Scrambling and decoding the charged quantum information

Some deep conjectures about quantum gravity are closely related to the role of symmetries in the gravitational background, especially for quantum black holes. In this paper, we systematically study the theory of quantum information for a charged, chaotic system. We show how the quantum information in the whole system has been represented by its charge sectors, using the theory of quantum chaos and quantum error correction, with concrete examples in the context of complex SYK model. We discuss possible implications for black hole thought experiments and conjectures about quantum gravity in the dynamical setup. We believe this work will have potential applications from theories of quantum gravity to quantum simulation in quantum devices.


Contents
1 Overview As C.N.Yang has stated, Symmetry dictates interaction. Symmetry often plays a fundamental role in modern quantum physics. However, symmetry is pretty hard to understand when we discuss theory with semiclassical diffeomorphism invariance, namely, gravity, especially with the existence of black holes. We will discuss two famous conjectures about symmetry in quantum gravity. Firstly, people believe that in a consistent quantum gravitational theory, there is no precise definition of global symmetry (see some early arguments, for instance, [1,2]). In fact, since black holes cannot distinguish global symmetry according to no-hair theorem, global symmetry in quantum gravity will lead to an infinite number of indistinguishable states and trouble for black hole remnants. This is called the no-global-symmetry conjecture.
Secondly, it is conjectured that gravity is the weakest force for allowed quantum gravity theories. Roughly speaking, for quantum gravitational theory associated with U(1) gauge symmetry, there always exist states whose charge-to-mass ratios are larger than a universal lower bound, which is equal to 1 M planck . This is called the weak gravity conjecture [3][4][5][6][7][8]. The original argument for this conjecture is also related to black holes: if all black hole states have small charge-to-mass ratios, those black holes are hard to decay, again causing a large number of states.
Those two conjectures put significant constraints on the space of effective field theories that are allowed by rules of quantum gravity, sharpening our understanding about the boundary of the string theory landscape [3,[9][10][11][12][13]. Furthermore, symmetry might also be important towards a resolution of the black hole information paradox [14]. For instance, There are proposals suggesting that supertranslation symmetry breaking may provide soft gravitons, and tracing out soft modes may provide a thermal spectrum during Hawking radiation process [15].
Recently, important progress has been made about symmetry in quantum gravity. Combining technologies from holography and quantum information science, people formulate a precise notion of global and gauge symmetries in gravitational theory, and moreover, give a physical proof of the no-global-symmetry conjecture in holographic theories [16,17]. The proof is based on quantum error correction theory of AdS/CFT: in AdS/CFT, the holographic dictionary is understood as an error correction code, where the code subspace corresponds to the low energy sector in CFT and effective field theory in the bulk [18][19][20]. From the quantum information point of view, the no global symmetry statement is shown to be closely related to Eastin-Knill theorem [21][22][23] in quantum error correction [24][25][26]: there is no exact covariant code associated with continuous global symmetry.
The novel proof given in [16,17] shows the power of quantum information science when applying to contexts of quantum gravity. However, the proof itself cannot manifestly make use of black holes and their radiation, which are closely related to early intuitions about both conjectures. Thus, it is natural to ask the following question: is it possible to make statements about symmetry in quantum gravity, manifestly following early intuitions about the black hole radiation (decay) process?
In fact, there are toy models in quantum information science about black hole radiation. A simple model made by quantum circuits, proposed by Hayden and Preskill [36] suggested that black hole scrambles information quickly during radiation, opening up the study of quantum chaos in the theory of quantum gravity. After realizing that black holes are fastest information scrambler in the universe [27,28], people use out-of-time-ordered correlators (OTOCs) [29,30] to quantify the speed of scrambling. In fact, black holes are shown to be maximally chaotic and saturate the early-time chaos bound for OTOCs [31]. The Sachdev-Ye-Kitaev model (SYK) [32,33], has been recently discussed as a simple model for quantum gravity, successfully reproducing maximal chaotic dynamics for black holes [34].
Those amazing developments motivate us to set up a toy model 1 of random unitary [39] with symmetry. We will make the following simple model: consider a given Hamiltonian H and a (global) charge operator Q, since we could decompose the whole Hilbert space into charge sectors, and they will separately evolve during time evolution. Namely, for a unitary operation U = e iHt , we have 2 where H i s are Hamiltonian blocks for each charge sector, and the whole Hilbert space dimension is L = 2 D (for qubit systems). For simplicity, we discuss the U(1) symmetry, the simplest continuous symmetry in the quantum system. Then, the charge operator is defined as a sum of Pauli Zs and the U(1) action is given by a representation from U(1) group, e iθQ , where q is a real number corresponding to the charge. The above settings are, of course, far from real black holes. The main problem with this setup is that it is not easy to define gauge symmetry and justify its difference from global 1 Toy models are proven to be very useful in these years. For instance, quantum error correction [35], quantum chaos [34,36] and black hole information paradox [37,38]. 2 It differs from usual unitary time evolution by a sign in this convention, where we could simply redefine the energy eigenvalues with a minus sign to flip it.
symmetry. However, here we are going to make the simplest setup: since in the context of Hayden-Preskill experiment, the unitary is completely random, there is no definition of locality. Thus we may not need to distinguish global and gauge symmetries. This type of model is allowed by the Eastin-Knill theorem, since quantum error correction is approximate for those random codes. Moreover, when discussing gauge symmetry, we could set the experiment with global symmetry in the boundary, which is dual to gauge symmetry in the bulk. When discussing global symmetry, we could directly put the experiment in the bulk. A very similar construction is made in [40], but with a slightly different motivation. Usually, people use random circuits with U(1) symmetry to construct models with energy conservation [40][41][42][43][44], while here we are mostly discussing charges. We also hope that our work is not only helpful for high energy physics and black holes, but also for quantum information science itself, for understanding the role of charge conservation in quantum information processing. This paper is organized in the following order.
• In Section 2, we will give an introduction to some basic concepts of quantum information theory, especially the theory of quantum chaos and quantum error correction. We will also set up some notations used for this work.
• In Section 3, we present a technical discussion on the theory of quantum chaos with U(1) symmetry, based on quantum information technology built mostly from [45]. We will present theorems and computations about spectral form factors, frame potentials, OTOCs, and decoupling properties. They are chaotic variables capturing scrambling properties of the system.
• In Section 4, we give an explicit example, the complex SYK model, to support theories developed in Section 3 by numerical computations.
• In Section 5, we discuss quantum error correction theory for random unitaries with U(1) symmetry. Based on discussions about Hayden-Preskill experiment, we present some arguments about the no-global-symmetry conjecture and the weak gravity conjecture.
• In Section 6, we comment on the role of discrete symmetries. We discuss fundamental differences between discrete and continuous symmetry in terms of quantum chaos, quantum error correction, and quantum gravity.
• In Section 7, we give an overview of potential research directions we are interested in related to this work. We mention further research in quantum gravity, conformal bootstrap, and quantum simulation/experiments in the real platform.
• In Appendices, we present a simple introduction of Mathematica package RTNI we are using for technical results in this work, an alternative representation of form factors, and a discussion about spectral decoupling in U(1)-symmetric random system.

Preliminaries
In this section, we will set up our notations and give some simple introductions to basic quantum information concepts for completeness.
Notations: We consider qubit system with dimension L = 2 D . When discussing black hole thought experiment, we also use n to denote the number of qubits and d = 2 n to denote the dimension of Hilbert space. We use (q, i) to denote the matrix elements, where q means the charge sector q, and i means the indices in the charge sector q. We sometimes ignore i and directly use q as a short hand notation, which means the matrix block q. For instance, Sometimes we also directly write A q as a short hand notation of A q q . We use H to represent the Haar ensemble.B means U BU † , andB q means U q B q U † q . We define In the main text, the expectation value means and we also use the notation Sometimes we use the subscript ⟨A⟩ E to denote that we are studying the average over ensemble E. Thus we might also write the above notation as ⟨A q ⟩ Eq .
Namely, the distribution is both left and right invariant, where the measure is normalized Using Haar randomness, we could compute the Haar integral where α, β are elements of permutation group S p over 1, 2, ⋯, p. If the numbers of U and U † in the integrand are not equal, the result of the integral is zero. The function Wg is called the (unitary) Weingarten function, which are computed in group theory. For instance, for S 1 we have while for S 2 and L ≥ 2 we have So we could derive the two most widely used Haar integral formulas For more detailed information, see [45]. We also give a brief introduction on the symbolic computation of Haar integral in Appendix A. We just need to discuss two cases. When q = p, namely charge sectors are equal, then naively we get the same formula we have before When q ≠ p, the integral has been factorized by two independent Haar integrals in different charge sectors, where each of them is zero. So we get Similarly, we could compute the higher moments There are the following non-vanishing situations.
Similar technics could be generalized to other cases we are interested in.
Form factor and frame potential: For a given random unitary ensemble E, we introduce the following two quantities. (Spectral) form factor R E 2k and frame potential F (k) E . Spectral form factors are widely used in random matrix theory, which are defined as the Fourier transform of spectral data for a given Hamiltonian ensemble. It is defined by Since it is only related to the trace, it only cares about the eigenvalue distribution of the system. Thus, if we diagonalize the unitary operator as 3 (2.20) and the eigenvalue measure of given ensemble is given by Dλ, then the spectral form factor is given by In the discussions later, we will switch the eigenvalue basis and usual matrix basis freely. Spectral form factor could successfully capture the spectrum distribution in the Fourier space. For instance, for k = 1 the form factor is just a Fourier transform of the spectrum distribution ρ(λ), For Haar randomness, the spectral form factor is given by the following theorem [46,47], Theorem 2.1. The 2k point form factor R H 2k (L) counts for the number of permutations of {1, 2, ⋯, k} whose longest increasing subsequences are smaller or equal to L, where for a given permutation π, the increasing subsequence means that i 1 < i 2 < i 3 < ⋯ such that π(i 1 ) < π(i 2 ) < π(i 3 ) < ⋯.
Then we immediately know that Some alternative expressions are summarized in [47]. Now we introduce frame potential. Frame potential characterizes the 2-norm distance between a given ensemble and Haar random unitary. It is defined as We have the following simple observations where we have used the property of Haar invariance.
Moreover, for Haar system, by Haar invariance we could simply observe that For further knowledge about form factor and frame potential, see [45,48].
In fact we define = 0, we say that the ensemble is k-invariant. A typical k-invariant system is the Gaussian Unitary Ensemble (GUE) [48].
Decoupling [49]: Consider a pure state ρ 0 = ψ 0 ⟩ ⟨ψ 0 , we make a Haar average U ρ 0 U † . Assume that the whole system is decomposed as A and B where d A = dim A and d B = dim B, we could define and I A is the identity operator on A. From Haar integral calculations we know that where ≈ means we take the limit where d = d A d B is large. This statement means that for Haar random pure state, when taking a small subsystem we will obtain a nearly maximally mixed state. This is called Page theorem.
Hayden-Preskill experiment: The decoupling inequality (see for instance [50,51]) is famously used in the Hayden-Preskill experiment [36]. We consider an initial product state in A and B, where A is sharing a Bell pair withĀ, and B is sharing a Bell pair withB (We assume that the dimension of the Hilbert space is the same for A andĀ, or B andB respectively). Then we apply random unitary for the system A, B. After the unitary operation, see Figure  1, we find that the state ρĀ C is nearly decoupled to ρĀ and a maximally mixed state on C, namely, where we assume that d A ≪ d D . One can show that decoupling betweenĀ and C ensures that one can reconstruct the original state in A with high fidelity. Quantum error correction: Quantum error correction is a well-established field in quantum information theory, focusing on studying how possible a quantum system could be protected against random errors. For generic introduction, see [50]. We briefly review the basic ingredients that might be used in this paper. An error correction code is made by a Hilbert space A, a noise channel (or noise combined with encoding) N ∶ S(H A ) → S(H B ), and a decoding map D ∶ S(H B ) → S(H A ). Here S(H) means the space of density matrix on the Hilbert space H. We usually have dim H A ≤ dim H B . The channel N could generically preserve the trace, Where I A means the identity operator on H A . Namely, the construction of the error correction code ensures that the information on code subspace H A has been protected. A useful necessary and sufficient condition for quantum error correction is given by Knill and Laflamme [52].
Theorem 2.7. Say that we have a code defined above. The necessary and sufficient condition for quantum error correction is given by where i⟩ and j⟩ are codewords, and C ab could be an arbitrary Hermitian matrix. This condition is required for every a, b and i, j.
For a given code, one could use several parameters to justify the capability of the code against given errors, for instance, the dimension of the code subspace, the distance of the code characterizing at most how large Paulis the code could correct, or fidelities that are often used for approximate quantum error correction.
When designing a code we often need to specify the noise. For instance, in AdS/CFT code, the code subspace is the bulk effective field theory, and encoding map is the AdS/CFT dictionary. The noise is specifically, erasing part of Hilbert space in the boundary (erasure). The AdS/CFT code is then protecting the bulk data from erasure errors, based on the claim of the entanglement wedge reconstruction.
Charge decoupling: Generically, although symmetries make the Hamiltonian decomposed, it does not mean that energy eigenvalues will be completely independent in different charge sectors.
One of the simplest example might be H = λI, where I here is the identity operator and λ is a disordered parameter follow a given distribution. In this example, obviously, the operator H commutes with every possible charge operator Q so any operation could define a global charge for such a system. However, in each random realization, all eigenvalues are highly correlated.
In the quantum information models we will talk about in this paper, we only consider models whose eigenvalues are not correlated in different charge sectors. We call it charge decoupling, if the corresponding Hamiltonian satisfies this property, or at least roughly satisfies for large L. We find it roughly holds for the complex SYK model in numerical simulation we have done. It will be interesting to study charged systems with highly correlated charge eigenspaces in the future.

Chaos
This is a technical section about computing chaotic variables in some charged random unitary ensembles. We will systematically compute spectral form factor, frame potential, OTOCs, and decoupling property using the theory of quantum chaos and Haar integral. We focus on three different types of models: generic charged systems with charge decoupling, U(1) Haar randomness, and k-invariant systems in each charge subspace. Some technical results are obtained using the Mathematica pakage RTNI, where we give a simple introduction in Appendix A. Computations of chaotic variables in general or specific charged systems have their own values, while the discussion of decoupling will directly lead to some quantum error correction interpretations about chaotic systems, where we will give a more detailed discussion in Section 5.

General result
To start, we make some general assumptions about charged system with charge decoupling. We consider the unitary is given by many charge sectors, and each charge sector acts independently on the state. In each charge sector, the unitary is generated by a chaotic Hamiltonian. Furthermore, we assume that each subspace H i has the eigenvalues λ p,a , where p, qs are denoting charges, and a is denoting the index of the eigenvalue inside the charge sector.
We will show the following theorem, For future convenience, we could also define Furthermore, we could consider R 4 , We write an identical but alternative expression for this form factor in Appendix B. Now we study the case in general. Generically, for 2k-point form factors, let σ be a partition of 2k different objects, while term i in the partition is specified as σ i , and i is ranging from 1 to (σ) (which contains σ i objects), the number of terms in the partition. Then the 2k point form factor is given by means that we assign α 1 , α 2 , . . . , α k , β 1 , β 2 . . . , β k to those 2k different objects, and compute form factors specified by σ i , in the p i charge sector E p i .

Haar randomness
Now we specify the system to be U(1)-symmetric Haar H. Firstly we could prove an asymptotic formula Proof. In fact, in large D, R 2k is dominated by Furthermore, for small k we could compute some examples,

General result
In general, the problem is highly simplified when operators commute with charge.
Proof. We could firstly look at two-point examples. We have Thus, in general, we have Since we know that U is block diagonal, so we have to force . . . (3.12) Since A and Bs are block diagonal, we have So every index should be equal, and we have as desired.
This theorem is simply expected since operators are decoupled also to different charge sectors. If we remove such assumptions, cases are a little harder. We will give the following simple example Example 3.2. We consider a generic two-point OTOC. We have The first term is the sum over all separate charge sectors. The second term is due to nonvanishing of mixing blocks A q p in matrix A or B.
This mechanism is easy to obtain in more general case, while similar with form factor calculation, we could compute partitions of 2k objects, and then assign each partition with known OTOs in charge sectors. We will leave those exercises to curious readers.

Haar randomness
Now we consider the case where the ensemble is the U(1)-symmetric Haar randomness. To avoid triviality, we could consider cases where operators are randomly assigned instead of block diagonal in charge eigenspaces. We proceed this analysis by examples. Example 3.3. We start from two-point function. We have Since U is block diagonal we have to force q 2 = q 3 and q 1 = q 4 , and to obtain a nontrivial Haar integral we have to force every charge index to be equal, thus we obtain Using Haar results we could proceed as Specifically, we could consider A and B are Paulis. So we have the following example. then we find Now we discuss higher-point functions.
Example 3.5. For four-point we have

k-invariant subspace
Now, we start to study charged systems with k-invariance. What is the practical quantum information model with conserved charge, following the spirit of k-invariance for a general random unitary system? Considering a real system with charge decoupling, we expect that each charge sector should be completely independent of other sectors. Thus, it is natural to assign k-invariance in each charge subspace. A practical model of this type is the complex SYK model: In each charge sector, the system looks like GUE, which is known to be kinvariant. Furthermore, the model almost has the property of charge decoupling generically in all time scale. Thus, we consider k-invariance in each subspace, as a generalization of k-invariance in the case of U(1) symmetry. Note that, k-invariance in each subspace, may not imply k-invariance for the whole random unitary. We start from the simplest case, where we assume that operators themselves are also independent in different charge sectors.
Proof. As proved before we have Now let us assume that in each sector it is highly deviated from Haar, and then the spectral form factors are large. In this case we have So the result is given by Now we give a two-point example Example 3.6. We start from two-point. It is given by The first term is Specifically, if each charge sector is just the Haar system, the R 1 part and the R 2 part are zero, so we recover the previous result for Haar randomness.
This is only the two-point function. For higher-point functions, it is much harder. We give explicit computations for general four-point function in Appendix C.
We say a few words here about spectral decoupling. Spectral decoupling is the phenomenon where late time correlation functions are given purely by the spectral data and time-independent details of the operators of the correlations in some specific k-invariant Hamiltonian [48,53]. In the calculation of the four-point function i the setup of k-invariant subspace, we observe a (at least formal) violation of this phenomenon. This means that the concept of spectral decoupling may not be easily generalized to the charged system at least in the sense of k-invariant subspace. It is not clear how generic the violation is, and it might be interesting to construct such an example in some systems with finite size.

General result
In general, frame potential is a much more complicated object. We have Theorem 3.5. Frame potential for a general charged system E with charge decoupling, could be written as a sum of variables inside charge sectors, although many of them cannot be represented as frame potentials in charge sectors.
Proof. We consider the first frame potential to start, we have In general, There are many possible terms in those constructions. We could extract terms that could be written as frame potential for charge sectors, where each U V terms are identified with V U . They look like Then the result is divided by partitions 5 , for instance we have For other terms, there is no naive conjugation that maintaining positivity in a single term, and cannot simply be written as frame potentials in charge sectors. 5 Note that the expression for the spectral form factor also has a similar partitioning

Haar randomness
For the U(1)-symmetric Haar system, frame potentials could simply be reduced to form factors.
Proof. Firstly we use Secondly, for Haar system all terms that are outside of formula 3.33 vanishes. Namely we have Combining the previous analysis we observe that Thus we could directly use the previous form factor results to predict frame potentials in Haar system.

k-invariant subspace
Now we consider the case for k-invariant subspace. For terms that are inside 3.33, the problem will be reduced to simplifying frame potentials in single charge sector, which has been already computed in [48]. For other terms, we give F (1) here as a example, while F (2) will be given in Appendix C.
We have [48]: The remaining terms are given by Haar invariance So we obtain We end this discussion by introducing the following simple observation.
Theorem 3.7. If we assume that in the expansion of frame potential into charge sectors, the contribution from the highest form factor dominates, we have

Decoupling
Here we restrict our discussion to the decoupling property where the system is U(1)-symmetric Haar. We consider the system is factorized by subsystems A and B. For a pure state ρ 0 , we average over some unitary U ρ 0 U † , and we consider partial trace We compare the state ρ A and maximally mixed state on A, generically we have So the one-norm is bounded by Thus, if we could bound ∫ dU Tr ρ 2 A , we could then bound one-norm. Now, we prove the following theorem, which will be used for the discussion of the Hayden-Preskill experiment.
Proof. We firstly try to denote the expression in the following form, Here the pair ab means the combined basis in the subsystem A and B. In order to proceed the computation, we introduce further notations. We write the indices a = (q a , j a ), where q a specifies the charge sector while j a specifies the indices with fixed charge sector q a . So we have The above expression looks very complicated. Thus here we only discuss a simpler situation, where we assume that the original state has fixed charge for subsystems A and B, q A and q B .
U is given by a direct sum of different charge sectors, (a 1 , b 1 ), (a 4 , b 1 ), (a 4 ,b 1 ) and (a 1 ,b 1 ) are still in the charge sector q, so we are free to use the Haar random formula in the charge sector q. For simplicity, we also assume that n A ≤ n B . Using the Haar randomness formula, we have four terms. Two of them are contractions between U and U † , while other two of them are swaps. We only write the derivation in detail for the first term as example, where U s are contracting with the nearest U † s. The rest of them are easy to generalize.
For the first term we have For the ρ part, we know that scanning over indices separately in A and B is equivalently scanning the indices for the whole system, and sum over them we just get 1 because it is the trace. The remaining part is equal to where q(a) will give the number of charge sector for indices a, and the result is One could show that for the swap terms, there is nothing but an extra smaller factor − 1 Thus swap terms are less dominated in the case of large d q . Thus we conclude A similar expression is derived by [40] in some cases of the U(1)-symmetric Hayden-Preskill experiment, which we will discuss later.

Example: complex SYK model
Now we discuss a standard example, the complex SYK model, a very good candidate for approximate charge decoupling and k-invariant subspace.
The complex SYK model is given by the following Hamiltonian where f s are Dirac fermions (f and f † are the annihilation and creation operators respectively). J is given by independent complex Gaussian distribution with constraints: and with zero mean. Sometimes we also include a fermion mass term ∑ i m f f † i f i but here we set the mass m f = 0. By construction, in the fermion number zero and one sector of this model, we have zero energy eigenvalues. So the spectrum in those fermionic charge sector is not random.
We plot the density of states for some single charge sectors and the whole sector for N = 14 complex SYK model in Figure 2 and Figure 3 respectively, where we shift the energy such that E = 0 is the ground state in each sector. A clear feature of those plots is the edge near the ground state, where for the single charge sector, we get ρ(E) ∼ E 1 2 and for the whole sector, we get ρ(E) ∼ E. This is a feature that is pointed out by a series of works [54][55][56][57]. The square root edge of ρ(E) ∼ E 1 2 is consistent with Gaussian random matrix theory and Schwarzian quantum mechanics, while the linear edge ρ(E) ∼ E is from an extra contribution of U(1) phase. In each charge sector (except fermionic number zero and one), the energy spectrum distribution, around the low energy, could be described by a Gaussian random matrix theory. From the classification in [58], N = 14 corresponds to Gaussian random unitary GUE. Those plots clearly show a square root edge near the ground state, which is a standard consequence of random matrix theory and Schwarzian quantum mechanics.
One could also study spectral form factor in such a theory in infinite temperature. Generically in the chaotic system, the spectral form factor starts from a slope down to a dip, and then a ramp towards a flat plateau. We plot the spectral form factor in Figure 4 for each charge sector, in Figure 5 for the whole charge sector. From these results, one could verify the formula approximately holds. The relative error is given in the Figure 6. The small deviation from this formula is because of hidden correlation between energy eigenvalues of different charge sectors. Thus, this indicates a small violation of charge decoupling.
������� �� ������ �� ��� ����� ������ There is another feature that around the time where each sector is approaching Haar randomness, there is a peak on the relative error. This feature shows that the correlation between charge sectors gets relatively large around the scrambling time.  Similarly, one can study the frame potential. In GUE and Majonara SYK model, we expect frame potential should decay towards a dip, and stay in the dip for a while, and then grows through a ramp, finally towards a plateau. In Figure 7 and Figure 8 we give plots for k = 1 frame potentials as example. Frame potentials in q = 2 ∼ 6 sectors are very close to systems with Haar invariance. (In fact, due to random matrix theory classification mentioned before, each sector is predicted by GUE level statistics, where GUE is an exact Haar-invariant system and its k-invariance is zero for all k). We compute 1-invariance in q = 2 ∼ 6 sectors in  Figure 9. Moreover, we check the following formula by plotting the relative error between two sides of the equation in Figure 10. This analysis shows that the error from the prediction 4.4 is small (smaller than 0.1), while the mismatch is again, due to correlations between energy eigenvalues in different charge sectors, namely a small violation of charge decoupling.
One another feature we could obtain is the time dependence. From 6 and 10 we see the deviation keeps growing until the system gets sufficiently scrambled, which implies that during ����� ��������� �� ��� ����� ������ scrambling the correlation between different charge sectors gets amplified. We also observe from 9 that 1-invariance keeps growing until scrambling, which means that in each sector, during scrambling, we get more deviation from exact invariance under Haar transformation, although measured in frame potential, the system gets more closed to Haar ensemble H following 7.

Codes
This section is written for quantum error correction interpretation of random unitaries. In this part, we will describe the theory of quantum error correction for Haar randomness and its U(1) extension. A standard application of the above theory is the Hayden-Preskill experiment. For U(1) Haar randomness, we show that it is consistent with the upper bound on the fidelities for covariant codes. Finally, we will address some possible implications for black hole thought experiments.

Quantum error correction for Haar randomness
The phenomenon of scrambling is widely used in the construction of quantum error correction codes, which could be understood as a connection between quantum error correction and chaos.
As a warmup example, we consider the following construction. Consider the system A and B, we put A with state basis α a ⟩ with 1 ≤ a ≤ d A , and B with a fixed state B⟩ (d A and d B are dimensions of A and B). We define the code subspace, Here, U is a sample from the Haar random ensemble with dimension d A + d B . We understand U as the encoding map. Thus, the dimension of the code subspace C is the same as the dimension of A.
In order to show the ability of quantum error correction of the above code, we try to do some heuristic computations about the Knill-Laflamme condition, and we take the average of it. We consider where O is a generic operator. We could compute Haar average which means that this condition is satisfied for an arbitrary operator. This is impossible in the usual definition of exact quantum error correction code without averaging, since code parameters should satisfy some bounds (for instance, the quantum singleton bound or the hamming bound).
Since it is a random average, we might also consider the variance of the Knill-Laflamme condition. Similarly, we compute This computation shows that the variance is approaching zero for large system L → ∞. As a clearer example, we consider the Pauli error. We set O = P † µ P ν . It is easy to show that Another aspects of showing quantum error correction ability of Haar randomness, is through the discussion of decoupling. It follows from the following simple facts. Consider a distance t + 1 code. Say that we have a state ψ⟩ in the code subspace with dimension n and we trace out n − t qubits, If it is a non-degenerate code, which means that the matrix C ab in the Knill-Laflamme condition is not degenerate, for any error operator with weight up to t. Then one can show that In fact, non-degeneracy implies that if O is not identity, since one can expand ρ (t) as where the sum is taken over Paulis. Then from the orthogonal property of Pauli operators we have all ρ O = 0, which implies that ρ is maximally mixed. This is another evidence supporting the fact that Haar random unitaries have good error correction properties, following from the decoupling property we have discussed above. One might note that maybe it is not very suitable to describe approximate, random quantum error correction codes in terms of code parameters like the distance. Thus, people usually rely on the definition of fidelity of recovery to define the decoding capability properly. This might be particularly important for applications in holography, since the concept of quantum error corrections usually hold approximately due to the leading order, semi-classical gravitational path integral. The phenomenon of non-zero variance has also been shown in the recent discussions about black hole information paradox and baby universe, where disordered average is shown explicitly in some examples of gravitational path integral and holographic dual (for instance, see some decent discussions [37,38,[59][60][61].).
The following theorem will somewhat make the above statement rigorous [62,63], Theorem 5.1. Consider a mixed state on the system Y Z, which is purified by X. The whole state ρ XY Z is a pure state. We assume that there are noises only acting on Z. Say that there are recovery maps R = D ○ N acting on ρ XY , then the supremum of the state fidelity over all possible recoveries (the fidelity of recovery) is bounded by the conditional mutual information where the fidelity is defined by and we have the definition of the conditional mutual information where S is the von Neumann entropy.
One such recovery map with high fidelity is so called the Petz map [64]. The condition where the conditional mutual information is zero, is called the quantum Markov condition for states. For chaotic randomness, like Haar distribution, the state is highly chaotic and thus Markov, ensuring a potential high fidelity of recoveries. The above condition is used also in the context of holography, see for instance, [65][66][67].
We will show later, that the above condition is exactly the decoupling condition in the Hayden-Preskill experiment.

Error correction in the Hayden-Preskill experiment
The above picture is closely related to the Hayden-Preskill experiment. We will do the Haar case with more detailed analysis here. the Hayden-Preskill experiment is used to model a black hole and discuss information transfer during Hawking radiation, which is described in Figure 1. In this diagram, Alice A has a small amount of information (a state), while B is a black hole which is connected (maximally entangled) with Bob's quantum computerB.Ā is a reference system, forming a Bell pair with A. After time evolution, C becomes the remaining black hole, and D is the Hawking radiation.
One could identify the above construction in the Theorem 5.1. After the encoding U , the reference system X is identified withĀ, and we also set DB as Y and C as Z. The noise is understood as the erasure of C. Thus, this procedure could be understood exactly as the error correction process above. Alice's diary is understood as codewords, random unitary U is understood as encoding map, while the error is understood as erasure. The fact that random unitary could serve as a good error correction code, ensures that one could correct the error (erasure noise) and recover the original codewords (Alice's diary). Note that in this example, the codewords are understood as the states of Bell pairs within AĀ, which has the Hilbert space dimension d A . In such construction, the codewords are fixed and we could choose the states in AĀ to be the computational basis, and we choose the random encoding U as a random matrix with respect to this basis. Since we are taking average over a given random state, Theorem 5.1 could apply 6 .
After applying the unitary operator, the state is given by where we use a, b to denote the basis fromĀ andB, while c, d to denote the basis from C and D. So we compute the reduced density matrix We define Then we obtain The Haar integral formula allows us to estimate the trace distance as where we drop the higher order terms in the 2-design formula. Note that the leading terms in the last two formulas are the same. Thus we arrive at 6 We might notice that the decoupling condition in the Hayden-Preskill experiment here is using the 2-Renyi entropy, while we are bounding the fidelity of recovery using the von Neumann entropy. When decoupling happens, the state is nearly maximally mixed. Thus we could ignore such a difference. However, we might also consider some general bounds from the 2-Renyi conditional mutual information towards the von Neumann conditional mutual information. (see some related works [68][69][70], or the appendices of [71]) This formula says that for a small input state d D ≫ d A , the total state is quickly decoupled. We could also use the 2-Renyi relative entropy and the 2-Renyi conditional mutual information If the 2-Renyi conditional mutual information is small enough, namely we arrive at an efficient decoupling. The requirement where the conditional mutual information is small, implies that we have a high fidelity of recovery based on the Theorem 5.1. Namely, one can reconstruct the Bell pair between A andĀ with high fidelity. Beyond the Petz map, one of the explicit decoding algorithm is recently constructed by Yoshida and Kitaev [72]. Thus we see again that the possibility of quantum error correction in this process, is based on the scrambling property of Haar random unitary.

Error correction and U(1)-symmetric Haar randomness
Now we could address the extension of the story about error correction and the U(1)symmetric Haar randomness.
We could do similar heuristic calculations as we have done before. We consider the states before encoding, α a , B⟩, to be charge eigenstates. We fix the total charge m to be m A + m B , where m A ≤ n A = log 2 d A is the fixed charge for A, and similar definitions apply for B. Now, the dimension of the code subspace isd We consider the Knill-Laflamme condition Since a and b have the same charge, we have Moreover, the variance is given by The above calculations imply a similar conclusion with Haar randomness: we will have good code property for large d m .
One consider the input state in the Hayden-Preskill experiment, A and B (with number of qubits n A and n B , Hilbert space dimension d A and d B ) to have fixed charge m A and m B . After a U(1)-symmetric evolution, we get systems C and D with Hilbert space dimensions d C and d D . We assume total charge m = m A + m B . In this setup, since we set the explicit charge for the system A and B, the actual dimensions we should consider in system A and B should be the dimensions of their charge eigenspaces. For convenience, we denotẽ Now, combining the tools we have before, one could show that in the current setup Part of the above calculations has been done in [40], which is in the special case where n C ≥ m ≥ n D . However, in our case we still wish to assume n C ≥ n D but we wish to keep the charge m to be more general, in order to inspire discussions about quantum gravity conjectures. Now we discuss the decoupling condition in this system. One could compute the conditional mutual information

So the decoupling condition isd
The decoupling condition in the case n C ≥ m ≥ n D , is the same as the criterion given in [40]. We will follow the same path but instead make a more general analysis, • In the case n C ≥ m ≥ n D , we have The approach of [40] is to show that for each f in the above sum, we wish to demand In the case where f ≥ m A , the right hand side is smaller than 1. For those f s we demand which is correct when n D ≫ n A . Although, for small f the above inequality may not hold, but one could argue that there are relatively small contributions from those f s since for large n C , it forms nearly a binomial distribution and the contribution f ∼ O(n D ) should be dominant. Thus, the conclusion is that in this case, n D ≫ n A will generically ensure the decoupling.
• We have noticed that the above computation could completely extend to the case where we have m ≥ n C ≥ n D , since thus we will still win to show the inequality in each term and the f ∼ O(n D ) dominance in the sum.
• The problem appears in the case of small charge n C ≥ n D ≥ m. We have Since the sum over f is restricted to the range between 0 and m, it has to be dominated by small f pieces if m is small enough. In this case, the condition n D ≫ n A may not ensure the decoupling. We will give a simple example. Consider m = 2 and m A = m B = 1. We have For large system, asymptotically we have Thus as long as we cannot arrive at the previous decoupling even if n D ≫ n A .
The decoupling condition beyond the usual n D ≫ n A criterion, in the case of small charge, is seemingly intuitive physically. If the system has a fixed small amount of charge, the code subspace has stronger restrictions and thus the decaying process is hard to proceed arriving at the approximate Markov condition. At this point we think it is safe to make the following remark, Moreover, if we assume the dominance of binomial distribution, we could arrive at the condition when it is safe to make m ∼ O(n D ). If so, the previous proof for larger charge could still apply, and we arrive at a safe decoupling at n D ≫ n A . We wish to interpret the above remark as a hint of weak gravity conjecture, although it is still far from explicit statement about real life of quantum gravity and black holes. We will make further comments at the end of this section.

Bounds from the approximate Eastin-Knill theorem
Here we discuss the connection between this work and the recently-proved approximate Eastin-Knill theorem [26].
As indicated from previous discussions, symmetry could provide significant constraint on the structure of quantum error correction codes. When the symmetry operators commute with the code subspace, we say that the code is covariant. To be more precise, we consider the encoding from the logical system S logical to the physical system S physical . The physical system is made by D subsystems Λ 1 , ⋯, Λ D . We assume that there is charge operator T logical acting on the logical system, while charge operator T physical acting on the physical system. Furthermore, we assume that the symmetry acts on the physical system transversely. Namely, we have and the condition of covariance requires that the symmetry operator should commute with the encoding. Take a look on the structure of the U(1)-symmetric Hayden-Preskill experiment, we could immediately realize that it is an approximate covariant code with U(1) symmetry. The symmetry operator is the charge which is definitely transverse. When we split the system by A and B, we restrict their charges to be m A and m B . The encoding map commutes with the symmetry operator, ensuring the covariance of the code. A celebrated result is obtained by Eastin and Knill [21], claiming that if the symmetry is continuous, then the exact covariant code does not exist. Recently, there are detailed discussions about extending this idea to approximate quantum error correction [26]. There are universal constraints written in terms of fidelity from properties of symmetries. for given noise and encoding, where C is the code subspace. Then we have where ∆T logical represents the difference between maximal and minimal values of charge operator eigenvalues in the logical system. ∆T i is the different between maximal and minimal values of charge operator eigenvalues for system i.
Since the theorem has been already proven, it implies that in the U(1)-symmetric Hayden-Preskill experiment there should exist the above universal constraint even if decoupling happens. Thus, we have the following statement, A directly computation about the above fact goes as the following. One can show easily that [26], the fidelity defined by inner product is the same as the fidelity defined by the trace distance on the density matrix. Furthermore, the worst-case fidelity searches for the worst performance in the code subspace, while in the U(1)-symmetric Hayden-Preskill we are using a specific state. Furthermore, we know from Theorem 5.1 that the conditional mutual information is bounded by the fidelity. Combining all the above facts together, we demand the consistency condition between the decoupling of the U(1)-symmetric Hayden-Preskill and the approximate Eastin-Knill theorem: The bound is following from the fact that in such system we have ∆T logical = m A and ∆T i = 1. When decoupling happens, we have Although we do not have a rigorous verification for the above combinatorial inequality, some heuristic interpretations might be given. The right-hand side is seeking for a bound on fidelity by looking at permutations using the rule of Haar randomness and performing some counting. The left-hand side is looking at a much weaker constraint by counting the number of qubits in the code subspace and the whole space directly. Thus heuristically, the specific construction based on Haar randomness should be weaker than the universal constraints. In the limit where n D ≫ n A and sufficiently large charge, where we expect decoupling should happen, both sides vanish. Thus, the above inequality will provide constraints on the limitation of decouplings that might happen in the U(1)-charged Hayden-Preskill experiment.

Comments on quantum gravity conjectures
In this part, we wish to present the study of the relationship between quantum information computations on the Hayden-Preskill experiment with the related quantum gravity conjectures. More precisely, we will discuss the existence and properties of global and gauge symmetries in quantum gravity. The above computations are far from real quantum gravitational systems like black holes. For instance, since we directly put a random unitary to represent black hole dynamics, there is no precise definition of the locality. Thus, it is hard to distinguish global and gauge symmetries, allowing us the opportunity to discuss both of them at the same time. However, we are still expecting that we could get some hints from toy models, following the spirit of several important works about toy models in quantum gravity (for instance, [36]).
People have long-time suspicion in the past that there is no precise notion of global symmetry in theories with consistent quantum gravity. An important argument is from the black hole no-hair theorem, stating that black holes are parametrized only by their mass, charge, and angular momentum. Since this charge means the local charge corresponding to gauge symmetries, there is no room for global charges to existing. Namely, suppose a precise notion of global symmetry exists, there is a huge amount of states which our descriptions about black holes cannot capture, causing large degeneracies and a thus large amount of entropy. If we don't wish it to happen, we should forbid the notion of global symmetry in the quantum gravitational system.
Recently, people make significant progress about global symmetries in quantum gravity, based on holography and quantum information theory. [16,17] presents a holographic physical proof of no-global-symmetry conjecture based on the following two steps: First, they use the tools from holography, so-called entanglement wedge reconstruction, to argue that bulk global symmetries should act on the boundary transversally in each subregion. Second, they argue that these symmetries acting on the boundaries are the logical operators that preserve the code subspace. From quantum information theory, they conclude that those symmetries must be the logical identity, justifying the no-global-symmetry conjecture in the bulk.
The proof made by [16,17], especially the first step, is closely related to the proof of Eastin-Knill theorem stating that there is no covariant code for continuous symmetries. Moreover, one could also construct a holographic proof of the Eastin-Knill theorem (see [25,26]). This follows from the fact that the holographic dictionary could be interpreted as a quantum error correction code [18,19], while covariant code means that the global symmetry exists in the bulk effective field theory, namely, the code subspace of the boundary Hilbert space.
Here, in the context of the Hayden-Preskill experiment, the existence of the (approximate) Eastin-Knill theorem has another interpretation: it could set a lower bound on the fidelity of recovery during charged black hole evaporates. The structure of charges provides crucial constraints on the code subspace, forbid the black hole system to decouple sufficiently, and arrive at a chaotic, Markov state.
Another side of the story comes from the claim about the (weak-form) weak gravity conjecture. While no-global-symmetry theorem suggests a fundamental property about global symmetry in the bulk spacetime, weak gravity conjecture is a claim about gauge symmetry. The slogan of the conjecture is very simple: gravity seems to be the weakest force in any theory that is consistent. More precisely, consider a quantum gravity theory associated with gauge symmetries, there always exists a state such that the charge-to-mass ratio is larger than some constants. Considering the existence of Reissner-Nordstrom metric in the semi-classical description of gravity, we could set this constant to be 1 M Planck in the natural unit.
We wish to argue that the existence of a lower bound on the charge in order to make the decoupling condition d D ≫ d A universal might be related to some heuristic arguments of weak gravity. In fact, it is pretty consistent with one of the earliest arguments that are presented in [3]: suppose that weak gravity conjecture is false, then there are many states that have relatively small charges. Considering that during the black hole decay process, the charge is conserved while mass is decreasing, then there exists at least one decaying product with a growing charge-to-mass ratio. Then we arrive at the conclusion that there are many stable states which completely cannot decay, which is not favored for a consistent quantum gravitational theory. This situation is pretty similar to what we have in the U(1)-symmetric Hayden-Preskill experiment. If the charge we have is sufficiently small enough, black hole quantum states are hard to decouple even in the case where d D ≫ d A , since small charge restrict the allowed code subspace. If we believe that the black hole decay process will end up with a highly chaotic state with a sufficiently large amount of decoupling, we have to make the initial state sufficiently charged. Thus the U(1)-symmetric Hayden-Preskill experiment could be regarded as a toy example about the decay of a charged black hole, supporting the statement of weak gravity conjecture. Furthermore, if we believe that this experiment happens in 1+1 dimension, we might regard the size of the qubit system as the mass. Thus the condition m ≥ O (n D ) suggests a bound relating charge and mass, sharpening the analogy between this experiment and weak gravity conjecture.
Finally, we briefly address issues about gauge/global symmetries and holography. Here our Hayden-Preskill experiment is at least manifestly, built in the bulk. We are not able to directly build gauge groups using random unitaries in this work. However, in the discussions about chaotic systems, we would imagine that the decoupling process could appear in the boundary. People believe that roughly speaking, gauge symmetries in the bulk are dual to global symmetries in the boundary. Thus we could, in principle, using constructions like complex SYK model to probe gauge symmetries in the bulk. We look forward to future studies about concrete models with gauge/global symmetry duality in the future.

Discrete symmetries
Although the main part of this paper is focusing on continuous symmetry, we wish to address a closely related issue, discrete symmetry. We claim that unlike continuous symmetries, discrete symmetries are more similar to systems without any symmetry: many properties are not far from the full Haar randomness.
The distribution of Haar randomness on the unitary group could be generalized easily to the unitary group with extra discrete constraints. For a D-qubit system, extra constraints induced by symmetry S will end up with a uniform distribution on the quotient space: U(2 D ) S. One of the earliest works about symmetries in random systems is due to Dyson [73] about the three-fold classification of random matrix theory. In his classification, we take S = O(2 D ), inducing the circular orthogonal ensemble (COE), or we could also take S = Sp(2 × 2 D ), inducing the circular symplectic ensemble (CSE). these two extended classes could be understood as random systems with time-reversal invariance T . COE corresponds to T 2 = 1 while CSE corresponds to T 2 = −1. A more detailed classification, so-called Altland-Zirnbauer [74] classification, involves ten classes in total involving at most two antiunitary operators, is widely used in the study of condensed matter physics, for instance, ten-fold classification of topological insulators [75,76].
Unlike continuous symmetries, discrete symmetries often reflect topological properties in the many-body system. In random systems in general, some chaotic properties, for instance, level distribution or spectral form factor, will show universal behaviors associated with corresponding discrete symmetries. A three(ten)-fold of random systems could naturally induce a three(ten)-fold classification of quantum circuits. One could generalize those Weingarten functions easily through group theory. Some earlier works have been done in [77,78], and those topics are recently summarized in [43,44,53,79] in terms of OTOCs, random quantum circuits and chaotic systems in general.
In the context of discrete symmetries, Weingarten calculations are very similar with Haar average without symmetry. For instance, we list some Weingarten functions for COE for large L: Comparing with the results with the whole random unitaries, we see that asymptotically, extra symmetries cannot affect the scaling on L. In fact, one could check that it is indeed a generic feature of discrete symmetries. With the help of earlier works (for instance [77]), we could easily check the following claim The above generic feature could also be reflected in the language of quantum error correction, showing the striking differences between continuous and discrete symmetries. Intuitively, from the scrambling point of view, since Weingarten functions have the same scaling behavior, there are no extra limitations for a code arriving at the end of decoupling. In fact, unlike Eastin-Knill theorem, discrete symmetries are allowed in quantum information theory, in the context of covariant codes. Theorem 6.1. There exist covariant codes with discrete symmetries.
In fact, from the earliest understanding about quantum error correction in AdS/CFT, the qutrit code discussed by [18], is an example of covariant code. In fact, there exist logical operators X logical and Z logical in the code subspace that could be viewed as global discrete symmetries. Some other examples and proofs are summarized in [24].
Note that such a situation is different from quantum gravity and holography. People believe that the no-global-symmetry conjecture is correct even for discrete symmetries in a consistent theory of quantum gravity. Thus, toy models constructed from covariant codes with discrete symmetries cannot capture features about global symmetries for AdS/CFT codes. So how it works from holographic statements such that it proves a stronger version of the Eastin-Knill theorem? Here, we wish to take the point of view from [26]. In the two-step proof by Harlow and Ooguri [16,17], the second step where symmetries are logical operators acting on each boundary, may not be preserved for generic codes that are used for fault-tolerant quantum computation. This is from the fact that we expect in the boundary quantum field theory, the code subspace is from the low energy spectrum of the corresponding CFT. Roughly speaking, symmetry operators acting on the boundary are required not to change the energy of the states too much, justifying that they are logical operators.
In the recent discussions about quantum gravity, people make extensive studies on the Jackiw-Teitelboim gravity in two dimension, as the dual gravitational theory of the SYK model. There are recently many discussions about discrete symmetries, and also continuous symmetries in such holographic context (see [54,57,[80][81][82]). It is interesting and important to explore symmetries in those concrete models, and their scrambling and quantum error correction properties during quantum information processing.

Outlook
In this section, we will point out some potential applications and possible future research directions along the line of this work.

Gauge and global symmetries
As we discuss in the main text, we could either build the Hayden-Preskill experiment directly with symmetries, or our stories about scrambling happen in the boundary. For the former, it will be interesting, in the future, to build concrete models about Hayden-Preskill experiment, or in general, quantum error correction codes with gauge symmetry. In the context of AdS/CFT code, considerable progress has been made along the line of [6,35,83]. For the latter, it might be interesting to consider the holographic Hayden-Preskill experiment following the setup of [38] with global symmetry in the boundary. The problem setup discussed in the main text is far from real, thus, more realistic models, especially with a concrete definition of gauge symmetries, will definitely be helpful for the understanding of black hole evaporation associated with symmetries in its low energy description. Moreover, the implementation of gauge symmetries may not only be helpful for theoretical studies of tensor networks, quantum gravity, and AdS/CFT, but also for machine learning and neural networks. For instance, see [84,85].

Black hole thought experiments from toy models
It will be helpful for considering more details in the U(1)-symmetric Hayden-Preskill experiment discussed in Section 5. For instance, what is the precise sense of the approximate Eastin-Knill theorem here in the high energy physics sense, without using AdS/CFT? How to connect the discussions about lower bound on the charge, to the evaporation experiments in the statement of weak gravity conjecture? Is it possible to address the entropic arguments used in [7] in the context of U(1)-symmetric Hayden-Preskill experiment? The last suggestion might require a generalization from Bell pair to thermofield double in the Hayden-Preskilltype experiment. It might also be interesting to discuss the connection between quantum gravity conjectures and the recovery construction, beyond Petz map or Yoshida-Kitaev decoding [40], moreover, in the context of traversable wormholes [86,87].
Recently, significant progress has been made along the line of Hayden-Preskill experiment, black hole evaporation and the information paradox. With the help of tools from gravitational path integral, holography, and entanglement, people reproduce the Page curve, a standard result of black hole unitarity from random unitary toy models [37,38,[59][60][61]. It might be interesting to address those calculations about black hole evaporation with global or gauge symmetries into account, in generic semiclassical descriptions following [37] or in explicit models like Jackiw-Teitelboim gravity or SYK model. Addressing those calculations in holographic context will be helpful for interpretations of symmetries in quantum gravity and quantum information processing in black hole dynamics.

Experimental platforms and large-scale bootstrap
Nowadays, there are rising interests from formal constructions of black hole thought experiment to quantum simulation in the analog or digital, real platform in code-atomic and condensed matter physics [88]. These studies might be helpful for the study of quantum information science itself, providing clear targets and motivations for benchmarking coldatomic devices or near-term quantum devices. Moreover, it might also be helpful towards a deeper understanding of black hole dynamics and holography in the near-term or long-term, and clarifying conceptual problems about computability and stimulability of our universe, namely, the quantum Church-Turing Thesis [89][90][91].
Charged systems in quantum many-body physics could provide fruitful platforms for exploring the propagation of quantum information in the quantum materials. For instance, significant progress has been made about thermoelectric transport in the complex SYK theoretically and experimentally [92]. Here, we wish to emphasize recent developments of conformal bootstrap for exactly solving CFTs with global symmetry. In the context of the four-point function, global symmetries will decompose crossing equations in different sectors based on their representation theory, causing generically different critical exponents for corresponding CFTs. With novel analytic and numerical tools developed recently [93], people could generate information about global symmetries automatically [94] for a large class of Lie group, and solve bootstrap equations in relatively large scale [95,96]. A particularly successful example recently is O(2) symmetric CFT in three dimension [96] (very similar to U(1)), which is widely applied in condensed matter and cold-atomic physics. Those studies about CFTs will potentially be very helpful for providing robust data for AdS/CFT and theoretical/experimental studies of quantum materials.

Quantum simulation
Here we say a few words about quantum simulation for charged systems. It is definitely interesting to construct further toy models in quantum circuits to simulate some specific charged black hole evaporation processes either in holography or in the bulk. A particularly interesting example is [97], where arguments about weak gravity conjecture could help prevent the appearance of the naked singularity. It might be interesting to address possible quantum information interpretations or check what happens for such processes in the context of holography.
Another interesting connection is about quantum circuits and hydrodynamics. Opposite to discussions mostly in the main text, people often treat U(1)-symmetric random circuits as a specific random circuit model for energy conservation. One could assign possible macroscopic hydrodynamical variables to quantum circuits, to study emergent phenomena of microscopic Haar randomness. This study could be helpful for understandings of quantum circuits themselves (for instance, the efficiency of approaching k-designs [98,99] or emergent phenomena like replica wormholes in open quantum systems [100]), but also be helpful for hydrodynamics itself. Can we simulate classical, emergent, novel hydrodynamical effects in quantum circuits? What is the meaning of concepts, like non-Newtonian fluid or Reynolds number in quantum circuits? Thanks to holography, it is not completely irrelevant to quantum gravity [101].

Other suggestions
Here we collect some other suggestions for future research: • Finite temperature. It might be interesting to generalize calculations in this paper to finite temperature, for instance, finite temperature version of spectral form factors, frame potentials or OTOCs, or Hayden-Preskill experiment with finite temperature (thermofield double). The Boltzmann distribution makes the calculation non-trivial, but more realistic to claim a connection between toy models and finite temperature black holes.
• k-invariant U(1) subspace. People generically believe that for chaotic systems with certain properties about operator averaging, which we call k-invariance, general chaotic data could be universally described by their spectral data, which is called spectral decoupling [48,53]. Here we find a situation where if the spectrum is k-invariant in each U(1) subspace, there are contributions that may not cause complete decoupling. Of course, if each sector is k-invariant, the whole system may not be k-invariant in total. Is this effect universal, or it could only appear in specific systems? It might be interesting to dig more about it, as a more detailed study about k-invariance in charged systems.
• More symmetries and exotic quantum matter. Here we only discuss U(1) symmetry as an example of continuous symmetries. However, it will be interesting to generalize it to more general, higher symmetries. Recently, people discuss several topics about anomaly, global symmetries, field theory dualities, connecting the study of QCD to topological materials (for instance, see [102]). It might be interesting to discuss the theory of quantum information in such systems with higher, more complicated symmetries (for instance, [81]

A Engineering Haar integrals
There are several programming packages for computing Haar integrals systematically (for instance [103,104]). Here, to be self-complete we will introduce some technical engineering details about the RTNI package appearing recently [105], which cover many technical computations about Haar integrals in this paper. As discussed in the main text, one could usually deal with problems about Haar integral in the following form, where α, β are elements of permutation group S p over 1, 2, ⋯, p. In many cases especially when U appears many times in the integral, it is hard to sum over so many δ algebras by hand. Thus, it will be much more efficient to consider computer algebra. The software [105] is especially useful for dealing the following types of integral in the compact form, where In this paper, we will only use the case for U † . One could use the function MultinomialexpectationvalueHaar to obtain analytic expressions for correlators in different types. For instance, we consider computing Here, d is the dimension of total Hilbert space, {X1, Y1, X2, Y2} means the operators appearing in the integral, {1, 2, 1, 2} is the index list (js) with the set U j ∈ U, U † , U T , U * , in order, of U s appearing in the integral, and False means computation without the trace (otherwise it will be True). The output is as expected, Furthermore, the package has nice graph representations provided associated with integrals, consistent with quantum circuit graphs appearing in the literature. For more details, see the introduction of the package itself in [105] or corresponding help documents.

B Form factors specifying diagonal and off-diagonal terms
This is a technical note on the alternative representation of spectral form factors. We could specify the type of sums in the spectral form factor by isolating diagonal and off-diagonal terms. We introduce the notation The form factors could also be represented in terms of P s for the U(1)-symmetric system. For instance, we have

C Spectral (un)decoupling for k-invariant U(1) subspace
Here we present a technical note about four-point functions and higher frame potential in the k-invariant U(1) subspace. In the continuously charged system, we cannot implement k-invariant condition for the whole system. Thus, it is natural to assign k-invariance in each charge subspace. The following analysis shows that this is a very different setup from kinvariance in the whole from the perspective of k-invariance for the whole system in terms of correlation functions, violating the assumption of spectral decoupling, where the whole timedependent contribution of the OTOCs is from the spectral data of the theory. This is, in fact, due to the mixing effects of the operator itself between different charge sectors.
Example C.1. We start to compute, For computing this we might try multiple partitions. Those partitions will split charge sectors, and in each sector, we take Haar average. The first term is that four indices are equal, where we use Haar integral formula to obtain  Another piece is referring to the charge sectors factorizes in the form 2-1-1, where we have six terms. We give two terms as example, while the remaining terms are permutations of them, dU Tr A q 4 1,q U q q B q 1,q U †,q q A q 2,q 3 U q 3 q 3 B q 3 2,q 4 U †,q 4 q 4 7 One might ask if it is legitimate to use this trick. Namely, can we remove the pole naively to get the correct moments of Haar integrals? This point is commented briefly in [78,106] and the answer is yes, although some clearer statements should be made. There is an alternative way to do this calculation beyond Weingarten calculus by geometric methods especially for small dimension of matrix, see for instance, [107]. 8 We make it sufficiently small, and it is suggested to appreciate it by zooming in. dU Tr B q 2,q 4 U †,q 4 q 4 A q 4 1,q U q q B q 1,q 2 U †,q 2 q 2 A q 2 2,q U q q = R †,Eq 3 1 d q 3 dU Tr B q 2,q 4 A q 4 1,q U q q B q 1,q 2 U †,q 2 q 2 A q 2 2,q U q q = R †,Eq 3 1 R †,Eq 2 1 d q 2 d q 3 dU Tr B q 1,q 2 A q 2 2,q U q q B q 2,q 4 A q 4 1,q U q q = R †,Eq 3 1 For four terms that are inequal, we have dU Tr A q 4 1,q 1 U q 1 q 1 B q 1 1,q 2 U †,q 2 q 2 A q 2 2,q 3 U q 3 q 3 B q 3 2,q 4 U †,q 4 Another type is the case where we have two operators belong to one charge sector, while another two operators belong to another one. It is given by the following three terms, dU Tr A q 1,p U p p B p 1,p U †,p p A p 2,q U q q B q 2,q U †,q q + dU Tr A q 1,p U p p B p 1,q U †,q q A q 2,p U p p B p 2,q U †,q q + dU Tr B q 2,p U †,p p A p 1,p U p p B p 1,q U †,q q A q 2,q U q q (C.6) The first and third terms are very easy to deal with and it is the same method from previous case, They are dU Tr A q 1,p U p p B p 1,p U †,p p A p 2,q U q q B q 2,q U †,q Tr[H q 1,q H q 2,q U †,q q U †,q q ] (C. 15) while the last term cannot be reduced to the spectral data.
We could also compute the frame potential F (2) E for k-invariant subspace.
Example C.2. We consider the second frame potential for k-invariant subspace. We have We give some final comments on the violation of spectral decoupling in the k-invariant subspace. Note that generically, the property of k-invariance in subsystems reduces the frame potential data, a package of averaged OTOCs, into form factor data in each subspace. However, we find that for generic OTOCs, there are terms that cannot be completely reduced to form factors. That means that during the process of averaging, some properties that are beyond spectral data have been lost. That is because frame potentials are cleaner objects, while individual OTOCs contain information beyond them.
The above calculation indicates that the setup where chaotic systems are k-invariant in each charge sector, is an example seemingly beyond the property of spectral decoupling that has been studied recently in [53]. However, it might also be possible that generically the violation is not generic in the actual system since it might be a small contribution at least in some specific time scales. It is interesting to see if it will happen in more specific physical models.