Refined symmetry-resolved Page curve and charged black holes

The Page curve plotted by the typical random state approximation is not applicable to a system with conserved quantities, such as the evaporation process of a charged black hole during which the electric charge does not radiate out with a uniform rate macroscopically. In this context the symmetry-resolved entanglement entropy may play a significant role in describing the entanglement structure of such a system. We attempt to impose constraints on microscopic quantum states to match with the macroscopic phenomenon of the charge radiation during black hole evaporation. Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes. We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation, and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve. We find the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time. Such refined Page curves may provide more realistic description for the entanglement between the charged black hole and radiation during the process of evaporation.

Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes.We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation, and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve.We find the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time.Such refined Page curves may provide more realistic description for the entanglement between the charged black hole and radiation during the process of evaporation.

I. INTRODUCTION
Quantum entanglement as one of the most prominent characteristics of a quantum system has been shown to play an important role in many fields such as quantum information, quantum computation, condensed matter physics as well as black hole physics [1][2][3][4][5][6][7].
It demonstrates the mysterious non-classical correlation between quantum subsystems.Entanglement entropy as an important measure of quantum entanglement has been extensively investigated.In particular, for a bipartite system Page [6] finds an important feature for the entanglement entropy between two subsystems, which now sometimes is referred to as "Page's theorem".It states that the average entanglement entropy of the smaller subsystem over random pure states is very close to its maximal value, which is constrained by degrees of freedom in the subsystem.It means that typically the smaller subsystem is almost maximally entangled with the other subsystem.Inspired by this observation, Page originally noticed that this picture may be applicable to the famous black hole information loss paradox [5] [7].The core issue about this paradox is whether the evaporation process of a black hole due to Hawking radiation1 , which is a semi-classical result in quantum field theory over curved but classical spacetime, is fundamentally unitary at the complete quantum mechanical level.Currently a complete quantum theory of gravity has not been established yet and the microscopic description of a black hole with quantum states are unknown.Nevertheless, Page suggests that one might choose random states as the approximation of black hole states during the evaporation process2 .Under the condition that the black hole evaporation process is unitary, it is believed that the Page's theorem is applicable to a system composed of the black hole and its radiation, and in principle one may find that the typical entanglement entropy of the radiation subsystem as the function of its size should follow the so-called Page curve.Such a picture proposed by Page stimulates a lot of work to further understand the black hole information paradox by investigating the entanglement between the black hole and radiation.In particular, since the island paradigm was proposed in the holographic approach [12][13][14], reproducing the Page curve for the evaporation process of the black hole by holography has been becoming a central task in order to argue that the information would be released in the later half stage of radiation.
Nevertheless, we know the approximation with totally random states sometimes is not precise enough to describe the evolution of quantum entanglement in practice, since many physically relevant systems usually have conserved charges, such as energy, momentum, and electric charge.In a system with conserved charges, the reduced Hilbert space is composed of quantum states which are subject to the conservation of charges, thus do not equal to the tensor product of the Hilbert spaces [15] [16], each of which is defined on individual subsystem separately.However, in this case one can still compute the entanglement entropy between two subsystems based on the reduced Hilbert space, which is the so-called "symmetryresolved (SR) entanglement entropy" [17] [18], and has been widely investigated in many theoretical aspects  and even in experiments [75] [76].It is important to notice that when computing the average SR entanglement entropy, the original Page's theorem in general is not applicable.So it is pretty interesting to investigate the Page curve for SR entanglement entropy (SR Page curve) and to compare it with the page curve of systems without conserved charges.Previously some relevant work on this topic can be found in [77,78].
It is well known that a stationary black hole is usually characterized by three conserved quantities classically, namely the mass E, the angular momentum J, and the electric charge Q.When considering the evaporation of such a black hole, it is also quite natural to assume that these three quantities are conserved during the process of evaporation.Therefore, one may apply SR entanglement entropy to describe the entanglement between the black hole and the radiation.In principle, a refined Page curve may be obtained with a similar method introduced by Page, except that one just considers the average value of entanglement entropy over the non-factorized reduced Hilbert space [79].However, we point out that for the evaporation of a charged black hole, such naive calculation based on random states does not align with the semi-classical calculation of Hawking radiation and Schwinger effect [8].The key point is that the charge does not radiate out at the same rate as that of the mass [8,[80][81][82][83].Specifically, if we were to randomly select states in the entire Hilbert space with a fixed global charge number, the average charge number of the subsystem (radiation part) would increase linearly with the number of particles (at least in the case of the usually simple qubit model, see Figure . 1).However, the semi-classical calculation of Hawking radiation and Schwinger effect reveal that the evolution of electric charge Q, the mass E and the angular momentum J exhibits distinct behavior during the evaporation [8,[80][81][82].In general, for a black hole the rate of losing energy and angular momentum changes relatively slowly in the entire evaporation process [80], but the radiation rate of electric charge Q depends on the stage of the black hole which is specified by the relations of parameters [8] [83].For different parameter values, the evaporation rate of the charge Q varies dramatically, which will be described with details in Sec.III.Consequently, the naive random model, even at a qualitative level, fails to capture the charge distribution during the course of evaporation for a charged black hole.In this paper, aiming to simulate the evaporation of a charged black hole, we analyze the SR entanglement entropy in a qubit model with conserved charges and obtain various refined Page curves that reflect the different behavior of the black hole radiating its charge.We will show that this model qualitatively captures the feature that the charge does not radiate out with a uniform rate during the evaporation of black holes.This paper is organized as follows.In Sec.II, we review the calculation of the average entanglement entropy over totally random states in a system without conserved charges and the average SR entanglement entropy in a system with conserved charges.In Sec.III, we introduce a qubit model for the evaporation of a charged black hole, and propose refined rules for selecting a random state with conserved quantity to simulate the distribution of charges during the different stages of evaporation, and then obtain refined SR Page curves.
In Sec.IV, we present some discussions and outlooks on the work in future.

II. THE AVERAGE ENTANGLEMENT ENTROPY AND PAGE CURVE BASED ON RANDOM STATES
In this section, we will review the general consideration about the entanglement between two subsystems in a bipartite system which is described by a random pure quantum state.We firstly compute the average value of the entanglement entropy in a system without conserved charges, and then turn to the SR entanglement entropy in a system with conserved charges.
A. The average entanglement entropy and Page curve in a system without conserved charges Given a bipartite system A ∪ B with the Hilbert space H AB = H A ⊗ H B , where H A and H B are the Hilbert space of subsystem A and B, respectively.Suppose the total system is described by a pure state |ψ⟩, then the entanglement entropy of A is defined by where ρ A is the reduced density matrix by tracing B. Suppose the dimension of H A and As a result, the average value of S A is obtained by integrating all the quantum states in One can also transform the integration variables into eigenvalues of ρ A , where the details can be found in [79].The final result is where Ψ(x) = Γ ′ (x)/Γ(x) is the so called Digamma function.The above result indicates that for a bipartite system described by pure states, the smaller subsystem is almost maximally entangled with the other subsystem.So the Page curve, which plots the entanglement entropy as a function of the size of the subsystem, will first increase with the size of the subsystem up to its maximal value at d A = d B , and then decrease with the size since for a pure system one always has ⟨S A ⟩ = ⟨S B ⟩, which now is constrained by the size of the smaller subsystem B. Obviously, when the Hilbert space of the system is large enough, the Page time is located at d A = d B , and the curve exhibits a symmetric behavior on both sides of the Page time.We remark that this result is rooted at the uniform measure over the Hilbert space, thus does not depend on the details of evolution and in this sense may be treated as a model independent result.
B. The average SR entanglement entropy and SR Page curve in a system with conserved charges In a system with conserved charges, only those quantum states subject to these constraints are allowable, leading to a reduced Hilbert space which may be much smaller than the total Hilbert space.For instance, a bipartite system contains a conserved charge Q, then the total Hilbert space can be decomposed into the direct sum of the eigenspace of Q, If the charge number Q is fixed and conserved in a system, then we just need to care about one sector H AB (Q).One immediate difference for H AB (Q) is that it can not be factorized into the tensor product of two Hilbert spaces of subsystems any more.Instead, it in general becomes the direct sum of tensor product of the Hilbert spaces of subsystems with fixed charges where s denotes the number of possible distributions of charges into two subsystems.In As for each distribution (q i , Q − q i ), the corresponding Hilbert space has the form of tensor product.Therefore in this situation, the entanglement entropy of subsystem A can be factorized into two parts Here, S A (q i ) represents the entanglement entropy within the factorized Hilbert space for the state |ϕ i ⟩, which can be readily computed using the formula discussed in the previous subsection.
The uniform measure is also factorized into two part [79] dµ where dν (p 1 , . . ., p s ) is the multivariate beta distribution [79].After the average integration we obtain the final result for the average SR entanglement entropy which reads as [79] ⟨S where Then in a parallel manner as described in the previous subsection we may obtain the SR Page curve as one changes the size of A and B.

III. A QUBIT MODEL FOR CHARGED BLACK HOLE EVAPORATION
In this section we consider a simple qubit model with conserved charges to simulate the evaporation of a charged black hole, with the assumption that the process of evaporation is unitary.As we mentioned in Sec.I, based on the analysis of Hawking radiation and Schwinger effect, charged black holes do not release their charge uniformly during evaporation [83] 3 .As a matter of fact, during evaporation the black hole may undergo different phases which depend on the black hole mass M and electric charge Q, which has previously been revealed in [8][83]4 .Specifically, for the situations that are of primary interest in this paper, when M > 2 × 10 7 M ⊙ , where M ⊙ denotes the mass of the Sun, the configuration space (M, Q) may be divided into two regions according to the characteristics of evaporation: the "mass dissipation zone" and the "charge dissipation zone" [83].When the charge-tomass ratio of the black hole Q/M is much less than one and M is large, the black hole will be in the "mass dissipation zone", then the black hole loses charges with a low rate dQ(t)/dM (t) < Q(t)/M (t).Thus the ratio Q(t)/M (t) becomes large as time t goes by.On the other hand, when Q/M is relatively close to one and M is small, the black hole will be in the "charge dissipation zone", then the black hole loses charges by Schwinger effect with a rate dQ(t)/dM (t) > Q(t)/M (t) such that the ratio Q(t)/M (t) becomes small with the evaporation.As a result, if a charged black hole starts to evaporate from a certain region within the "mass dissipation zone", its electric charge may remain nearly unchanged until more than half of its mass is lost.Only when the mass decreases to an order similar to the charge (in natural units), does significant charge release begin and the black hole enters the "charge dissipation zone" 5 .Therefore, we conclude that the non-uniform release of charge is a common phenomenon during the evaporation of charged black holes.Next, we apply a micro-level qubit model to simulate the evaporation of a charged black hole and propose refined rules to describe the charge release with a non-uniform rate, and investigate how the Page curve in this scenario differs from that evaluated in the case with completely random states.
We consider a qubit model which is composed of N qubits to simulate a charged black hole radiating out particles.We divide the system into two subsystems A and B, corresponding to the radiation and the charged black hole itself, respectively.For numerical analysis, we set the total number of qubits to be N = 20, and require N A + N B = 20 where N A and N B are the number of qubits in A and B, respectively.Thus, the different partitions with N A from 0 to 20 represent the different stages of evaporation from the initial state to the final state.Roughly speaking, the number N B can be considered as a quantity analogous to the mass value M of the black hole while N A is the energy of radiation.Alternatively, since the black hole loses its mass and the energy of radiation becomes larger during evaporation, the number of qubits N A may play a role of time as well.Next, we need to introduce a quantity to simulate the charge of a black hole.Just similar to the notion of spin, we assume each qubit may have a charge of either +1 or −1, so there is a 2-dimensional Hilbert space for each qubit.The total charge Q of the system is defined as the eigenvalue of the operator As a typical pattern of evaporation, we consider the system to have a total charge of Q = 4, which is aimed to simulate a black hole with an initially small ratio of the charge-to-mass.
In the configuration with (N = 20, Q = 4), at the early stage of evaporation, subsystem B may fall into the "mass dissipation zone" due to Q B ≪ N B .In this case, as a toy model, we stipulate that the subsystem B does not release any charge until it shrinks to its half size, namely the half number of the total qubits.This condition implies that when N A is less than or equal to half of the total particles ( N 2 ), the Hilbert space of radiation is H A (q i = 0), and correspondingly d(H A (q i = 0)) = . On the other hand, the Hilbert space of the black hole is H B (Q − q i = 4), and correspondingly d(H B (Q . After half of the black hole has evaporated, namely N A ≥ N 2 , we consider the subsystem B has entered the "charge dissipation zone" since Q B /N B is large enough.For a qualitative description, we could require the charge to evaporate uniformly from the black hole afterward, therefore, the corresponding Hilbert space for radiation is )) with N A from N/2 to N .We remark that for a finite N , a technical problem may arise when N A jumps to N with integer steps.Since for some N A , the corresponding possibly is not an integer, then we have to skip this step or demand its integer part as q i .When plotting the figure to illustrate the evolution of entanglement entropy with small N , it may cause visible imprecision.But we stress that such imprecision does not appear anymore in the thermodynamic limit, which requires N th , Q th → ∞, but keeping the radio 2 , while after that moment at N A ≥ N 2 , the charge is released with a constant rate.The purple solid line plots the expected charge profile in the thermodynamic limit, with the same rules.It is noticed that for N = 20, the deviation from the case of thermodynamic limit with perfect constant rate for N A ≥ N 2 is visible, but the deviation would gradually disappear as N get bigger.As a comparison, the green dotted line illustrates another pattern where the charge is released uniformly throughout the process of evaporation, which is obtained by computing the average value in total Hilbert space H AB (Q = 4).Our main results are demonstrated in  non-uniform charge release from the black hole, we apply different rules to pick out different sectors of the total Hilbert space before and after the middle point N A = N 2 , which disrupts the symmetry of the dimensions of Hilbert space before and after the middle point N A = N 2 .In addition, in the plot of the case with N = 20 (the blue dotted line), we applied integer approximation to generate more data for drawing this plot.
In the end of this section, we remark that we have only considered some typical patterns of charged black hole evaporation with a specific set of parameters.Definitely one may perform a similar analysis for other patterns (which we have mentioned in the first paragraph of Sec III and footnote 4) with different discharge behavior.One need impose different constraints on original Hilbert space, and then pick up reduced Hilbert space to match different macroscopic patterns of evaporation.Hilbert space H AB in the thermodynamic limit, where the coordinates (N a , ⟨S A ⟩) correspond to (N a , ⟨S A ⟩)× 20 N th as a result of normalization.

IV. CONCLUSION AND DISCUSSION
In this paper we have made the first step to apply symmetry-resolved entanglement entropy to plot the Page curve for understanding the evaporation of charged black holes.Due to the non-uniform rate of discharge during evaporation, we should not simply evaluate the average entanglement entropy over random states in a single Hilbert space for all time, as this would lead to a uniform charge release.As a toy model, we have computed the SR entanglement entropy in a qubit system that simulates the charge distribution at different stages of evaporation for a charged black hole.After imposing restrictions on the Hilbert space of the system, we have obtained a reduced Hilbert space for each stage of the evaporation and plotted the refined SR Page curve for this qubit system.We observe that the refined SR Page curve exhibits two distinctive features compared to the random cases: it has a different Page time and displays asymmetric behavior on either side of the Page time.
Although the qubit system as a toy model is too simple to describe the quantum states of a genuine charged black hole, we stress that considering the entanglement structure of random states in the qubit model has grasped the spirit of the unitary evolution of a quantum chaotic system, which exhibits highly entangled behavior and may be viewed as an analogy of a black hole.The refined SR Page curve obtained in this paper matches with the macroscopic phenomenon of the discharge during black hole evaporation, and has shed light on understanding the release procedure of information from a charged black holes at a microscopic level.Definitely, many aspects of this model can be improved in future.Firstly, the analysis presented in this paper for charged black hole is somewhat at a qualitative level.For a more realistic black hole, various kinds of particles may evaporate, whose effects can not be ignored.The evaporation rate for different particles can be derived from semi-classical calculations of radiation [8].Secondly, to simplify the analysis, we have only considered a linear relationship between charge and particle number release, while the actual quantitative relationship(e.g.dQ A (t) dt and dM A (t) dt ) would also depend on the specific calculations of evaporation details [81][82][83].Thirdly, it is also interesting to consider a system with multiple conserved charges, such as energy and angular momentum.We expect the Page curve will differ quantitatively in such situations.
such a system, due to the fact that H AB (Q) ̸ = H A ⊗ H B , we need do more to figure out the uniform measure over the Hilbert space.By virtue of the direct sum structure of H AB (Q), we may write a random state in H AB (Q) as |ψ⟩ = s i=1 √ p i |ϕ i ⟩, with p i ≥ 0 and s i=1 p i = 1, where

20 .
We can just follow the calculations as above mentioned and then properly normalize the results.Now we demonstrate our numerical results for N = 20 and Q = 4.For comparison we depict the average charge and average entanglement entropy as the function of N A for different patterns of evaporation in Figure (1) and Figure (2).In Figure (1), three different patterns for the charge evaporation are demonstrated.The blue dotted line plots the numerical results for N = 20, where we require no charge is released before N A ≤ N

Figure ( 2 )
, in which various Page curves are plotted for different patterns of evaporation.Evidently, in comparison with the Page curves with random state approximation, the refined Pages curves exhibit two prominent features.Firstly, the refined Page time may shift from the middle point N A = N/2.Secondly, the refined Page curves exhibit asymmetric behavior on both sides of the Page time.Such features are understandable since in order to match the

FIG. 1 :
FIG. 1: The average charge number ⟨Q A ⟩ over random states with particle number N A in subsystems A under different evaporation patterns.(i) The green dotted line represents the average value over random states in Hilbert space H AB (Q = 4) ; (ii) The blue dotted line represents the average value over random states in the refined Hilbert space H AB ; (iii) The purple solid line represents the average value over the random states in the refined Hilbert space H AB in the thermodynamic limit, where the coordinates (N a , ⟨Q A ⟩) correspond to (N a , ⟨Q A ⟩)× 20N th as a result of normalization.

FIG. 2 :
FIG. 2: The average entanglement entropy ⟨S A ⟩ over random states with particle number N A in subsystems A under different evaporation patterns.(i) The red solid line represents the average value over random states in the total Hilbert space H AB = H A ⊗ H B = {| − 1⟩, |1⟩} ⊗20 ; (ii) The green dotted line represents the average value over random states in the Hilbert space H AB (Q = 4) ; (iii) The blue dotted line represents the average value over random states in the refined Hilbert space H AB ; (iv) The purple solid line represents the average value over random states in the refined Now we intend to compute the average value of the entanglement entropy ⟨S A ⟩ for random states in H AB .We need first find the uniform measure in H AB .For this purpose we pick out an orthogonal basis {|n⟩} for a random state |ψ⟩ = d AB n=1 c n |n⟩, and then the measure is just the uniform measure on the unit sphere of C d AB , which is