Pith. sign in

REVIEW 3 major objections 6 minor 78 references

Measurement-induced entanglement Hamiltonian

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Partial measurements leave the entanglement Hamiltonian local, with the measurement outcome stored in a position-dependent chemical potential.

desk verdict Post-selected entanglement Hamiltonians carry outcome-dependent chemical potentials; the paper is right in spirit, but the locality assumption needs a direct RDM-level test before I'd fully sign off. read the letter →

arxiv 2608.06006 v1 pith:7RCDXEJ7 submitted 2026-08-06 cond-mat.stat-mech hep-thquant-ph

Viktor Eisler , Erik Tonni This is my paper
classification cond-mat.stat-mechhep-thquant-ph
keywords entanglementHamiltonianmeasurement-inducedconformalfieldtheoryfreefermionsprojectivemeasurementlocalchemicalpotentialentropypost-selectedstates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies what happens to the entanglement between two halves of an infinite free-fermion chain when projective measurements are made in two intervening regions. It tries to show that, for the most probable measurement outcomes, the reduced state of the unmeasured segment is still described by a local entanglement Hamiltonian: a position-dependent inverse temperature times the energy density, minus a position-dependent chemical potential times the particle density. The inverse temperature profile is universal, independent of which outcome occurred, and vanishes as a square root near the entangling points. The chemical potential, by contrast, encodes the measurement outcome through the charge density it induces inside the segment. If correct, this means the entanglement Hamiltonian of a post-selected state is a much finer diagnostic of the measurement than the entanglement entropy alone.

What carries the argument

The central object is the conformal map $w(z)$ — a composition of a Möbius transformation, an elliptic-sine ($\operatorname{arcsn}$) map, and a logarithm — that sends the complex plane with two slits onto a cylinder of height $h$ and circumference $2\pi$. Derivatives of this map produce the local inverse temperature $\beta(x) = 2\pi/w'(x)$, giving Eq. (6). A local gauge transformation $g_m(x)$, constructed from the transformation properties of the stress tensor and U(1) current, then converts the twisted-boundary thermal problem into a grand-canonical one with homogeneous inverse temperature $\beta_0 = 2\pi$ and chemical potential $\mu_0 = \delta/(2h)$; demanding that the gauge transformation conserve total charge fixes $g_m(\ell)=0$ and leads directly to the outcome-dependent chemical potential formula (8). This ansatz (5) is the load-bearing structure: it asserts the post-measurement EH is a local first-order Dirac operator.

What would settle it

Compute the exact lattice entanglement Hamiltonian (23) for a generic high-probability measurement outcome that is not one of the periodic patterns, extract $\beta$ and $\mu$ via (24), and check whether the residual nonlocal part of the matrix vanishes in the continuum limit; any surviving longer-range hopping would show that (5) does not describe the actual reduced density matrix.

Watch

Extended reading notes

Core claim

The central claim is that for a critical hopping chain, after partial projective measurements in the occupation basis, the entanglement Hamiltonian of the unmeasured interval $A$ takes the local form $H_m = \int_0^\ell \beta(x)\bigl(T_{00}(x) - \mu_m(x)\rho(x)\bigr) dx$, with $\beta(x)$ given by a square-root profile derived from a conformal map and independent of the outcome $m$, and with $\mu_m(x)$ a local chemical potential tied to the induced density via $\mu_m(x) = \pi\langle\rho(x)\rangle_m + \frac{\pi}{\beta(x)} h(\delta - 2\pi Q)$. The authors establish this by mapping the post-measurement path integral on a plane with two slits to a cylinder, treating the measurement outcome as a twisted boundary condition, and matching the resulting partition function to a grand-canonical state after a local gauge transformation. Lattice free-fermion calculations for selected measurement records confirm both profiles, with the entropy forming a periodic function $S(Q)$ of the induced charge whose upper edge agrees with the CFT prediction.

Load-bearing premise

The whole argument rests on assuming that, in the continuum limit, the post-measurement reduced density matrix of the unmeasured segment is exactly a local first-order differential operator of the form (5); this is verified only for a handful of specially chosen measurement records.

Editorial extensions

If this is right

  • The entanglement entropy of a post-selected state depends only on the total induced charge $Q$, through a periodic function $S(Q)$ of period one; outcomes with the same $Q$ are indistinguishable at the entropy level.
  • The entanglement Hamiltonian resolves this degeneracy: its local chemical potential $\mu_m(x)$ tracks the induced density, so the EH carries more information about the measurement outcome than the entropy alone.
  • The local inverse temperature $\beta(x)$ is independent of the measurement outcome and exhibits a square-root profile near the entangling points, replacing the linear Bisognano–Wichmann profile of the unmeasured chain.
  • Because the post-measurement EH remains local, it can in principle be measured by entanglement-Hamiltonian tomography on quantum simulators, and the calculation extends to finite chains and finite temperature.
  • The lattice data match the CFT prediction on the upper edge of the entropy–charge curve, while low-probability outcomes that do not flow to a conformal boundary condition fall below the universal curve.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension would be to test whether the density ansatz (10), fitted on periodic outcomes, also describes generic high-probability bit strings; a universal prefactor would mean the chemical potential, like $\beta(x)$, is essentially fixed by the geometry of the measurement regions.
  • The thermodynamical picture (entropy as a function of total charge, EH as a grand-canonical state) suggests that Born-averaged entanglement entropies in monitored circuits could be decomposed sector by sector in induced charge, with each sector carrying its own local chemical potential.
  • Because the local chemical potential is proportional to the induced density, a gravitational dual of this setup would likely encode the post-selected outcome as a localized charge distribution near the entangling surface rather than as a global conserved charge.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies the entanglement Hamiltonian (EH) of an infinite free-fermion hopping chain in its ground state, after projective measurements are performed in the occupation basis on two finite segments M1 and M2. The authors focus on the unmeasured segment A and propose that, for measurement outcomes that flow to conformal boundary conditions, the continuum entanglement Hamiltonian is local and takes the form H_m = ∫_0^ℓ β(x)(T_00(x) − µ_m(x)ρ(x)) dx, with β(x) fixed by the conformal map and independent of the outcome, and µ_m(x) related to the induced charge density by Eq. (8). The CFT derivation combines a conformal mapping from the two-slit geometry to a cylinder, a twisted torus partition function, and a local gauge transformation. Lattice free-fermion calculations of entropy, charge, and the continuum limit of the EH are used to support the prediction. The main conclusion is that post-selected entanglement Hamiltonians contain outcome-dependent information—encoded in a local chemical potential—beyond what is visible in the entanglement entropy.

Significance. If the central claim holds, the paper is significant: it extends the known locality of entanglement Hamiltonians from ground states to post-selected states under projective measurements, and it gives a thermodynamic interpretation with an inhomogeneous chemical potential. The explicit closed-form expression for β(x), the exact lattice update rules for the correlation matrix, and the all-outcome entropy/charge comparison in Fig. 2 are clear strengths and make the predictions falsifiable in quantum-simulator experiments. The main caveat is that the locality assumption is not tested directly at the level of the reduced density matrix; the numerical evidence is partly self-consistent by construction because the extraction formula Eq. (24) already assumes the local first-order form. A direct RDM-level test would substantially increase confidence. With that caveat, the paper is a valuable contribution to the measurement-induced entanglement literature.

major comments (3)
  1. [End Matter, Eq. (24)] The load-bearing claim is the locality of H_m, Eq. (5). The numerical check of this claim is weakened by the fact that Eq. (24), used to extract β(x_i) and µ_m(˜x_i) from H_m, is itself derived in Ref. [65] under the assumption that the continuum EH has the local first-order form (5). Figs. 3 and 4 therefore demonstrate that the lattice EH can be consistently projected onto a local template, but they do not rule out surviving nonlocal or higher-derivative corrections. I ask for a direct RDM-level test: for example, compare the exact reduced correlation matrix C_{m,A} with the correlation matrix of the local Gibbs state defined by (5), or show that the difference between H_m and the discretized local expression vanishes in the continuum limit. This is essential because the paper's central conclusion depends on locality, not merely on the value of the extracted coefficients.
  2. [Fig. 3 and text after Eq. (6)] The outcome independence of β(x) is asserted but not demonstrated: Fig. 3 shows profiles only for the Q=0 outcome, and the accompanying statement that 'for various other outcomes we find only tiny deviations' is not supported by data or a quantitative bound. Since the independence of β from m is one of the two main features of Eq. (5), the paper should either plot β for representative extreme outcomes (e.g., δ=π) or report the maximal deviation over all high-probability outcomes as a function of lattice spacing. Without this, the claim is under-supported.
  3. [Fig. 4 and Eq. (8)] The comparison in Fig. 4 omits the largest boundary points, where the continuum limit is least reliable, and the authors note that deviations increase with δ. This is reasonable, but it leaves the chemical-potential prediction tested only in the bulk and for moderate twists. In addition, Eq. (8) is derived by combining the assumed local form (5) with the gauge-transformation identity (7), so it is partly a consistency condition of the ansatz rather than an independent consequence of the measurement. The lattice comparison partially compensates for this because Eq. (9) uses a different smoothing of the exact density, but a direct test of the local Gibbs form would make the logic fully independent.
minor comments (6)
  1. [Eq. (2)] The statement 'One can verify that the result is identical to the one obtained via bosonization in [48,49]' hides a non-trivial identity; a short derivation or an appendix entry would help the reader understand the relation between the twisted fermion partition function and the bosonized result.
  2. [Fig. 2] The color code for Born probabilities is described only verbally; please add a color bar or an explicit logarithmic mapping from probability to color, since the figure is central to the claim that high-probability outcomes form the upper edge.
  3. [Fig. 3 caption] The caption should define the scaled variables more precisely: the horizontal axis is i/ℓ and the vertical axis is β(x_i)/ℓ, but the text says 'scaled CFT result (6)' without stating the scaling convention used for the comparison.
  4. [Eq. (10) and Fig. 5] The fitted constant C≈0.04 is reported without a fitting range or an uncertainty estimate; since the density ansatz is used to support the chemical-potential discussion, please provide these details and clarify the dependence of C on ζ.
  5. [Reference [69]] The boundary-localized charge contribution is attributed to a reference 'in preparation'; if this contribution is used to explain why Eq. (10) accounts only for a fraction of the total charge, the paper should either supply the derivation in an appendix or remove the reliance on an unpublished reference.
  6. [Notation, Eq. (8)] The denominator in the second term of Eq. (8) is ambiguous in the typeset text; please write it explicitly as \(\frac{\pi}{\beta(x)\,h}(\delta-2\pi Q)\) so that the dependence on h is unambiguous.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the local EH ansatz is explicit, CFT predictions are compared with independent free-fermion lattice data, and the numerical locality check via Eq. (24) is partly self-consistent but not definitionally circular.

full rationale

The paper's central claim is the local form (5) for the post-measurement entanglement Hamiltonian. This is introduced explicitly as an ansatz, not as the output of a fit or as a renamed input. The inverse-temperature profile beta(x) is computed from the derivative of the conformal map in the End Matter (Eqs. (16)-(18)/(6)) and is not adjusted to data. The chemical-potential relation (8) is derived from the combined conformal and gauge transformations together with the U(1) current algebra; it is a conditional consequence of the local ansatz, but it is a genuine internal derivation and could in principle fail if the gauge-transformation construction were inconsistent. The lattice checks use independent data: the post-measurement correlation matrix is built by the exact single-site update rules (20)-(21), and the numerically extracted EH matrix is given by the free-fermion formula (23), which does not assume locality. The continuum-limit identification (24) from Ref. [65] is a standard gradient-expansion procedure used to read off beta(x_i) and mu_m(x_i); because that identification is itself phrased in terms of the local operator form (5), the resulting comparison is partly a consistency check and may under-test the absence of nonlocal corrections, especially near the endpoints, where the authors also omit the largest data points and note that Eq. (10) accounts only for part of the total charge. Those are limitations of the numerical evidence rather than instances of the prediction being equivalent to its inputs by construction. No fitted parameter is renamed as a prediction, and the self-citations to [41,64,65] supply established technical tools rather than an unverified uniqueness argument. The central derivation therefore is not circular, although the numerical support for locality is somewhat weaker than the phrase 'nicely confirmed' suggests.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper relies on standard CFT tools (conformal mapping, theta functions, current algebra) and on the assumption that projective measurements at half filling are captured by conformal boundary conditions. The only genuinely fitted quantity is the amplitude C in the empirical density ansatz (10); the rest of the derivation is analytic or checked against lattice data.

free parameters (1)
  • C = 0.04
    Amplitude in the empirical density ansatz (10), quoted as C≈0.04, fitted to lattice data; the paper states it is essentially independent of ζ.
assumptions (6)
  • domain assumption The post-measurement RDM is described by a path integral on the two-slit geometry with conformal boundary conditions at the slits.
    Invoked in the setup after Eq. (1) and in the conformal mapping section; this is the bridge between lattice measurements and CFT.
  • domain assumption To preserve conformal symmetry, the U(1) vector current vanishes at the boundaries, and antiperiodic (NS) boundary conditions are imposed along imaginary time.
    Stated in the paragraph before Eq. (2); these boundary conditions determine the partition function and the form of the EH.
  • domain assumption The gauge transformation g_m(x) can be chosen small, with g_m(ℓ)=0, so it does not change the total charge.
    Used after Eq. (7) to fix the chemical potential; relies on charge conservation and the local EH ansatz.
  • domain assumption The continuum limit of the lattice EH, via the identification (24), yields the local Dirac operator (5).
    This is the assumption that no nonlocal terms survive in the scaling limit; verified numerically for selected outcomes but not proven in general.
  • standard math Wick's theorem and Gaussianity of the post-measurement state.
    Used in End Matter Eq. (20); standard for free-fermion states under projective density measurements.
  • domain assumption The lattice calculations are performed at half filling with symmetric measurement intervals.
    The comparison to CFT relies on the half-filled identification (9) and symmetric geometry; the paper does not test other fillings.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Measurement-induced entanglement Hamiltonian." pith.science (2026). https://pith.science/paper/7RCDXEJ7

@misc{pith2026260806006,
  author       = {Pith},
  title        = {Pith review of: Measurement-induced entanglement Hamiltonian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7RCDXEJ7}},
  note         = {Machine review of arXiv:2608.06006}
}
read the original abstract

We study the entanglement Hamiltonian of an infinite hopping chain in its ground state, after partial projective measurements in the occupation basis. For a segment separated by two measurement regions from the rest of the chain, we show that the reduced density matrix can be related to a grand-canonical state via a conformal mapping and a gauge transformation in the underlying field-theory description. The entanglement Hamiltonian is then described by a local inverse temperature that vanishes as a square root around the endpoints and is independent of the particular measurement outcome. In sharp contrast, the local chemical potential is shown to be related to the induced charge density in the segment. Hence the entanglement Hamiltonian of a post-selected state contains much more information on the measurement outcome than the respective entropy.

Figures

Figures reproduced from arXiv: 2608.06006 by the authors.

Figure 1
Figure 1. FIG. 1. Measurement setup (top) and conformal map from [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Entanglement entropy against induced charge for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Inverse temperature profiles [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Chemical potential [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

78 extracted references · 34 canonical work pages

  1. [37]

    Najafi and M

    K. Najafi and M. A. Rajabpour, Entanglement entropy after selective measurements in quantum chains, JHEP 12, 124, arXiv:1608.04074 [cond-mat.str-el]

    arXiv

  2. [65]

    Eisler, E

    V. Eisler, E. Tonni, and I. Peschel, On the continuum limit of the entanglement Hamiltonian, J. Stat. Mech. 1907, 073101 (2019), arXiv:1902.04474 [cond-mat.stat- mech]

    arXiv 2019

  3. [1]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2000)

    2000

  4. [2]

    C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky- Rosen channels, Phys. Rev. Lett.70, 1895 (1993)

    work page 1993

  5. [3]

    P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A52, R2493 (1995)

    1995

  6. [4]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett.86, 5188 (2001)

    2001

  7. [5]

    Raussendorf, D

    R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A68, 022312 (2003), arXiv:quant- ph/0301052

    arXiv 2003

  8. [6]

    H. J. Briegel, D. E. Browne, W. D¨ ur, R. Raussendorf, and M. Van den Nest, Measurement-based quantum com- putation, Nature Phys.5, 19 (2009), arXiv:0910.1116 [quant-ph]

    arXiv 2009

Show all 78 references
  1. [7]

    Calabrese, J

    P. Calabrese, J. Cardy, and B. Doyon, Entanglement en- tropy in extended quantum systems, J. Phys. A A42, 500301 (2009)

    2009

  2. [8]

    Laflorencie, Quantum entanglement in con- densed matter systems, Phys

    N. Laflorencie, Quantum entanglement in con- densed matter systems, Phys. Rept.643, 1 (2016), arXiv:1512.03388 [cond-mat.str-el]

    2016 arXiv

  3. [9]

    Casini and M

    H. Casini and M. Huerta, Lectures on entanglement in quantum field theory, PoST ASI2021, 002 (2023), arXiv:2201.13310 [hep-th]

    2023 arXiv

  4. [10]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- Induced Phase Transitions in the Dynamics of Entangle- ment, Phys. Rev. X9, 031009 (2019), arXiv:1808.05953 [cond-mat.stat-mech]

    2019 arXiv

  5. [11]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018), arXiv:1808.06134 [quant-ph]

    2018 arXiv

  6. [12]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019), arXiv:1901.08092 [cond-mat.stat-mech]

    2019 arXiv

  7. [13]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum cir- cuits, Phys. Rev. B101, 104302 (2020), arXiv:1908.08051 [cond-mat.stat-mech]

    2020 arXiv

  8. [14]

    Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B101, 104301 (2020), arXiv:1908.04305 [cond-mat.stat-mech]

    2020 arXiv

  9. [15]

    A. C. Potter and R. Vasseur, Entanglement Dynamics in Hybrid Quantum Circuits (2022) arXiv:2111.08018 [quant-ph]

    2022 arXiv

  10. [16]

    Noelet al., Measurement-induced quantum phases re- alized in a trapped-ion quantum computer, Nature Phys

    C. Noelet al., Measurement-induced quantum phases re- alized in a trapped-ion quantum computer, Nature Phys. 18, 760 (2022), arXiv:2106.05881 [quant-ph]

    2022 arXiv

  11. [17]

    J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nature Phys.19, 1314 (2023), arXiv:2203.04338 [quant-ph]

    2023 arXiv

  12. [18]

    J. C. Hokeet al., Measurement-induced entanglement and teleportation on a noisy quantum processor, Nature 622, 481 (2023), arXiv:2303.04792 [quant-ph]

    2023 arXiv

  13. [19]

    Ashida, S

    Y. Ashida, S. Furukawa, and M. Ueda, Quantum crit- ical behavior influenced by measurement backaction in ultracold gases, Phys. Rev. A94, 053615 (2016), arXiv:1605.06899 [cond-mat.quant-gas]

    2016 arXiv

  14. [20]

    Buchhold, Y

    M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl, Ef- fective Theory for the Measurement-Induced Phase Tran- sition of Dirac Fermions, Phys. Rev. X11, 041004 (2021), arXiv:2102.08381 [cond-mat.stat-mech]

    2021 arXiv

  15. [21]

    Minoguchi, P

    Y. Minoguchi, P. Rabl, and M. Buchhold, Continu- ous gaussian measurements of the free boson CFT: A model for exactly solvable and detectable measurement- induced dynamics, SciPost Phys.12, 009 (2022), arXiv:2108.04256 [cond-mat.stat-mech]

    2022 arXiv

  16. [22]

    Agrawal, A

    U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Pot- ter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, En- tanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits, Phys. Rev. X 12, 041002 (2022), arXiv:2107.10279 [cond-mat.dis-nn]

    2022 arXiv

  17. [23]

    S. J. Garratt, Z. Weinstein, and E. Altman, Mea- surements Conspire Nonlocally to Restructure Criti- cal Quantum States, Phys. Rev. X13, 021026 (2023), arXiv:2207.09476 [cond-mat.stat-mech]

    2023 arXiv

  18. [24]

    Murciano, P

    S. Murciano, P. Sala, Y. Liu, R. S. K. Mong, and J. Al- icea, Measurement-Altered Ising Quantum Criticality, Phys. Rev. X13, 041042 (2023), arXiv:2302.04325 [cond- mat.stat-mech]

    2023 arXiv

  19. [25]

    Tang and X

    Q. Tang and X. Wen, A critical state under weak mea- 6 surement is not critical, JHEP09, 168, arXiv:2411.13705 [cond-mat.stat-mech]

    arXiv

  20. [26]

    Tantivasadakarn, R

    N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases, Phys. Rev. X 14, 021040 (2024), arXiv:2112.01519 [cond-mat.str-el]

    2024 arXiv

  21. [27]

    Verresen, N

    R. Verresen, N. Tantivasadakarn, and A. Vishwanath, Ef- ficiently preparing Schr¨ odinger’s cat, fractons and non- Abelian topological order in quantum devices (2021), arXiv:2112.03061 [quant-ph]

    2021 arXiv

  22. [28]

    T.-C. Lu, L. A. Lessa, I. H. Kim, and T. H. Hsieh, Measurement as a Shortcut to Long-Range Entangled Quantum Matter, PRX Quantum3, 040337 (2022), arXiv:2206.13527 [cond-mat.str-el]

    2022 arXiv

  23. [29]

    Iqbalet al., Non-Abelian topological order and anyons on a trapped-ion processor, Nature626, 505 (2024), arXiv:2305.03766 [quant-ph]

    M. Iqbalet al., Non-Abelian topological order and anyons on a trapped-ion processor, Nature626, 505 (2024), arXiv:2305.03766 [quant-ph]

    2024 arXiv

  24. [30]

    Foss-Feiget al., Experimental demonstration of the advantage of adaptive quantum circuits (2023), arXiv:2302.03029 [quant-ph]

    M. Foss-Feiget al., Experimental demonstration of the advantage of adaptive quantum circuits (2023), arXiv:2302.03029 [quant-ph]

    2023 arXiv

  25. [31]

    G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori’s Cat: Stable Long-Range Entanglement from Finite-Depth Unitaries and Weak Measurements, Phys. Rev. Lett.131, 200201 (2023), arXiv:2208.11136 [quant-ph]

    2023 arXiv

  26. [32]

    Chan and A

    A. Chan and A. De Luca, Projected state ensemble of a generic model of many-body quantum chaos, J. Phys. A 57, 405001 (2024), arXiv:2402.16939 [quant-ph]

    2024 arXiv

  27. [33]

    Milekhin and S

    A. Milekhin and S. Murciano, Observable-projected en- sembles, Quantum9, 1888 (2025), arXiv:2410.21397 [quant-ph]

    2025

  28. [34]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and D. Senechal,Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997)

    1997

  29. [35]

    M. A. Rajabpour, Post measurement bipartite entan- glement entropy in conformal field theories, Phys. Rev. B92, 075108 (2015), arXiv:1501.07831 [cond-mat.stat- mech]

    2015 arXiv

  30. [36]

    M. A. Rajabpour, Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact results, J. Stat. Mech.1606, 063109 (2016), arXiv:1512.03940 [hep-th]

    2016 arXiv

  31. [38]

    J. L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B240, 514 (1984)

    1984

  32. [39]

    J. L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B324, 581 (1989)

    1989

  33. [40]

    J. L. Cardy, Boundary conformal field theory (2004), arXiv:hep-th/0411189

    2004 arXiv

  34. [41]

    Cardy and E

    J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612, 123103 (2016), arXiv:1608.01283 [cond-mat.stat- mech]

    2016 arXiv

  35. [42]

    Hoshino, M

    M. Hoshino, M. Oshikawa, and Y. Ashida, Entanglement swapping in critical quantum spin chains, Phys. Rev. B 111, 155143 (2025), arXiv:2406.12377 [quant-ph]

    2025 arXiv

  36. [43]

    Verstraete, M

    F. Verstraete, M. Popp, and J. I. Cirac, Entanglement versus Correlations in Spin Systems, Phys. Rev. Lett. 92, 027901 (2004), arXiv:quant-ph/0307009

    2004 arXiv

  37. [44]

    M. Popp, F. Verstraete, M. A. Martin-Delgado, and J. I. Cirac, Localizable entanglement, Phys. Rev. A71, 042306 (2005), arXiv:quant-ph/0411123

    2005 arXiv

  38. [45]

    C.-J. Lin, W. Ye, Y. Zou, S. Sang, and T. H. Hsieh, Probing sign structure using measurement-induced en- tanglement, Quantum7, 910 (2023), arXiv:2205.05692 [quant-ph]

    2023 arXiv

  39. [46]

    McGinley, W

    M. McGinley, W. W. Ho, and D. Malz, Measurement- Induced Entanglement and Complexity in Random Constant-Depth 2D Quantum Circuits, Phys. Rev. X15, 021059 (2025), arXiv:2410.23248 [quant-ph]

    2025 arXiv

  40. [47]

    Cheng, R

    Z. Cheng, R. Wen, S. Gopalakrishnan, R. Vasseur, and A. C. Potter, Universal structure of measurement- induced information in many-body ground states, Phys. Rev. B109, 195128 (2024), arXiv:2312.11615 [quant-ph]

    2024 arXiv

  41. [48]

    Khanna and R

    K. Khanna and R. Vasseur, Measurement-Induced En- tanglement in Conformal Field Theory, Phys. Rev. Lett. 136, 160402 (2026), arXiv:2508.02788 [quant-ph]

    2026 arXiv

  42. [49]

    Khanna and R

    K. Khanna and R. Vasseur, Universal Statistics of Measurement-Induced Entanglement in Tomonaga- Luttinger liquids (2025), arXiv:2512.13809 [quant-ph]

    2025 arXiv

  43. [50]

    Dalmonte, V

    M. Dalmonte, V. Eisler, M. Falconi, and B. Vermer- sch, Entanglement Hamiltonians: From Field Theory to Lattice Models and Experiments, Annalen Phys.534, 2200064 (2022), arXiv:2202.05045 [cond-mat.stat-mech]

    2022 arXiv

  44. [51]

    Nehari,Conformal Mapping, Dover Books on Mathe- matics (Dover Publications, New York, 1952)

    Z. Nehari,Conformal Mapping, Dover Books on Mathe- matics (Dover Publications, New York, 1952)

    1952

  45. [52]

    Alvarez-Gaume, G

    L. Alvarez-Gaume, G. W. Moore, and C. Vafa, Theta Functions, Modular Invariance and Strings, Commun. Math. Phys.106, 1 (1986)

    1986

  46. [53]

    We use the conventionϑ 3(z|τ) = P∞ n=−∞ eiπτ n2 e2πinz

  47. [54]

    C. G. Beneventano and E. M. Santangelo, Finite temper- ature properties of the Dirac operator under local bound- ary conditions, J. Phys. A37, 9261 (2004), arXiv:hep- th/0404115

    2004

  48. [55]

    Ogawa, T

    N. Ogawa, T. Takayanagi, and T. Ugajin, Holographic Fermi Surfaces and Entanglement Entropy, JHEP01, 125, arXiv:1111.1023 [hep-th]

    arXiv

  49. [56]

    Aguilera-Damia, M

    J. Aguilera-Damia, M. Sol ´ ıs, and G. Torroba, Nonrel- ativistic Dirac fermions on the torus, JHEP12, 060, arXiv:2309.03302 [hep-th]

    arXiv

  50. [57]

    Peschel, Calculation of reduced density matrices from correlation functions, J

    I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A36, L205 (2003), arXiv:cond-mat/0212631

    2003 arXiv

  51. [58]

    Peschel and V

    I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A42, 504003 (2009), arXiv:0906.1663 [cond-mat.stat- mech]

    2009 arXiv

  52. [59]

    Eisler, E

    V. Eisler, E. Tonni, and I. Peschel, Local and non- local properties of the entanglement Hamiltonian for two disjoint intervals, J. Stat. Mech.2208, 083101 (2022), arXiv:2204.03966 [cond-mat.stat-mech]

    2022 arXiv

  53. [60]

    J. J. Bisognano and E. H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16, 985 (1975)

    1975

  54. [61]

    J. J. Bisognano and E. H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys.17, 303 (1976)

    1976

  55. [62]

    P. D. Hislop and R. Longo, Modular Structure of the Local Algebras Associated With the Free Massless Scalar Field Theory, Commun. Math. Phys.84, 71 (1982)

    1982

  56. [63]

    Casini, M

    H. Casini, M. Huerta, and R. C. Myers, Towards a deriva- tion of holographic entanglement entropy, JHEP05, 036, arXiv:1102.0440 [hep-th]

    arXiv

  57. [64]

    Eisler and I

    V. Eisler and I. Peschel, Analytical results for the en- tanglement Hamiltonian of a free-fermion chain, J. Phys. 7 A50, 284003 (2017), arXiv:1703.08126 [cond-mat.stat- mech]

    2017 arXiv

  58. [66]

    V. G. Knizhnik and A. B. Zamolodchikov, Current Alge- bra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B247, 83 (1984)

    1984

  59. [67]

    Moosavi, Inhomogeneous Conformal Field Theory Out of Equilibrium, Annales Henri Poincare25, 1083 (2024), arXiv:1912.04821 [math-ph]

    P. Moosavi, Inhomogeneous Conformal Field Theory Out of Equilibrium, Annales Henri Poincare25, 1083 (2024), arXiv:1912.04821 [math-ph]

    2024 arXiv

  60. [68]

    Mintchev, D

    M. Mintchev, D. Pontello, and E. Tonni, Modular trans- port in two-dimensional conformal field theory, JHEP08, 021, arXiv:2503.16368 [hep-th]

    arXiv

  61. [69]

    Eisler and E

    V. Eisler and E. Tonni, in preparation

  62. [70]

    Kokail, R

    C. Kokail, R. van Bijnen, A. Elben, B. Vermersch, and P. Zoller, Entanglement Hamiltonian tomography in quantum simulation, Nature Phys.17, 936 (2021), arXiv:2009.09000 [quant-ph]

    2021 arXiv

  63. [71]

    M. K. Joshi, C. Kokail, R. van Bijnen, F. Kranzl, T. V. Zache, R. Blatt, C. F. Roos, and P. Zoller, Exploring large-scale entanglement in quantum simulation, Nature 624, 539 (2023), arXiv:2306.00057 [quant-ph]

    2023 arXiv

  64. [72]

    Dalmonte, B

    M. Dalmonte, B. Vermersch, and P. Zoller, Quantum Simulation and Spectroscopy of Entanglement Hamilto- nians, Nature Phys.14, 827 (2018), arXiv:1707.04455 [cond-mat.str-el]

    2018 arXiv

  65. [73]

    Giudici, T

    G. Giudici, T. Mendes-Santos, P. Calabrese, and M. Dal- monte, Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem, Phys. Rev. B98, 134403 (2018), arXiv:1807.01322 [cond-mat.stat-mech]

    2018 arXiv

  66. [74]

    Antonini, G

    S. Antonini, G. Bentsen, C. Cao, J. Harper, S.-K. Jian, and B. Swingle, Holographic measurement and bulk tele- portation, JHEP12, 124, arXiv:2209.12903 [hep-th]

    arXiv

  67. [75]

    NIST Digital Library of Mathematical Functions, Landen Transformation, DLMF,§19.8(ii)

  68. [76]

    For these Jacobi theta functions, we adopt the convention given byϑ 2(z|τ) = P∞ n=−∞ eiπτ(n+1/2) 2 e2πi(n+1/2)z and ϑ4(z|τ) = P∞ n=−∞(−1)n eiπτ n2 e2πinz , consistently with the one employed forϑ 3(z|τ)

  69. [77]

    Najafi and M

    K. Najafi and M. A. Rajabpour, Formation probabilities and shannon information and their time evolution after quantum quench in the transverse-field xy chain, Phys. Rev. B93, 125139 (2016). 8 END MA TTER Conformal map and local inverse temperature The conformal mapz7→w(z) empl...

    2016

  70. [78]

    9 Correlation matrices and lattice EH Here we show how to construct the EH for the ground state|ψ 0⟩of the hopping chain, after a particular outcome of the partial measurements

    withτ= i π h , and the first identity of this sequence leads to the expression (6) in the main text. 9 Correlation matrices and lattice EH Here we show how to construct the EH for the ground state|ψ 0⟩of the hopping chain, after a particular outcome of the partial measurements...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.