Quantum error correction and large $N$

In recent years quantum error correction(QEC) has become an important part of AdS/CFT. Unfortunately, there are no field-theoretic arguments about why QEC holds in known holographic systems. The purpose of this paper is to fill this gap by studying the error correcting properties of the fermionic sector of various large $N$ theories. Specifically we examine $SU(N)$ matrix quantum mechanics and 3-rank tensor $O(N)^3$ theories. Both of these theories contain large gauge groups. We argue that gauge singlet states indeed form a quantum error correcting code. Our considerations are based purely on large $N$ analysis and do not appeal to a particular form of Hamiltonian or holography.


Motivation
Seminal paper [1] by Almheiri, Dong and Harlow demonstrated that quantum error correction(QEC) naturally emerges in the AdS/CFT correspondence. The idea is simple: the same bulk region can be reconstructed using different parts of the boundary. So if some part of the boundary is lost or subject to quantum noise, information in the bulk is perfectly preserved and can be recovered using a different part of the boundary. This has lead to a variety of interesting results, such as entanglement wedge reconstruction [2] and derivation of Ryu-Takayanagi formula [3].
Several toy models [4], [5] of error-correcting bulk-boundary correspondence were constructed using perfect and random tensor networks in the bulk. In these examples the boundary has one spatial dimensional and the bulk is two-dimensional Poincare disk. One drawback of these models is that they do not have a Hamiltonian, so they are not dynamical. These constructions resemble approximate wave function constructions for quantum many-body systems using matrix product states(MPS) [6,7], projected entangled pairs(PEPS) [8,9], tree tensor networks [10] and multi-scale entanglement renormalization ansatz(MERA) [11]. However, it is not clear how they are related to conventional holographic systems, such as N = 4 super Yang-Mills or at least Sachdev-Ye-Kitaev(SYK) model [12,13,14,15].
The purpose of this paper is two-fold. The first goal is to examine quantum error correcting properties in more standard holographic systems. As was anticipated and as we will see shortly, this quantum error correction is only approximate in the large N limit. The second goal is to see how small this error can be. Recently it was proven that in the presence of continuous global symmetries there is a lower bound on recovery error [16,17,18], so the recovery cannot be perfect. In this paper we will prove an upper bound on error, thus demonstrating that it has to be small in the large N limit. Let us describe these two goals in more detail.
Before considering a full quantum field theory problem, it is natural to study first quantum mechanical systems. In this paper we study generic SU (N ) matrix models and O(N ) 3 tensor models. As an example, one can keep in mind Banks-Fischler-Shenker-Susskind(BFSS) [19] matrix model or Carrozza-Tanasa-Klebanov-Tarnopolsky (CTKT) [20,21] tensor quantum mechanics. BFSS is a dimensional reduction of N = 4 super Yang-Mills to quantum mechanics and it provides some of the strongest evidences for the holographic correspondence, as the gravity predictions from black hole thermodynamics have been matched with numerical Monte-Carlo simulation of this model [22,23,24]. CTKT model has the same large N limit as SYK, but does not have any disorder.
However, we would like to emphasize once again that our results are based on certain large N properties, and are not tied to any particular Hamiltonian. This may sound surprising for the following reason. In the original paper [1] on QEC in AdS/CFT the error correcting properties were tightly bound with bulk locality. It is true that BFSS at low energies does have a description in terms of ten-dimensional supergravity, but we do not expect bulk locality for a generic matrix quantum mechanics. However, it has been argued [25,26,27,28] that various large N matrix models, even a harmonic oscillator, do have a description in terms of string theory. The corresponding geometry has a string size curvature. Our results suggest that error correction may be a generic feature of string theories.
Investigating QEC in generic matrix/tensor has one important drawback: they do not have a spatial structure. Nonetheless, in the spirit of holographic quantum error correction we can ask: how robust are the code states in these models against erasures of fermions? The Hilbert space of scalar fields is infinitely-dimensional, so we will postpone the investigation of the scalar sector to future work.
What is the error correcting code subspace in matrix/tensor models? It was suggested earlier [29] that the redundancy in holographic QEC maybe tied 1 the gauge redundancy. As another motivation, let us recall that both BFSS and CTKT(or SYK) have similar features: in both of them classical gravity description is expected to arise at low energies and large N and they both have large internal symmetry groups. In the case of BFSS it was conjectured [30] that SU (N ) non-singlet in BFSS are gapped and therefore absent from low-energy gravity description. This statement was later corroborated by Monte-Carlo simulations [31]. Similarly, it was conjectured that O(N ) 3 singlets form holographic states in CTKT model [32]. Therefore, we are going to study error correcting properties of singlet states in large N matrix/tensor models. Despite focusing on SU (N ) or O(N ) 3 singlets, the states might be charged under other groups. This brings us to the second goal of this paper. This interpretation of AdS/CFT also fits well with the expectation that the full theory of quantum gravity does not have any internal continuous global symmetries [33,34,35]. It has been known for a long time that quantum error correcting codes do not allow the presence of continuous global symmetries [36]. However, this statement, known as Eastin-Knill theorem, has a few crucial assumptions. The most important assumption for the applications in QFT and quantum gravity is that the quantum systems are finite dimensional. Obviously this is not the case in QFT. Also, the theorem states that exact quantum error correction is not possible. Recent papers have proven numerous bounds on approximate error correction [37,38] in presence of continuous symmetries [16,17,18] and Haar-random charged systems [39]. Very roughly, one of the bounds states [16,17,18] that the recovery error 2 can not be smaller than where Q is total charge of the original state and it is assumed that the system is built from n elementary physical subsystems. For singlet states Q = 0, so this bound does not rule out perfect error correction. Moreover, we should mention right away that this bound is not directly applicable for our case, since the action of the symmetry group is not transversal 3 . Notice that eq. (1.1) has the total number of degrees of freedom n in the denominator. So in large systems the error can be small. How small could it be?
Despite being quantum mechanical systems, matrix/tensor models have a huge number of degrees of freedom in the large N limit 4 . Our second goal is to show that there is an upper bound for recovery error in these models as long as the number of erasures is not too big.
Last, but not least, let us point out a resemblance between gauge singlets and the so-called stabilizer codes in quantum information theory. Code states in these codes are build as invariant states under a certain subgroup {G i } of the full Pauli group 5 : In the present paper we essentially studied a version where stabilizer group is a continuous gauge group, with gauge group charges Q i annihilating the code states: There have been proposals to make decoherence-free subsystems [40] and scar-states 6 [41] using group charges Q i . However, in the current paper we will study the QEC properties and the action of generic operators.

An illustration
Let us illustrate the problem we are addressing. Suppose we have a set of Majorana fermions ψ a ij in the adjoint representation(indices ij) of SU (N ). For our purposes index a can be treated as "flavor" index. In principle we can have some other fields as well. Imagine that all the states we are interested in are SU (N ) singlets. We start from a singlet state |s and add non-singlet perturbations. They can be seen either as "errors" or excitations deliberately added by an observer. Specifically consider two states: Is there a measurement which can distinguish |ξ 1 and |ξ 2 ? If there is, then we can act by another non-singlet operator ψ and return back to the original state |s , since Majorana fermions square to one. In other words, we could say that these two types of errors are correctable.
In general, the answer is no. The fundamental fact of quantum mechanics is that states can be distinguished(with probability 1) by a measurement only if they are orthogonal. In the above example, the overlap ξ 1 |ξ 2 is not zero even if we take into account the singlet condition as we can form a singlet from three ψ 1 : Since |s is a singlet, we kept only the singlet channel 7 in the product of three ψ 1 : Tr ψ 1 ψ 1 ψ 1 + [non-singlets and 1/N terms] (1.6) Factor 8 N 3 comes from putting j = k, l = p, i = q in both sides and summing over kpq. Also we have not specified anything about the flavor index, so in general we do not expect that this matrix element is zero. However, notice that there is a factor of 1/N 3 . If N is large and we manage to show that the matrix element is small compared to N 3 , it would imply that we can distinguish the above two states with probability which is close to 1. This way we would have approximate error correction 9 .
Naively, the expression Tr ψ 1 ψ 1 ψ 1 contains N 3 terms and if Majorana fermions square to one, we expect that the corresponding matrix element will be of order N 3 . In this paper we are going to show that one can bound this matrix by So we are saved by a factor of √ N and in the large N limit states |ξ 1,2 are indeed almost orthogonal, so the two errors can be corrected with very high probability.
Interestingly, scaling N 5/2 is what one expects from 't Hooft scaling which we review in Appendix D. This scaling is expected to hold in low-energy sector of BFSS matrix model. However, in the present paper we are going to derive similar bounds for any singlet states, not just for low-energy ones. Also for more complicated operators our bounds will be less strict than 't Hooft scaling.
It is known [42] that generalized coherent states in vector/matrix models are orthogonal in the large N limit. It implies that their dynamics is classical in the large N limit. However, these coherent states are singlets under the gauge group, whereas we are discussing non-singlets.

Outline of the paper
Most of this paper is devoted to showing that this trend continues to hold: there is indeed approximate error correction in the large N limit as long as there are not too many errors. Essentially this will follow from the fact that generic non-singlet states are orthogonal in the large N limit. We are going to introduce the formalism of error operators and explain why correcting certain errors is equivalent to correcting erasures of fermions.
First we will have to generalize the bound (1.7) for more general operators and tensor models. Our main tool is the observation that we can bound operators using elementary SU (N ) representation theory. By repeatedly using Cauchy-Schwartz inequality and cutting and gluing operators we can bound arbitrary fermionic operators by certain quadratic Casimirs of auxiliary SU (N ).
For example, we can introduce SU (N ) 1 which rotates only ψ 1 ij . Then operator Tr ψ 1 ψ 1 ψ 1 ψ 1 is proportional to quadratic Casimir of SU (N ) 1 : Singlets under SU (N ) are not necessarily singlets under SU (N ) 1 . However, because of Pauli principle we can not have arbitrary "large" SU (N ) 1 representations, so that the Casimir can be bounded 10 by ∝ N 3 . More precisely, using ψ anti-commutation relations and cyclicity of the trace, one can easily obtain that it is proportional to identity operator 11 : For both matrix and tensor models we find that the error can be bounded by a power of 1/N : where S 0 is the number of affected fermions. However, this set will be a bit different in matrix and tensor models: in the matrix case we allow up to S 0 N 1/10 erasures, whereas in the tensor case we can allow up to S 0 N 1/6 errors. These powers of N do not look very impressive, but in many places we used very generous upper bounds, so the above powers of N can probably be easily improved. Also, from the illustration above it is clear that error-correcting properties are going to hold as long as the number of affected fermions is parametrically smaller than N . However, in matrix models the number of singlet operators build from k fermions grows exponentially [43] with k, whereas in tensor models this growth is even factorial [44], so naively one might expect that gauge-singlet codes can correct only ∼ log N errors. So it will require some effort to show correctability of ∼ N # errors. The idea of our derivation is elementary. We will borrow a simple formalism of tracepreserving quantum operations from quantum information theory. This formalism can describe erasures and more general quantum noise. Knill-Laflamme(KL) [45,46] condition states that the set of error operators {E α } is correctable if and only if: where P code is the projector onto the code subspace. In our case P code is the projector on singlets. This means that we are interested in the singlet channel in the product E † α E β . Had it contained only the identity operator, it would have led directly to (1.12). However, in general we will also have a non-trivial singlet operator O αβ : We show that the singlet channel O αβ is suppressed by 1/N . Secondly, it is a long way between 1/N term in the KL condition (1.13) and the error bound (1.11). The meaning of the original KL condition is that errors are orthogonal on the code subspace. It means we can come up with a measurement telling us with probability 1 which error has occurred(syndrome measurement) so that we can fix it. In our case errors will be almost orthogonal: Therefore for a syndrome measurement we will have to find the nearest orthogonal basis. This problem is known in mathematics as orthogonal Procrustes problem, which we will discuss in Appendix A.3. It has a very simple explicit solution in terms of the square root of the above Gram matrix: So the existence of an effective syndrome measurement is tied to bounding various norms of V . The construction we present is known as "pretty good measurement" [47,48,49,50], which we tailor to large N situation. The paper is organized as follows. In Section 2 we briefly review the motivation why quantum error correction emerges in AdS/CFT. Section 3.1 is dedicated to our main technical tool: tracepreserving quantum operations. We explain in details what kind of quantum operations we will be using and state Knill-Laflamme theorem which gives necessary and sufficient conditions for a perfect error correction. In Section 3.2 we explain that erasure of a subsystem, commonly discussed within the AdS/CFT, can be described as a special quantum operation.
In Section 4 we discuss the QEC properties of SU (N ) matrix quantum mechanics. Section 5 is devoted to rank-3 tensor models. We describe the models and state our main results. Also we explain why error correction eventually breaks down.
In Conclusion we summarize our results and discuss numerous open questions.
Appendices are dedicated to technical proofs. In Appendix A we prove a general bound on recovery fidelity using some mild assumptions on singlet operator spectrum. In Appendix B we prove the necessary bounds for matrix models and in Appendix C we prove similar bounds for tensor models. In Appendix D we explain what we mean by "'t Hooft scaling" and give some evidence why it should hold in the low-energy sector of BFSS.

Holographic error correction
In this Section we very briefly review the motivation for holographic error correction following the original paper [1]. Results from this Section will not be used in the rest of the paper.
The motivation comes from HKLL AdS-Rindler bulk reconstruction [51,52]. HKLL provides an explicit formula for bulk fields in the leading order in 1/N . Namely, studying free field equations of motion in Rindler wedge of AdS d one finds that one can reconstruct the bulk fields using only the data on the boundary of AdS. More precisely, given a boundary region A, one can reconstruct the bulk fields in the casual wedge W C [A] [53,54] of A.
Consider a Cauchy slice of AdS 3 and separate the boundary into three regions, A, B and C - Figure 1. Figure 1: Bulk Cauchy slice of AdS 3 and three boundary regions A, B, C. Gray regions are the intersections of the corresponding causal wedges W C and the Cauchy slice. Bulk fields within them can be reconstructed using the respective boundaries via HKLL. Bold lines χ are the causal surfaces. For empty AdS, χ actually coincides with Ryu-Takayanagi surface. However, in general χ A is larger [54].
We see that point O can not be reconstructed given any single region A, B or C. However, it can be reconstructed from any pair AB, AC or BC. Therefore, if any of the three regions is lost(erased) we can still reconstruct point O. This is exactly the setup of quantum error correction.
As we will see in the next Section, it would be more convenient to study the protection against more general errors and derive the protection against erasure as a by-product.

Quantum operations
In this Subsection we will introduce the formalism of error operators which is standard in quantum information theory. We will describe the type of errors we will consider later in the paper. In the next Subsection we will argue that this set is enough to describe erasures at known locations.
How do we describe noise in a quantum system? [46] We can consider the situation when the original system, with density matrix ρ is coupled to an environment in a state |e 1 . Without loss of generality we consider the environment in a pure state. Then together they undergo a "normal" unitary evolution: Suppose that the environment has orthonormal basis |e α , α = 1, . . . , M 0 . Then tracing out the environment yields: Introducing operators E α = e 1 |U |e α we see that a generic quantum operation can be described by a set of operators E α , α = 1, . . . , M 0 acting on the system's density matrix ρ as: Imagine that the above operation is an undesirable noise and we want to correct it: come up with another operation(a set of R µ ) restoring the original ρ: Suppose we have a code subspace and a projector operator P code onto this subspace. One simple theorem from the quantum information theory is the following: [46,45]: Quantum operation (3.3) can be corrected if and only if there is a complex matrix N αβ such that So far we have not specified what kind of operators E α we are considering, α is just an index to enumerate them. We will study E α consisting of a single product(no contractions) of Majorana fermions ψ I , where I is could be a multi-index 12 : For example, There are two reasons for that. Majorana fermions ψ I can be represented as spins using Jordan-Wigner representation: where X, Y, Z are Pauli matrices. With this normalization, all Majorana fermions square to one: Therefore products of ψ I form conventional quantum noise operators such as bit flip X and phase flip Z. More importantly, we will show in the next section that this is enough to describe erasures of subsystems.
Another reason is that we want to preserve the trace in the evolution (3.3). We can ensure that by requiring If E α is a simple product of ψ I (up to a constant), then E † α E α ∝ 1, so after a simple normalization the trace is preserved. However, if E α contains contractions then its square might involve complicated operators and there is no simple argument why the trace will be preserved.
In our case, if we treat the singlet states as a code subspace, it means that E † α E β should act as a scalar N αβ on singlet states. The product E † α E β can be decomposed into irreducible representations under the corresponding gauge group. Now the theorem can be reformulated as:errors are correctable if and only if the singlet channel consists of identity operators 1 and gauge group Casimirs (as they annihilate singlets).
One last comment is that we will consider the case of erasures at known locations. It means that operator indices I l in α can be drawn from one fixed set of indices.

Erasure and approximate error correction
In this subsection we will connect the discussion about the error operations in the previous subsection with holographic error correction described in Section 2. Following [3], we will argue that the ability to correct any error defined by a product of Pauli matrices leads to the ability to correct erasures of known fermions. We will also discuss this in the setting of approximate error correction.
How do we describe erasure of a subsystem E with quantum operation (3.3)? One option is to use the so-called depolarization channel 13 which makes the reduced density matrix on E maximally mixed 14 : In th above equations we split the system into E and the compliment of E which we call E. ρ EE is the original density matrix: If E is a one qubit, then E is given by: where Pauli matrices 1, X, Y, Z act on E. Obviously, if E consists of more than one qubit, we can depolarize all the qubits in E one by one. Such operation will involve all possible Pauli strings supported on E. It was proven in [3] that this is equivalent to correcting the erasure of E. This is why we dedicated a lot of time to Majorana operator strings in the previous section. Suppose now depolarization (3.12) is only approximately correctable. Meaning that there is recovery operation R: R : ρ EE →ρ EE (3.16) 13 In quantum information theory "erasure channel" means a different thing [55]: it replaces a qubit by an erasure state |2 which is orthogonal to both spin-up |1 and spin-down |0 .
14 Usually people consider depolarization with probability p: but here we want to enforce the depolarization, thus p = 1.
such thatρ is close to the original ρ. What do we mean by "close"? One measure is the conventional L 2 matrix norm, often called the trace distance: Another important measure is fidelity. Fidelity F between two density matrices are defined by 15 : In fact, these two are closely related. One can show that So fidelity and trace distance are equivalent. In this paper we will mostly concentrate on fidelity.

Matrix models
In this Section we will discuss error correcting properties of fermionic singlet sector in various matrix quantum mechanical models. Our discussion will not be tied to a particular Hamiltonian. This Section is organized as follows. In Section 4.1 we describe the field content of matrix models we study and how we build error operators. In Section 4.2 we discuss KL condition and how it is related to operator spectrum. Also in this section we describe our main observation, which is the suppression of singlets in KL condition. In Section 4.3 we formulate our main result. In Section 4.4 we explain how error correction breaks for large amount of erasures.

Setup and error operators
We will consider a collection of Majorana fermions in the adjoint representation of SU (N ): where a = 1, . . . , D is "flavor" index and ij are matrix indices of SU (N ) adjoint matrix, which is traceless hermitian matrix. For BFSS D = 16. We may want to define error operators in terms of ψ a ij strings: However, this is a bad choice because E † α E α is not proportional to identity operator 16 In other words, ψ a ij are not Pauli strings. Moreover they are not independent 17 . What we can do is to 15 Uhlmann's theorem states that where the maximum is over all possible purifications |ξ , |ζ of ρ and σ. 16 ψ a ij have anti-commutation relations of Clifford algebra: The ijkl color structure in the right hand side is specific to SU (N ). 1/N term is needed to make it traceless in ij and kl. Also ψ a ij are hermitian, but not as operators, but in the matrix sense: Because of the traceless condition: i ψ a ii = 0 introduce a proper orthogonal basis T (ij) in the space of SU (N ) algebra adjoints: Variable (ij) should be understood as a single variable taking dim su(N ) = N 2 − 1 values. Now and ψ a (ij) are hermitian operators ψ a We focus on erasures of fermions at S 0 known locations. As we have explained in the previous section it means that the error operators E α are given by products of ψ a ij : where index l enumerates pairs I = (a, (ij)) in the string α. Having erasures at known locations means that pairs I are to be drawn from a particular fixed set of S 0 triples. This way M 0 = 2 S 0 .

Operator spectrum
According to KL theorem we are interested in projecting the product E † α E β onto singlet(code) subspace: If α = β then due to Majorana relation (4.5) we have only identity operator: E † α E α ∝ 1. More generally, for α = β, we will have a bunch of non-trivial singlet operators which we collectively call O αβ : Our main observation is that matrix elements of O αβ in singlet states are small.
Let us see what singlet operators the product E † α E β may have. The fermions have peculiar (ij) indices, which label orthogonal su(N ) algebra generators. How do we contract them to form a singlet operator?
The inverse of eq. (4.4) is After that we can easily form a singlet with matrices ψ a ji . The matrices T (ij) will automatically form the same singlet. For example: In the large N limit we can neglect the difference between N 3 − 1/N and N 3 . Also in the above example we had a mixing between Tr ψ 1 ψ 2 ψ 3 and Tr ψ 1 ψ 3 ψ 2 . 18 Obviously if N is large and the number of operators is not too big, such "wrong contractions" will be suppressed. We will comment on this more in the next Section. So we simply write As we have mentioned in the Introduction, we can bound various fermionic operators by considering a representation theory of auxiliary SU (N ) groups. In particular, consider SU (N ) a which rotates fermion ψ a ij only. Singlet under the original gauge SU (N ) are not necessarily singlets under SU (N ) a . But since we are dealing with fermions, we would not have arbitrary "big" representations. Therefore the Casimir will be bounded. In Appendix B we show that the quadratic Casimir C 2 (SU (N ) a ) of SU (N ) a is proportional to Tr (ψ a ) 4 . Whereas this operator can be bounded by 2N 3 : Notice that naively we expect that this operator scales as N 4 . By using various tricks with cutting and gluing ψ we can bound any fermionic operator by an appropriate combination of Casimirs. The only exception are bilinears like Tr ψ 1 ψ 2 . One can easily check that for them the naive expectation N 2 is true: (4.14) We will make sure we avoid them. Essentially it means that the set of erasures/errors can contain each SU (N ) adjoint index (ij) only once, because bilinears come with a color factor Tr T (ij) T (kl) = δ (ij),(kl) . One final comment is that color factors like Tr T (ij) T (kl) . . . are not large. In Section B.3 we demonstrate that they can be bounded by √ 2. In Appendix B we prove the following bound: Theorem 2. For any single-trace fermionic operator O k made from 3 ≤ k ≤ 2 √ N fermions, there is a bound on the matrix elements in singlet states: where the | · | mean the element-wise matrix norm in the singlet subspace: 18 Since ψ 1 ij ψ 2 kl ψ 3 pq ∝ δ jk δ lp δ qi Tr ψ 1 ψ 2 ψ 3 − δ jp δ qk δ li Tr ψ 1 ψ 3 ψ 2 it happened because the two delta symbols can be both non-zero for some small amount of indices.
In terms of fermionic number k the above bounds can be concisely formulated for any operators, not necessarily single-trace: as long as O k does not contain bilinear operators, such as Tr ψ 1 ψ 2 , since they are of order N 2 , and therefore are not suppressed compared to naive N counting.

Main result
Let us return to the error correction and formulate our main theorem. One has to be careful about translating 1/N correction in (4.8) to the actual fidelity of recovery R. Appendix A is dedicated to this question. Finally, we state our main result: Theorem 3. (Error correction in matrix models) Gauge-singlet sector of matrix models forms an approximate error correcting code against any erasure of S 0 fermions at known locations as long as and the set of erased fermions does not contain one SU (N ) index (ij) more than once. Recovery fidelity can be bounded by: This theorem is proved in Appendix A. C MM is a product of three different factors and the fact that it is huge stems from using loose upper bounds for all of them.

Large operators
As we mentioned in the Introduction, O αβ P code might not be suppressed for complicated operators.
This happens because O αβ might contain a lot of singlets. Let us explain how it might happen. This will give us an estimate of how many erasures we can correct. Considerations below are very crude, so they could probably be refined. Let us consider O αβ build from k fermions. We need to estimate its matrix elements in singlet states.
1. First of all, in the previous Section we neglected operators with "wrong contractions".
Thanks to Theorem 2 they have the same scaling with N . So a problem might occur only if they are too many. In Appendix B.4 we show is indeed not a problem as long as k ≤ N/2. For such k they introduce an extra factor of 3/2 at worst.

In Appendix B.3 we show that color factors Tr
3. Product E † α E β of length k typically contain several singlet operators. Expectation value of each of them is suppressed by 2 k /(N k/10 ) due to eq. (4.18). 4. The number of different singlet operators build from k operators ψ a ij can be generously bounded 19 by the number of singlet representations in the tensor product of k adjoint representations of SU (N ). This can be computed by the following integral over unitary matrices: where Tr U Tr U † − 1 is the character of the adjoint representation. Using the following large N result [56] SU (N ) We see that this number is bounded 20 by k!.
We conclude that for αβ of length k, the singlet contribution is bounded by It is clear that this number is small as long as errors E α include no more than fermions.
Unfortunately, for our proof presented in Appendix A, it is not enough that matrix elements of O αβ P code are small. We will need to show that sums like are small, where |s 1,2 are singlet states. Notice, that this can not be small for errors at unknown locations. For example, β can be empty string. Then even if α includes only two fermions, the sum over α will have N 2 terms, overcoming 1/N suppression.

Tensor models
In this Section we analyze error correcting properties of fermionic tensor models. Our results and the general discussion are very similar to the Section 4 about matrix models. This Section is organized as follows. In Section 5.1 we describe the field content of tensor models and state various elementary properties of CTKT model. In Section 5.2 we formulate our main result. In Section 5.3 we discuss KL condition and operator spectrum. In Section 5.4 we explain how error correction breaks for large amount of erasures. Finally, in Section 5.5 we will discuss the correction of less general errors which are specific to tensor models.

Setup of the model
Unlike matrix model case, they are truly hermitian: This model has very large symmetry group And the corresponding quadratic Casimir operators C 2 are As was mentioned in the Introduction, the number of singlet states is exponentially big: Also there are a lot of singlet operators. The number of singlet operators build from 2k fermions grows factorially [44]: In principle, for singlet states one can exclude operators containing O(N ) charge operators 5.3. However, the above asymptotic formula still holds even for this subset. Although we would not assume any dynamical information, it is still worth recalling some properties of the CTKT Hamiltonian.
The CTKT Hamiltonian has the following explicit expression: In the large N limit this model is also dominated by melonic diagrams, leading to SYK q = 4 GΣ action. One can think about the CTKT Hamiltonian as N SYK = N 1 N 2 N 3 SYK model with very sparse J ijkl . Namely, there are only N 2 SYK non-zero terms out of N 4 SYK possible.

Main result
As in the case of matrix models we need to study the matrix elements of product of fermionic operators. We focus on erasures at known locations, such that error operators E α are given by products of ψ abc : where index l enumerates triples I = (a, b, c) in the string α. Having erasures at known locations means that triples in α are to be drawn from a particular fixed set of S 0 triples. This way Similarly to matrix-model case, first we will need a bound on matrix elements of singlet operators. In Appendix C we prove the following bound: Theorem 4. For any singlet state |s in tensor quantum mechanics and any singlet operator build from 2k fermions ψ abc : Similarly to the matrix model case, this bound is based on simple properties of O(N ) Casimirs. First we bound Casimir operators by N 5 and then we use cutting and gluing to bound all other operators by powers of Casimirs. Now we can formulate the statement about the error correction: Theorem 5. (Error correction in CTKT) All singlet states in CTKT model form an approximate quantum error correcting code against errors {E α } at S 0 known locations as long as Recovery fidelity can be bounded by This theorem is proved in Appendix A. Like C MM , C CTKT is a product of three different factors and that fact that it is huge stems from using loose upper bounds for all of them.

Operator spectrum
As in the case of matrix models, the previous theorem uses various facts about N scaling of singlet operators.
Let us discuss the behavior of matrix elements of E † α E β in singlet states. For example, if operators E α include only single fermion operators, then s 1 |E † α E β |s 2 will vanish unless α = β, since there are no non-trivial singlets formed from two ψ operators: ψ abc ψ a b c ∼ 1 × δ aa δ bb δ cc + non-singlets such as Q 1 aa (5.12) However, if E α are two-fermion operators then their product 21 might contain the Hamiltonian and Casimir operators 5.4: The numerical factor in the above expression have 1/N corrections and tensor δ with multiple indices is defined as Casimir operators act by zero on the singlet sector, so their presence does not cause any problems. However, the Hamiltonian does not act as identity. Moreover, if we increase the number of ψ, there is a factorially growing number of singlet operators [44].
Actually, we do not need to get rid of all the singlets. Casimir operators and identity operators are good, so we can keep them. Notice, that all operators come with certain δ a a . . . structure. One obvious thing to do is to make sure this structure is zero in front of bad singlet operators. This requires restricting the set of a, b, c indices in E α . By doing this we allow E α to have length ∼ N , but restricting the set of color indices. We will present the details in Subsection 5.5 Now we are going to bound the singlet contribution to E † α E β by 1/N as long as E α are not very long. The key element is the bound on matrix element we mentioned earlier.

Large operators
Again parallel to matrix model case, for large k there could be too many different singlets in E † α E β and they can overcome 1/N suppression. Now we need to compute the number of singlets arising in the product of 2k 3-tensor fermionic fields. This will include both different singlet operators and different delta-function color factors. Like in the matrix model case, we neglect the fermionic symmetry and obtain the following upper bound: Using the following large N result [56] SO(N ) [dM ] (Tr M ) 2k = 2 k 1 √ π Γ k + 1 2 (5. 16) we get that the number is bounded by Finally, using the Stirling approximation we see that the total singlet contribution in the product E † α E β can be bounded by: So that this contribution is small for k N 1/6 /2.

Exact error correction for less general errors
Now we consider possible restrictions on E α to get rid of "bad" singlet operators. In deriving the expansion (5.13) we essentially used Wick theorem. Notice, if we have two ψ-operators, ψ abc and ψ a b c , contracting just one index, for example a − a , leads to a non-singlet operator ψ abc ψ ab c from which one can build either a charge operator or a more complicated (bad) singlet. Charge operators(and identity) are characterized by the property that once we contact one pair of indices(a in this case) for two particular ψ, at least one other pair (b or c) will be contracted between them too.
This observation leads to the following naive solution. We restrict the set of ψ in E α such that contraction of one index in a pair of ψ will lead to the contraction of another pair. This can be done as follows. Suppose that instead of generic O(N 1 ) × O(N 2 ) × O(N 3 ) model we have O(N ) 3 model. Then we restrict the set of ψ abc in E α , from N 3 to N 2 by selecting two indices, for example a and b, and a permutation σ of N elements, such that the only allowed ψ are ψ a σ(a) c . However, it is not enough to ensure that contraction in a will lead to a contraction in b, since there might be several ψ with the same a. So we further require that all a are different. We have essentially proven (by construction) the following theorem. such that The number of such operators is N 2d d! (1 + O(ld/N )). Then quantum operations build from E α are correctable.
Obviously, such E α can not include more than N different fermions.

Conclusion
In this paper we investigated error correcting properties of the singlet subsector of matrix and tensor models. Our results are purely "kinematical": they rely on large N limit only, and do not involve any dynamical information. For example, they are applicable to the well-studied BFSS matrix model and CTKT tensor quantum mechanics. This provides further evidence that AdS/CFT is tightly related to QEC. For simplicity we concentrated on fermionic subsector which is finite dimensional. We have found that singlets indeed can correct for erasures at known locations provided that there are not too many of them.
As was expected from holographic picture, the error correction is exact only at infinite N limit. In general, we have managed to bound the recovery fidelity by 1/N # (i.e. 1/N in some fractional power).
Unfortunately, the models we studied do not have any spatial structure, so it is very difficult to probe the bulk in them. Moreover, for generic matrix/tensor models the bulk is highly stringy. BFSS does have a well-defined ten-dimension bulk, but it is still not clear what is the boundary counterpart of extremal surfaces in the bulk [57].
Obviously it would be very interesting to study QEC in systems with spatial structure. One possibility would be to study coupled SYK models or coupled tensor models, since at low temperatures they are dual to traversable wormholes [58] and in this case one can even talk about RT surfaces. The mass deformation of BFSS model, known as BMN possesses M2/M5 brane vacua which are realized as fuzzy-spheres. It would be interesting so study the error correction in this case and involve some dynamical input. It is expected that low-energy states of BMN/BFSS have 't Hooft scaling (D.4) in the matrix elements. As was noted in the main text, 't Hooft scaling grows slower with N , than the bounds we used, so assuming 't Hooft scaling would yield better bounds.
It might seem that our results are quite weak: we can correct errors involving only 1/N fraction of fermions at best. However, in our setup we can reconstruct the original state completely. Recall the Figure 1 which was the starting point for holographic error correction. In this setup erasure of A, B, C should not affect point O. However, upon erasure of A, for example, the bulk information inside its casual wedge W C [A] is lost. So from the boundary perspective we should not expect to reconstruct the original state after big erasures, since some information is really lost. It was proposed in [3] that this should correspond to the so-called subsystem quantum error correction. It would be interesting to study this setup for singlets and see how much the space of errors can be enlarged.

Acknowledgment
I am indebted to J. Maldacena for numerous comments and discussions. I am grateful to I. Danilenko for answering numerous questions about Lie algebras, A. Alfieri for pointing out orthogonal Procrustes problem and C. King for helping to find the proper cuts for 10-particle operators in tensor models. Also I would like to thank A. Almheiri, X. Dong • Small Latin indices from the beginning of the alphabet, a, b, c, . . . denote the orthonormal basis in the code subspace. For example, the projector P code can be written as: • Greek indices α, β denote a string of ψ I 1 ψ I 2 . . . of length no more than S 0 , with indices drawn from a particular subset of S 0 elements. • We consider the error operators E α , α = 1, . . . , M 0 acting on density matrices as • Up to a normalization, E α is a unitary operator. We denote to corresponding unitary by • Operators E α satisfy the following analogue of Knill-Laflamme condition: Where O ij commutes with P code . As follows from this requirement: O αα = 0 • Finally, states |c α denote normalized states: These states are charged and do not belong a code subspace. Unfortunately, these states are not orthogonal. However, they are almost orthogonal.
We will make this statement precise later.

A.2 Idea of the proof
The proof is inspired by the proof of Theorem 10.1 in [46]. The main challenge is the fact that naively defined syndrome operators do not form a well-defined quantum operation. Let us define naive projectors P α : And naive recovery R: We will need to find projector P R which completes the set of projectors P α . This recovery recovers well: in the sum over α and β: the term with α = β is exactly the original density matrix and the terms with α = β are suppressed by 1/N . However, this is not a trace-preserving operation: trace preservation requires the operator P R to be: However, because of (A.7), different projectors P α are not orthogonal. Therefore the left hand side is not even a positive operator. Despite the fact that P α are almost orthogonal, we need them to be exactly orthogonal. We will show in the next subsection (Section A.3) that there exists an orthogonalization P α of P α which is close to P α . After that we will define an improved recovery R: After obtaining simple bounds on distance between P α and P α in Sections A.6.2 and A.7.2, it will be a simple algebraic exercise to show that R has high fidelity.

A.3 Orthogonal Procrustes problem
So we are facing the following geometric problem: we have a set of normalized vectors |a α , which are almost orthogonal. We want to define a new set | a α which are normalized and orthogonal. Obviously there are infinitely many choices of orthogonal basis | a α . We want the one which maximizes | a α| a α |. This problem can be easily solved explicitly. In mathematical literature this is knows as orthogonal Procrustes problem. To simplify our notation a bit, let us define v i = |a α , so that small Latin indices from the middle of the alphabet denote the states: i = {a α}. We want to find orthonormal basis e i closest to v i . Let us denote the Gram matrix of vectors v i as V ij : Matrix V ij is hermitian and positive-definite(since the scalar product is positive-definite). It can be diagonalized with the help of unitary U , where diagonal matrix D has positive entries: Denote the scalar products of e i and v j by M ij : In general M is not hermitian. But obviously we have the following matrix equation:

General solution of this equation is
where K is another unitary. We need to minimize the distance from M to identity matrix: It is easy to see that the extremum of this problem is equivalent to extremizing the trace of M + M † . Recalling eq. (A.17), we see that the extremal K = U † , so that M is a square-root of V : This answer make sense: if V is close to identity, as we expect, M is close to identity too. Namely, denote δV = V − 1. Then Returning to the previous notation, vectors e i are |a α . So we see that It would be convenient to introduce the following bounds for the Gram matrix δV : where η and are small. We will prove in Sections (A.7) and (A.6) that

A.4 Gram matrix bounds
This Section is dedicated to the following technical Corollary: Corollary 1. Assume < 1. Then we have the following bounds: • Sum of the diagonal elements: Factor 2 S 0 comes from summing over α.
• Similar sum of diagonal elements squared: • Even if we have a sum over singlets states |a a| we can use the definition of δM and turn it into identity: a,α δM s α a α δM a α s α = a,α γ,µ;k,l=2 We can insert more projectors onto code subspace to turn this sum into matrix powers of δV , since (δV ) a α bβ = a|U † α U β |b = a|O αβ |b (A.30) For two norms involving O αγ 1 and O αµ 1 we will use the basic bound with η(Corollary 1, k = 1), so that we can freely sum over all γ and µ: • We will need to bound the sum involving two set of fermionic strings, α and β: Again, expanding each δV in terms of norms O γ i γ i+1 , we use the η bound for the norm with O αγ 1 . After that we can perform the sum over all index strings including β. Notice that now the sum over k, l starts from 1: • Using the same reasoning we straightforwardly obtain the following bounds for various products of three δM : • And finally, the sum involving four δM :

A.5 Finishing the proof
Having obtained the orthogonal basis | a α , we can defined enhanced projectors: And the projector orthogonal to all of them: We have to introduce it to make the following recovery operation well-defined, but it is not going to play any significant role, since it is orthogonal to all |a and |a α . We define the enhanced recovery operation R as: Theorem 7. Suppose that the Gram matrix δV obeys the bounds (A.22) with , η < 1. Then the recovery operation R has following lower bound on fidelity for ρ in singlet subspace.
Proof. Assume ρ = |s s|. Then the fidelity is Now we expand M = 1 + δM . The term with four 1 yield 1, the rest of the terms we can bound with Corollary 1. Putting all the terms together we have: The last inequality holds for , η < 1. The last step is to consider a generic density matrix Then after the recovery the density matrix R(E(ρ)) is given by: To obtain the desired bound we use the strong concavity of the fidelity: • If the total length of E † α E β is 2k, then the matrix elements of the singlet operator O αβ are bounded by: where state |s is O(N ) 3 singlet.
• If function l(α) denote the length of a string α, the above bound can be rewritten as: The right hand side is a decreasing function of l since we are considering l ≤ S 0 ≤ N 1/6 /2. Coefficient 512 comes from l = 4.

A.6.2 Gram matrix bounds
Lemma 1. The following inequality holds for fixed a, b, α: as long as S 0 ≤ N 1/6 /2. Also in the above expression we used the bound for singlet operators in the form (A.47).
Proof. It is not difficult to see that above sum goes over all non-empty strings γ, regardless of the string α: Notice that although γ is a combination of strings α and β it may not be longer than S 0 . The last inequality we used comes from eq. (A.46). Binomial coefficient comes from choosing 2k fermionic sites from available S 0 . It is easy to see that as long as S 0 ≤ N 1/6 /2 the terms in the sum are decreasing with k. So we can bound the sum by taking the biggest term k = 2 and multiplying it by S 0 /2. This way we obtain the bound (A.48). This seems like a very crude approximation, but in fact as can be seen from Stirling approximation, for k ∝ eN 1/6 , a single matrix element 8 k k! 3 /N k/2 becomes of order one, so S 0 O N 1/6 is a tight bound in terms of powers of N .
Recall that the bound (A.46) or equivalently (A.47) is true for any singlet state |c . Also recall that O αβ is either hermitian or anti-hermitian. It implies that it is actually a bound for the (absolute value of) maximal eigenvalue. From that we infer that Again we would like to stress that this bound hold for any singlet |s . Now we will use this fact to bound δV k for arbitrary k. Let us recall that δV is the Gram matrix without diagonal elements: Proof. Explicitly the above matrix element is: Note that since singlet states |a are orthogonal, the sum over the fermionic strings goes over After that the sum over c i converts |c i c i | into identity operator. Here it is important that the code subspace coincides with the whole singlet space. So we get At first sight it seems that for very large k we are dealing with a very long string of fermionic operators, which expectation value might not be suppressed by 1/N . But in fact we have a product of well-known singlet operators O αβ for which we have a nice bound given by Lemma 1. We can rewrite the matrix element in (A.54) as: where |a O α,γ 1 is a normalized singlet state given by: So we could lower the number of singlet operators by introducing the state |a O α,γ 1 . Repeating this procedure for other operators and using the bound (A.50) we obtain: If we do not have a sum over α, we can use the bound (A.46) for αγ 1 pair and after repeatedly using Lemma 1 for γ 1 , . . . , γ k−1 we get the statement of the Lemma. Otherwise we start the sum from α, yielding extra power of (22S 5 0 )/N .

A.7 Specific bounds for matrix models A.7.1 Basic bounds
• For matrix models, Majorana index I is I = (a, (ij)) Where (ij) is SU (N ) adjoint index and a is arbitrary "flavor" index taking D values.
• If the total length of E † α E β is k, then the matrix element of O αβ in a (normalized) singlet state |s is bounded by where the last inequality comes from putting k = 3 and demanding k ≤ N 1/10 (A.59) • If function l(α) denote the length of a string α, then one can rewrite the above bound as:

A.7.2 Gram matrix bounds
Lemma 3. The following inequality holds for fixed a, b, α: as long as the lengths of α, β are less than ≤ N 1/10 . Also in the above expression we used the bound for singlet operators in the form (A.60).
Proof. One can use the fact that the sum where Γ(1 + S 0 , M ) is incomplete Gamma-function: for which the following bounds exist [59]: where C is a number such that M > C C−1 S 0 . In our case we can put C = 1 + 2S 0 /M if M > 2S 0 . This way we obtain the bound as long as S 0 ≤ N 1/10 .
Also recall that O αβ is either hermitian or anti-hermitian. It implies that it is actually a bound for the (absolute value of) maximal eigenvalue. From that we infer that Now we will use this fact to bound δV k for arbitrary k. Let us recall that δV is the Gram matrix without diagonal elements: Lemma 4. For fermionic matrix models Proof. Again, the proof is almost the same as in the CTKT case(Lemma 2). We again has to use the property that we can transform the matrix element

B.1 Main argument
In this Section we will prove the following bounds on singlet fermionic operators O k made from 3 ≤ k ≤ 2 √ N fermions: In this Section, "bounds" mean bounds on matrix elements in singlet states. In other words, for any normalized singlet states |s 1,2 . It would be useful to introduce auxiliary operators Q a 1 a 2 ij as: One simple, but central fact is that the operator Tr (Q aa Q aa ) = Tr (ψ a ) 4 is proportional to identity operator: One can see this by commuting the first ψ to the last position and using the cyclicity of the trace. This agrees with group-theoretic expectations: operators Q aa are proportional to generators of SU (N ) a . This group rotates fermions ψ a only. Because they anti-commute the corresponding representations include Young diagrams with rows and columns of length no more than N . It means that the quadratic Casimir is bounded by const ×N 3 . The quadratic Casimir is proportional to Tr (Q aa Q aa ).
We can obtain a similar bound for other four-fermion operators. Suppose we have Tr ψ 1 ψ 2 ψ 3 ψ 4 . This operator is not hermitian, so we will bound the hermitian part. The anti-hermitian part can be bounded using exactly the same approach. The hermitian part is Tr ψ 1 ψ 2 ψ 3 ψ 4 /2 + Tr ψ 4 ψ 3 ψ 2 ψ 1 /2. For any two set of operators A I , B I , there is the following inequality: In our case A I ≡ A ij = ψ 1 ψ 2 ij and B † ij = ψ 3 ψ 4 ij . Hence the hermitian part can be bounded by Repeating this procedure one more time we see that any four-fermion operator can be bounded by 2N 3 . Let us first consider a single-trace operator O 3m containing 3m fermions. We can write it as product of the form QψQψQψ . . . : We are interested in evaluating the matrix element between the singlet states |s 1,2 : Our main tool is the following trick first used in [32]. Since |s 1,2 is invariant under any SU (N ) rotations we can force some of the indices to have specific values. For example, for each separate term with some i, we can use SU (N ) rotation to make it i = 1. So we can put i = 1 everywhere and multiply the sum by N . We can continue doing this for other indices. However, for index j now we have two choices: either j = 1 and we leave it alone or j = 1 and using a different SU (N ) rotation we can make it j = 2. Notice that since j = 1 this additional rotation does not change the value of i = 1. In the second case we multiply the sum by N − 1. We can continue doing this. As long as 2m < N the sum over indices is dominated by configurations where all indices are different. We will give an additional argument for this below eq. (B.14). So SU (N ) rotations can transform it into s 1 |O 3m |s 2 = N 2m s 1 |Q a 1 a 2 12 ψ a 3 23 Q a 4 a 5 34 . . . ψ a 3m 2m,1 |s 2 (B.10) up to subleading 1/N corrections. Now we can use Cauchy-Schwartz-Popov inequality 22 . We will have a product of ψ a 3m 2m,1 ψ a 3m 1,2m in the middle. For the reasons explained in Section 4.1 this 22 For arbitrary states |ξ 1,2 and operator A: This is standard Cauchy-Schwartz inequality. Popov part was specifying indices in eq. (B.10).
product is not proportional to identity operator. However it is a positive hermitian operator and its eigenvalues can be bounded 23 by 2. Moreover, we have two hermitian conjugate operators on the sides, so s|Hψ a 3m 2m,1 ψ a 3m 1,2m H † |s ≤ 2 s|HH † |s (B.11) Moreover, due to results of Section B.2, we can commute conjugated ψ to the middle ignoring arising anticommutators. Repeating this many times, in the end we obtain: | s|O 3m |s | 2 ≤ N 4m 2 m s|Q a 1 a 2 12 Q a 2 a 1 21 . . . |s (B.12) We can convert the right hand side into a singlet operator: | s|O 3m |s | 2 ≤ N 2m 2 m s| Tr (Q a 1 a 2 Q a 2 a 1 ) Tr (Q a 3 a 4 Q a 4 a 3 ) . . . |s (B.13) Operator Tr (Q a 1 a 2 Q a 2 a 1 ) is four-fermion, so it can be bounded by 2N 3 . Therefore in the end we get | s|O 3m |s | 2 ≤ N 2m 4N 3 m (B.14) Now let us explain why cases with coincident indices in eq. (B.8) are subleading in N . Essentially every coincidence will bring to eq. (B.10) an extra term with 1/N suppression and slightly different singlet operator spectrum. For example, leading to operators like Tr (Q a 1 a 2 Q a 3 a 4 Q a 5 a 6 Q a 7 a 8 ) instead of Tr (Q a 1 a 2 Q a 3 a 4 ) Tr (Q a 5 a 6 Q a 7 a 8 ) in eq. (B.13). For l coincidences, the number of such extra operators can be bounded by l!2 l−1 since every coincidence can be used to produce a different singlet and also we can contract coincident indices in different ways.
The scaling of these operators will be the same as in the case of no coincidences. Let us explain this in more detail. Unless we have more than 3m/2 coincidences, the operators appearing there will have less than 3m operators, so we can use induction and (B.1) to bound them. If we have 3m/2 coincidences or more, then these contributions can be bounded by N 3m/2 (the number of terms in the sum), which is already better then (B.1).
Also, coincidences can occur in many different places, leading to an extra binomial coefficient C l 2m . In the end, the relative contribution of coincident configurations is which is small since we are interested in 3m ≤ N 1/4 . Now let us consider a single-trace operator with uneven number of fermions. One can split one fermion: If we are studying an expectation value in some singlet state |s we can use SU (N ) rotation to make index j to be 1. Then there are two possibilities: either i is 1 or not 1. If it is not, we can use another SU (N ) rotation to make it equal 2: We can again use Cauchy-Schwartz inequality. In our case we have two pairs: O 12 3m , ψ a 12 and O 11 3m , ψ a 11 . For the first pair we get which is single-trace operator with 6m fermions, which we can bound with 4 m N 5m . For the second pair we get a similar expression: we omitted Tr (O 3m ) because in the original expression O 3m was convoluted with trace-free elementary fermion ψ a ij , so we could subtract the trace part from O 3m . Combining the two contributions we get Let us consider O 3m+2 . We can split two ψ operators: Using the same logic as above, specifying ijk to be some fixed indices and using Cauchy-Schwartz inequality, we conclude that we can bound the original O 3m+2 by square root of some O 6m times 2N 2 : this proves the last inequality in (B.1).

B.2 Resolving anti-commutators
Let us consider a single trace operator O k made with k fermions ψ a ij . Suppose we want to rearrange positions of operators ψ. In this subsection we are going to demonstrate that additional operators arising from anti-commutators are suppressed by 1/N . Intuitively, it happens if k is not too large, so that the index sum is dominated by different indices. Namely, we will show that k √ N . This implies that in rearranging ψ operators inside the trace we can neglect all (anti-)commutators. Our task is greatly simplified by the fact that anti-commutators are c−numbers, so we will simply reduce the fermionic number and then use the proposed bound (B.1). However, we should keep in mind that there could be k(k − 1)/2 places where ψ failed to anti-commute, we have to take it into account in order to prove that anti-commutators are subleading. Also we have to watch carefully if our manipulations are going to introduce factors ij ψ a ij ψ a ji = N 2 − 1 (B.23) as they potentially can spoil the scaling (B.1). Essentially, there are two cases when anti-commutators appear, in other words indices "collide": • Collision of (i 1 , j 1 ) and (i 2 , j 2 ) In this case the extra anti-commutator term is Where each O # is a single trace operator. l is determined by the distance between ψ operators under the trace. Could it be that we have introduced a disconnected piece like (B.23)? It is not possible for O k−2 since we have already bounded all k = 4 operators. It is possible for O k−l−2 O k . But in this case we do not have to multiply by k(k − 1) since such occurrence might happen only twice for each ψ. So we have to multiply by 2k instead. So in the end we may have ∝ k 2 O k−2 or 2kN 2 O k−4 . But it is still consistent with the scaling (B.1) as long as k N .

B.3 Color factors
We want to bound expressions like We can do it again using elementary algebra and the completeness relation: We will use induction in k: C 1 = 0 and C 2 ≤ 1 due to normalization (4.3). We can consider the square of C k and take a sum over (ij) k : The first term can be rewritten as Now, T (ij) are hermitian and due to normalization (4.3) has eigenvalues smaller then one. Therefore the first term can be bounded by Repeating this for other operators we arrive at Hence, all C k can be bounded by for N ≥ 2.

B.4 Wrong contractions
In this Section we are going to estimate the relative contribution of "wrong contractions" (like Tr ψ 1 ψ 3 ψ 2 /N 2 in eq. (4.11)). Due to the bound (B.1) these operators have the same scaling with N . So a problem may arise if the number of such extra operator overcome the 1/N suppressions. We can bound their number from above 24 , by noticing that "wrong contractions" are given by all possible permutations of fermionic operators. A trivial permutation gives unsuppressed operator, whereas other permutations are suppressed by N p , where p is the number of permuted fermions. If we fix p, there are !p non-trivial 25 permutations. We can bound this number simply by p!. So the relative contribution of "wrong contractions" can be bounded by where Γ(k + 1, N ) is incomplete Gamma function 26 : which has the following bounds [59]: where constant C is such that N ≥ C C−1 k. In our case assuming k ≤ N/2 we choose it to be C = 1 + 2k/N . We conclude that the sum (B.35) can be bounded by

C Bounding singlet operators in CTKT model
In this Appendix we will prove the bound (5.9).

C.1 Hamiltonian and Casimirs
As a warm-up, let us repeat the calculation from [32] and bound 27 the Hamiltonian by N 5 . This is illustrated by Figure 2. 24 It is an upper bound because we are neglecting the cyclicity of trace, thus adding extra operators 25 Number !p, known as derangement, is the number of permutations of p elements which leave no element in its original place. For large p, !p ≈ p!/e. 26 Similar computation is done in Appendix A.7.2 but we repeat it here to make different sections more self-contained. 27 Unlike [32] we have ψ 2 abc = 1. Diagrammatically, a singlet operator is a 3-regular graph, where vertices are fermion operators ψ abc and tri-colored edges are index contractions. Splitting an operator into two corresponds to a bi-partition of vertices into black and gray.
We can rewrite the Hamiltonian as: Where A are (Hermitian) generators of O(N 2 ), since we can combine a pair (b, c) into a single index running from 1 to N 2 : Ignoring the constant in (C.1), we can rearrange the terms in H CT KT : Therefore the Casimirs and the Hamiltonian are bounded by N 5 .

C.2 Main argument
It is easy to generalized this bound for bigger singlet operators. We will work inductively in the number of fermions 2k. We want to show that for each 2k-fermion operator O 2k there is 2k − 4 fermion operator O 2k−4 such that s|O 2k |s ≤ N 5 s|O 2k−4 |s (C.6) Consider an operator O 2k consisting of 2k fermions. It involves the sum over k triples (a, b, c). We split it into two operators: Multi-index M represents a collection of elementary indices. In Section C.3 we prove a lemma that up to 1/N corrections, O M k and O M k are hermitian or (anti-hermitian) and they commute with each other. We estimate these 1/N corrections to show that they do not spoil the desired inequality (C.6). Intuitively, it happens because for small enough k, the sum over triples (a, b, c) is dominated by configurations where all triples are different. Without loss of generality we assume that they are both hermitian 29 . Therefore we can again use Cauchy-Schwartz inequality 30 : So in order to bound matrix elements we can cut an operator in two pieces and glue them to their respective copies. Imagine we managed to cut in such a way that both O M k , O M k contain a ψ-operator with two dangling lines - Figure 3  the number of ψ-operators to 2k − 2 with an extra N 2 in front. Naively, we got a even a better bound compared to (C.6): s|O 2k |s ≤ N 2 s|O 2k−2 |s (C.9) Unfortunately, unlike the original O 2k , new operator O 2k−2 can contain 31 abc ψ abc ψ abc = N 3 taking us back to the proposed bound (C.6). Indeed, it is allowed to have a ψ abc ψ abc after we eliminate the new bubble. In fact, this is how a bound on Casimir operators can be obtained. So we must allow this situation. However, we must cut carefully and do not introduce an extra "stray" ψ with all lines dangling, as its square will result in a extra N 3 from diagram on Figure  6.
For example, consider 8-fermion operator on   2 30 For brevity we omit dressing the expression with ξ| · |ξ 31 This is discussed in detail in Section C.3.
Do we always have a proper cut? By proper we mean that both O M k , O M k have a ψ with two dangling lines, and, moreover, there are no disconnected ψ abc . We have proven by hand that such cut is possible for all connected singlet operators containing 4,8,10 fermions. We omitted 6-fermion operators, because there are no "true" six-fermion operators: all of them reduce to 4-fermion operators.
It would be convenient to contract all a indices right away and so O 2k can be thought of as build from A b 1 c 1 b 2 c 2 : One way to make sure there are no hanging ψ abc is to cut using A only. This is possible if k is even. If it is odd, we will need to cut one A in half. Let us start from proving that one can always make a proper cut if 2k ≥ 12. The proof is illustrated by Figure 5.  Stage I: we pick up a A combination(two ψ connected by a "red" a index) and declare it black. We need to consider two cases.
• I(a): Suppose one of black ψ is connected to two different A, which we declare gray.
• I(b) Alternatively, assume that it is impossible. Then for each A, both of its ψ are connected to the same A. Repeating the same argument for lower A in I(b) we arrive at the Hamiltonian operator in I(c).
Stage II: consider one of the gray ψ in I(a) which is connected to the initial black A. There are three possibilities: • II(a): Its other leg is connected to a new A. We declare new A to be black.
• II(b): Its other leg is connected to the initial black A.
• II(c): Neither of the above is true. Then the two gray A are connected to themselves. To be more precise, gray ψ, connected to the initial black A, have to connect to the other gray A. Imagine this situation happens to all triples of A and cases (a) and (b) never realize. Then we can re-paint the A and repeat the above conclusion that gray A are connected to themselves. This way we first arrive at II(d) and then to II(e), which is an isolated 6-fermion operator.
Cases II(a), II(b) are good, for each black and gray parts, we have a ψ with two dangling indices. If k is even we can paint other A as we want. Obviously, this will not introduce "stray" ψ. However, if k is odd, we need to make sure we can cut one of the A in half without introducing strays.
Stage III: suppose k is odd. Since we are working with 2k ≥ 12, we have at least three extra A -III(a). We call them new A as in contrast to old A shown on II(a) and II(b). We paint all new A except one or two into black and gray. We do this arbitrarily, but maintaining the balance, such that in the end we need to split one remaining blank A into a gray or black ψ.
• III(b), III(c), III(d) or symmetric situations: obviously we can properly split the remaining A(white circles) into black and gray.
• III(e): we can not do that, since it will lead to a stray black ψ(there is also a symmetric situation with black and gray exchanged). To order to recover good situations like III(b,c,d) we need to change the color of at most one fermion. We can assume that all gray ψ belong to the old A depicted on II(a) or II(b). The reason for this is that we have at least one new black A and one new gray A which we can always repaint and exchange their colors. Formal proof of this assumption is the following. Assume that the blank A is connected to one new gray A. This connection involves 1,2 or 3 out of 4 color lines emanating from the blank A. In principle it is possible that some of the gray ψ in Stage III are the same. Notice that the blank A can not be completely connected to another A, as it leads to charge bubbles. We exchange this gray A with another black one, leading to III(b,c,d). The remaining step is when the blank is completely connected to the old gray A. In this case this choice of the blank A is not good and we need to pick up another one. We have at least three A to choose from and they can not be all connected to old gray A, since there are not enough color lines there.
As we briefly mentioned above, we have checked by hand that the proper cut exists for 2k = 4, 8, 10. The above proof covers 2k ≥ 12. The proper cut always exists, so we have proven inequality (C.6). Also, as we have said before, all six-fermion operators reduce to the fourfermion operators up-to an extra N 2 in front. Therefore s|O 6 |s ≤ N 2 s|H CTKT |s ≤ N 7 .

(C.12)
This is the origin of 1/ √ N for odd k in the bound (5.9).

C.3 Resolving anti-commutators
Consider a singlet O 2k build out of 2k ψ-operators. We have a sum over k triples I = (a, b, c). In this Section we argue that commuting fermions past each other will yield extra anti-commutator terms which are suppressed in N . This result implies right away that O 2k and its sub-parts are (anti-)Hermitian and so we can use inequality (C.8). Let us consider case by case when indices of fermions may collide, i.e. fermion anticommutator is not trivial. Luckily, since we have Majorana fermions, collision of indices will produce operators with less fermion operators. We should keep in mind that fermionic operator with 2k fermions can have k(2k − 1) anti-commutator terms.
Suppose that in O 2k two ψ-operators with no common indices, e.g. ψ a 1 b 1 c 1 , ψ a 2 b 2 c 2 have, in fact, the same indices. This question arises, for example, if we want to commute ψ a 1 b 1 c 1 past ψ a 2 b 2 c 2 . It means that we substitute the product of these operators by a delta function: {ψ a 1 b 1 c 1 , ψ a 2 b 2 c 2 } = 2δ a 1 a 2 δ b 1 b 2 δ c 1 c 2 (C. 13) This implies that instead of O 2k we have k(2k−1)O 2k−2 , which is consistent with inequality (C.6) if k N 5/4 • Collision of (a, b 1 , c 1 ) and (a, b 2 , c 2 ). Now suppose that ψ with one common index collide. In this case we have a {ψ ab 1 c 1 , ψ ab 2 c 2 } = 2N δ b 1 b 2 δ c 1 c 2 (C.14) So now we extra factor of in front: N 2k(k − 1)O 2k−2 . This is still consistent with (C.6) if k N 3/4 .
Since we are interested in matrix elements in singlet states, in the very beginning we can exclude the charges (5.3): bc ψ a 1 bc ψ a 2 bc = −iQ 1 a 1 a 2 + N 2 δ a 1 a 2 (C.15) Since we are interested in the singlet states, the charge operator can be commuted to bra or ket, producing a number of the order of the number of fermions and reducing the number of operators from 2k to 2k − 2. So both terms effectively reduce the number of fermions to 2k − 2. We will assume that we have an operator build from no more than N 2 fermions. This way the second part dominates. It means that instead of bounding O 2k we have to bound kN 2 O 2k−2 (C. 16) This is a factor of k because there are at most k such pairs with two indices contracted. This is consistent with eq. (C.6) if k ≤ N 1/2 .
• Collision of (a, b, c) and (a, b, c) This is a drastic situation when O 2k contains a disconnected piece - Figure 6 abc ψ abc ψ abc = N 3 (C. 17) In fact, this is not possible. In the original problem of decomposing a product of ψ into a b c Figure 6: Diagrammatic representation for abc ψ abc ψ abc singlets such combinations are absent. Indeed, if there is two identical ψ abc , their product can be substituted by 1 and we need to decompose a product of 2k − 2 fermions into singlets. In our manipulations with the operators in the main part of the proof, we made sure not to introduce such disconnected factor. We might worry that while computing anticommutators, when we substitute a product ψψ by a delta function, we might accidentally create such disconnected pieces. But this is actually not possible because ψ share at most one common index as we have explained above.

D An argument for 't Hooft scaling
Let us argue why 't Hooft scaling holds in the low energy sector of BFSS. Field content of BFSS/BMN has 16(spinor of SO (9)) real Majorana fermions ψ a , a = 1, . . . , 16 and 9(vector of SO(9)) real scalars X µ both in the adjoint representation of SU (N ). BFSS model is essentially a dimensional reduction of N = 4 SU (N ) super Yang-Mills theory from 3 + 1 dimensions to one dimension. BMN is a special massive deformation. Yang-Mills coupling constant g 2 Y M in 1 dimension has dimension of mass 3 . Therefore the system is strongly coupled in the IR. In this regime, when the 't Hooft constant λ = g 2 Y M N is large BFSS is conjectured to be dual to Einstein-dilaton gravity in 10 dimensions [60]. However, since this theory does not have conformal symmetry, the geometry is not AdS 2 × S 8 : the S 8 radius is not constant and there is a large curvature region corresponding to the UV regime where 't Hooft coupling is small.
The Lagrangian of BFSS is the dimension reduction of four-dimensional N = 4 SU (N ) super Yang-Mills to one dimension: where γ ν are 10-dimensional Gamma-matrices D t = ∂ t + i[A t , ·] is the covariant derivative in the adjoint representation. There is one coupling in this model: g Y M is the gauge coupling constant.
First of all, let us explain why the power 5/2 in (1.7) is consistent with 't Hooft scaling. Naively, one expects a factor of N for each trace. However, in our case fermion operators are normalized differently. In the 't Hooft argument one considers Lagrangian in the form where we have omitted the interaction terms, and X and χ schematically refer to a collection of scalar and fermionic fields. In this normalization, inspecting Feynman diagrams one indeed concludes that single-trace expectation values of χ scale as N Tr χ k ∼ N (D.3) However, χ have non-canonical commutation relations, because of the factor N/λ in front of the Lagrangian: χ 2 ∼ λ N . In the present paper we work with fermions in the canonical normalization ψ 2 ∼ 1, hence ψ ∝ χ √ N . Hence for canonically normalized fermions Tr ψ k ∼ N 1+k/2 (D.4) However, in quantum mechanics we solve Schrödinger equation instead of computing Feynman diagrams. So it is not obvious that 't Hooft scaling should hold. However, in case of BFSS model it was argued that it holds at least for simple expectation values such as Tr X 1 2 .
Using supersymmetry and virial theorem, one can rigorously show [61] that the "typical size" R of the vacuum state is R 2 = 0| Tr (X 1 ) 2 |0 N λ 2/3 (D.5) It is believed that this inequality is saturated. For example, one can study non-singlet excitations in this model [30]. In can be shown that the energy gap between the vacuum and the lightest SU (N )-charged(adjoint) state is 't Hooft scaling (D.5) implies that non-singlets are separated by a finite gap from the ground state. Monte-Carlo simulations support this conjecture [31].