n < m + n 0 , and n ≥ m + n 0 . The two parts that extends to ±∞ are symmetric with the same contributions, and in the middle part the entropy is maximal with a uniformly distributed probability ensemble. We find The sum is clearly convergent, so we get an upper bound for S m (g), which is actually independent of the string mode m. For large n 0 ∼ ∞, the the infinite sum is infinitesimally small, and the dominant contribution comes from the second term log(2n 0 + 1). To find the optimal upper bound, we look at the difference of the right hand side of (8) at n 0 + 1 and at n 0 ).
The above expression is positive for large n 0 , so we get better bounds when we decrease n 0 from infinity. The difference (9) as a function of n 0 may cross zero multiple times. To get a minimal value on the right hand side of (8), we need to check the integers n 0 where the expression (9) is positive at n 0 and non-positive at n 0 − 1, in the range (7) up to a sufficiently large value. String dynamics is usually quite difficult to analyze in the strong coupling limit, with notable exceptions due to many revolutionary strong-weak dualities discovered in the 1990's, see e.g. [17]. Here due to the availability of convergent results up to all string loops, we can extrapolate to strong string coupling limit g → ∞ and analyze the asymptotic behavior. In this limit the optimal upper bound (8) is simply obtained by choosing the integer n 0 ∼ g 2+ , with an infinitesimal positive parameter . For this choice the infinite sum in (8) barely becomes infinitesimal. For sufficiently large coupling constant g, we can write a simple bound Usually, the maximal entanglement entropy is interpreted as the logarithm of the "effective dimension" of the Hilbert space. So we arrive at an interesting conclusion that although there are infinitely many BMN strings, the effective dimension is actually finite and grows at most a little more than quadratically as g 2+ with the string coupling g. Our analysis can be further applied to BMN strings with more stringy modes. The next simplest example is the BMN operator with 3 different stringy modes with the closed string level matching condition m 1 +m 2 + m 3 = 0. The probability amplitude and entanglement entropy are defined similarly as before. There is a similar upper bound for two-point function. For generic case m i = n i , i = 1, 2, 3, we have In the strong coupling limit, the upper bound for probability amplitude scales like g 3 instead of g 2 in (5). We also dissect the summation range for the entanglement entropy where now the dominant middle part is a twodimensional domain bounded by g 3+ . Skipping the details, we derive S (m1,m2,m3) (g) < (6 + ) log(g), g ∼ ∞.
In general we expect a universal logarithmic upper bound for entanglement entropy with larger coefficients for BMN strings with more oscillator modes.  (8) with the choice of optimal integer n0 in the range (7).

SOME NUMERICAL ANALYSIS
In this section, we perform some numerical analysis to learn more about the entanglement entropy of BMN strings. In our previous paper [8], we have computed the two-point function Ō J −m,m O J −n,n h up to genus h = 3. We can use the vacuum correlator Ō J O J h = g 2h 2 2h (2h+1)! as a gauge of the numerical accuracy of the weak coupling approximation. For example, keeping up to h ≤ 3 contributions, we get 99.8% of the total contributions of the vacuum correlators for g = 5, or 90.8% for g = 10. So we expect our available data are good for some precise analysis up to g ≤ 5, and for some rough analysis up to g ≤ 10.
As an illustration example we plot the entanglement entropy S m (g) for for two case m = 1, 100 and 0 < g < 10 in Figure 1. We use the data up to genus 3 and also truncate the sum in (3) at |n| < 10000. The numerical accuracy is sufficient for our purpose. We see that the plots of actual values are consistent with and not too far off the upper bound derived analytically in (8).
We discuss two other salient features of the plot. Firstly, for a fixed mode m, the function S m (g) appears to be a monotonically increasing function of g. This is intuitively easy to understand. A BMN string O J −m,m generally has stronger interactions with another string O J −n,n of nearby modes, i.e. smaller |m−n|, than those with faroff modes. As the string coupling constant g increases, the interactions have longer range and are more evenly distributed among strings with far-off modes, so the entanglement entropy should increase. We have checked numerically many other examples that this monotonicity seems to be universally true at least for weak coupling g. However, it seems difficult to give a rigorous analytic proof.
A second feature is that for a fixed string coupling g, the function S m (g) seems to depends very weakly on the string mode m. In Figure 1 we see the plots for m = 1 and m = 100 are only slightly distinguishable. The function S m (g) for fixed g has larger fluctuations around small modes, e.g. m ≤ 5. However, as the mode number increases, the fluctuation becomes much smaller. The dependence on string mode m is not monotonic. We check many examples that this is true at least for weak coupling. The intuitive explanation is that when we compute the two-point function as in the integrals in (1), the dominant contributions come from the difference between mode numbers which appears in the integrals over stringy oscillator modes of e.g. y 1 , y 2 in (1), while the absolute mode number just contributes some oscillatory phases that give small fluctuations to the entanglement entropy.

CONCLUSION AND FURTHER QUESTIONS
We have studied the entanglement entropy of BMN strings by analytic and numerical methods. It would be interesting to further improve the results. In particular, is our logarithmic bound (10) close to optimal, or can be much improved? For example, the entanglement entropy may actually turn out to have a finite upper bound in the strong string coupling limit. We think this is not likely but can not rule out this possibility. More elaborate analysis are needed to answer these questions.
Another interesting research direction is to explore whether there are some kind of geometric interpretations of our results, or some interesting connections to (entanglement) entropy in other contexts. It is well known that although they are not exactly the same, entanglement entropy is related to black hole entropy in some aspects. Motivated by the area law of black hole entropy, Bekenstein proposed a universal upper bound on entropy for bounded systems [1]. The Bekenstein bound was later improved and generalized in many contexts. It would be interesting to explore whether our bound e.g. (10) is somewhat related to this line of works.