Charge-Resolved Entanglement and Charged Moments

Charge-resolved entanglement asks how a symmetry-invariant reduced state is distributed among subsystem-charge sectors and how much quantum uncertainty remains after a sector is fixed. Charged moments are the generating functions, Fourier projection isolates each sector, and division by the sector probability produces the normalized conditional entropy. These three steps must not be conflated.

Required background. Symmetry-constrained operations fixes the charge action and invariant algebra.

Helpful background. Rényi analytic continuation supplies the replica and continuation cautions.

Chapter map. The overview gives the task-to-resource map, the comparison table, and the three gates for an operational claim.

Charged and projected moments

Work first with a normalized density matrix in a finite regulator. Assume that the subsystem charge has integer eigenvalues and that

$$[\rho_A,Q_A]=0, \qquad Q_A=\sum_q q\Pi_q.$$

The state is block diagonal,

$$\rho_A=\bigoplus_q p_q\rho_{A,q}, \qquad p_q=\operatorname{Tr}(\Pi_q\rho_A), \qquad \rho_{A,q}=\frac{\Pi_q\rho_A\Pi_q}{p_q}$$

for every sector with $p_q>0$. Define the charged moment

$$Z_n(\alpha) :=\operatorname{Tr}\!\left(\rho_A^n e^{i\alpha Q_A}\right), \qquad \alpha\sim\alpha+2\pi.$$

Fourier inversion gives the unnormalized projected moment

$$Z_n(q) :=\operatorname{Tr}(\Pi_q\rho_A^n) =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\, e^{-i\alpha q}Z_n(\alpha).$$

Because each block of $\rho_A^n$ is $p_q^n\rho_{A,q}^n$,

$$Z_n(q)=p_q^n\operatorname{Tr}(\rho_{A,q}^n), \qquad S_n(q)=\frac{1}{1-n}\log\frac{Z_n(q)}{p_q^n}.$$

Thus $p_q=Z_1(q)$, whereas $Z_n(q)$ for $n>1$ is not a normalized density-matrix moment. The literature has also used “symmetry-resolved entropy” for an unnormalized sector contribution; the conditional convention used here is the normalized one in Murciano, Di Giulio, and Calabrese 2020, § 2.1, Eqs. (2.1)–(2.7).

Sector moments reconstruct the total Rényi moment as

$$\operatorname{Tr}\rho_A^n =\sum_q Z_n(q) =\sum_q p_q^n e^{(1-n)S_n(q)}.$$

They do not give a probability-weighted sum of Rényi entropies. The linear decomposition occurs at $n=1$:

$$S(\rho_A)=H(p)+\sum_qp_qS(\rho_{A,q}), \qquad H(p):=-\sum_qp_q\log p_q.$$

Replica interpretation and continuation

In a replica path integral, $Z_n(\alpha)$ inserts a symmetry flux through the replicated geometry. Goldstein and Sela 2018, Eqs. (1)–(6) give the charged moment, Fourier projection, and flux-defect construction in a $1+1$-dimensional critical system. The insertion is topological only when the regulator and contacts preserve the relevant symmetry, and its phase depends on the origin chosen for $Q_A$.

There are two logically different continuations. If the finite density matrix is known, $\rho_A^n$ for real $n>0$ is fixed by spectral calculus. If only integer-replica path integrals or an asymptotic saddle are known, those integer values do not determine a unique analytic function without additional growth and regularity conditions. Fourier inversion, a large-size saddle, and $n\to1$ can then fail to commute. The explicit nonuniform replica limits in Murciano, Di Giulio, and Calabrese 2020, Appendix B and §4.3 illustrate why the order of limits must be reported.

Entanglement equipartition is likewise asymptotic. In several $1+1$-dimensional critical systems, the leading term in $S_n(q)$ is independent of $q$ in the central window, but subleading logarithms break that equality. Charge deviations in the central window scale with

$$\sigma_Q=\sqrt{\operatorname{Var}Q_A},$$

not with the variance itself. For a critical free-fermion interval, $\operatorname{Var}Q_A$ grows logarithmically with interval length; fixed-charge, central-window, and large-deviation limits therefore probe different regimes. The leading and first equipartition-breaking terms are derived in Murciano, Di Giulio, and Calabrese 2020, §§3.3 and 4.2.

Exact free-fermion interval benchmark

For a number-conserving Gaussian state, let $C_A$ be the restricted one-body correlation matrix. If its eigenvalues are $\nu_j$, then

$$Z_n(\alpha) =\prod_j\left[(1-\nu_j)^n+e^{i\alpha}\nu_j^n\right] =\det\!\left[(1-C_A)^n+e^{i\alpha}C_A^n\right].$$

This is Goldstein and Sela 2018, Eq. (9). It uses the unshifted number $Q_A=\sum_{j\in A}c_j^\dagger c_j$. If instead $Q_A$ is centered by subtracting a constant, $Z_n(\alpha)$ acquires the corresponding overall phase, as in Murciano, Di Giulio, and Calabrese 2020, Appendix C, Eqs. (C.2)–(C.4). A correlation-matrix treatment that also explains what sector data can and cannot be reconstructed appears in Monkman and Sirker 2023, §§ 2–3.

For an exact finite-region benchmark, take four regulated fermion modes and the two-particle Slater determinant

$$|\Psi\rangle =\left(\frac12a_1^\dagger+\frac{\sqrt3}{2}b_1^\dagger\right) \left(\frac{\sqrt3}{2}a_2^\dagger+\frac12b_2^\dagger\right)|0\rangle.$$

The two adjacent regulated modes $A=\{a_1,a_2\}$ form the interval, while $b_1,b_2$ are its complement. The state has fixed total number and is Gaussian. Its restricted correlation matrix is

$$C_A=\operatorname{diag}\!\left(\frac14,\frac34\right).$$

Exact Fourier projection gives the following benchmark in natural-log units.

$q$	$p_q=Z_1(q)$	$Z_2(q)$	$S_2(q)$
$0$	$3/16=0.1875$	$9/256=0.03515625$	$0$
$1$	$10/16=0.625$	$82/256=0.3203125$	$0.1984509387$
$2$	$3/16=0.1875$	$9/256=0.03515625$	$0$

The checks close numerically:

$$\sum_qp_q=1, \qquad \sum_qZ_2(q)=\frac{25}{64}=0.390625 =\operatorname{Tr}\rho_A^2.$$

For the von Neumann entropy,

$$S(\rho_A)=1.1246702892, \qquad H(p)=0.9214934309, \qquad S(\rho_{A,1})=0.3250829734,$$

and $H(p)+\tfrac58S(\rho_{A,1})=1.1246702892$. In bits, the total, sector-Shannon, and averaged conditional contributions are respectively $1.6225562489$, $1.3294340029$, and $0.2931222460$. The $q=0$ and $q=2$ conditional states are pure. This explicitly verifies both moment reconstruction and the von Neumann sector decomposition.

For a block of $\ell$ sites, $Z_n(\alpha)$ is a polynomial of degree $\ell$ in $e^{i\alpha}$. With $M$ equally spaced phases $\alpha_m=2\pi m/M$, the discrete transform returns

$$Z_n^{(M)}(r) =\frac1M\sum_{m=0}^{M-1}e^{-i\alpha_m r}Z_n(\alpha_m) =\sum_{q\equiv r\ ({\rm mod}\ M)}Z_n(q).$$

Therefore $M\ge\ell+1$ reconstructs every sector exactly up to numerical roundoff. This is stronger than merely observing positive, normalized coefficients.

Adversarial control: Fourier aliasing

Deliberately undersample the two-site benchmark with $M=2$. The transform merges $q=0$ and $q=2$:

$$p_{\rm even}^{(2)}=\frac38=0.375, \qquad Z_{2,{\rm even}}^{(2)}=\frac9{128}=0.0703125.$$

Both reconstructed probabilities remain nonnegative and normalized, and the total second moment is still correct. Nevertheless, treating the merged bin as one sector gives

$$S_{2,{\rm even}}^{(2)} =-\log\frac{9/128}{(3/8)^2} =\log2,$$

while the two true conditional entropies are both zero. The adversary therefore establishes a precise limit: positivity, normalization, and recovery of the total moment $Z_n(\alpha=0)=\operatorname{Tr}\rho_A^n$ do not certify sector resolution. The charge support or an anti-aliasing convergence test must also be supplied.

For larger intervals, a reproducible scaling study can use

$$(C_A)_{jk}=\frac{\sin[\pi(j-k)/2]}{\pi(j-k)}, \qquad (C_A)_{jj}=\frac12,$$

at $\ell=8,16,32$ with $M=\ell+1$, followed by deliberate reductions of $M$. This is a lattice CFT scaling limit at fixed filling and $1\ll\ell\ll L_{\rm sys}$. A relativistic continuum limit instead holds physical length, mass, and momentum scales fixed while the bare lattice parameters are tuned as the spacing decreases; the two limits should not be described interchangeably.

Common pitfalls

Using number variance as sector entanglement. Variance and $H(p_q)$ characterize the charge distribution. $S(q)$ characterizes the normalized conditional state.

Forgetting sector normalization. $Z_n(q)$ is unnormalized. Divide by $p_q^n$ before taking the Rényi logarithm.

Trusting normalization as an anti-aliasing test. A discrete transform can merge sectors while preserving positivity, total probability, and total moments.

Calling equipartition exact. Leading large-size equality can be broken by subleading terms and by sectors outside the central fluctuation window.

Exercises

1. Derive the sector reconstruction formula

Starting from $\rho_A=\bigoplus_qp_q\rho_{A,q}$, derive $Z_n(q)$ and reconstruct $\operatorname{Tr}\rho_A^n$.

Solution

Orthogonal charge blocks multiply independently, so

$$\rho_A^n=\bigoplus_qp_q^n\rho_{A,q}^n.$$

Projecting and tracing gives

$$Z_n(q)=\operatorname{Tr}(\Pi_q\rho_A^n) =p_q^n\operatorname{Tr}\rho_{A,q}^n =p_q^n e^{(1-n)S_n(q)}.$$

Summing the mutually orthogonal blocks yields

$$\operatorname{Tr}\rho_A^n =\sum_qp_q^n e^{(1-n)S_n(q)}.$$

At $n=1$, differentiating the block eigenvalues instead gives $S(\rho_A)=H(p)+\sum_qp_qS(\rho_{A,q})$.

2. Reproduce the two-site conditional Rényi entropy

Show that the normalized $q=1$ state has eigenvalues $9/10$ and $1/10$, and recover the value of $S_2(1)$ in the table.

Solution

The one-particle sector contains the two occupation patterns $(1,0)$ and $(0,1)$ in the entanglement-mode basis. Their unnormalized probabilities are

$$\frac14\left(1-\frac34\right)=\frac1{16}, \qquad \left(1-\frac14\right)\frac34=\frac9{16}.$$

Their sum is $p_1=10/16$, so normalization gives eigenvalues $1/10$ and $9/10$. Hence

$$\operatorname{Tr}\rho_{A,1}^2 =\left(\frac1{10}\right)^2+\left(\frac9{10}\right)^2 =0.82,$$

so

$$S_2(1)=-\log(0.82)=0.1984509387$$

in nats, or $0.2863041852$ bits.

3. Diagnose discrete-Fourier aliasing

Prove the congruence-class formula for $Z_n^{(M)}(r)$ and explain why $M=2$ cannot distinguish the two even sectors of the benchmark.

Solution

Insert $Z_n(\alpha_m)=\sum_qe^{i\alpha_mq}Z_n(q)$ into the discrete transform. The phase sum is

$$\frac1M\sum_{m=0}^{M-1}e^{2\pi i m(q-r)/M} = \begin{cases} 1,&q\equiv r\ ({\rm mod}\ M),\\ 0,&\text{otherwise}. \end{cases}$$

Therefore all charges congruent modulo $M$ are added. For $M=2$, $q=0$ and $q=2$ both enter the even bin. No calculation using only those two phase samples can recover how much weight belonged to each sector.

References

• Goldstein, Moshe, and Eran Sela. “Symmetry-Resolved Entanglement in Many-Body Systems.” Physical Review Letters 120 (2018): 200602. DOI. Open PDF.
• Monkman, Kyle, and Jesko Sirker. “Symmetry-Resolved Entanglement: General Considerations, Calculation from Correlation Functions, and Bounds for Symmetry-Protected Topological Phases.” Journal of Physics A: Mathematical and Theoretical 56 (2023): 495001. DOI.
• Murciano, Sara, Giuseppe Di Giulio, and Pasquale Calabrese. “Entanglement and Symmetry Resolution in Two Dimensional Free Quantum Field Theories.” Journal of High Energy Physics 2020, no. 8 (2020): 073. DOI. Open PDF.