pith. sign in

arxiv: 2512.08150 · v1 · submitted 2025-12-09 · 🪐 quant-ph

Detecting quantum many-body states with imperfect measuring devices

K. Uriostegui , C. Pineda , C. Chryssomalakos , V. Rasc\'on Barajas , I. V\'azquez Mota This is my paper

Pith reviewed 2026-05-17 01:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coarse-grainingimperfect addressingmany-body statesprobability densitymaximally mixed statepartial tracesrandom matrix methods
0
0 comments X

The pith

As the number of qubits grows, coarse-graining from imperfect addressing concentrates states sharply around the maximally mixed state.

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

The paper models a coarse-graining channel that arises when particles in a multipartite quantum system cannot be perfectly addressed. For two qubits this channel is a convex mixture of the two possible partial traces, and the authors compute the probability density over the possible output states using geometry for small cases and random-matrix techniques for larger ones. They show that the density concentrates around the maximally mixed state with increasing qubit number, so that nearly pure coarse-grained states become rare. A reader would care because this sets a fundamental limit on what can be reliably inferred about the underlying many-body state from realistic, imperfect measurements.

Core claim

The coarse-graining map produces a convex mixture of the two partial traces for two qubits and, when extended to larger systems, yields a probability density that concentrates around the maximally mixed state, making nearly pure coarse-grained states increasingly unlikely.

What carries the argument

The coarse-graining channel, realized as a convex mixture of partial traces for two qubits and extended by random-matrix methods for many qubits.

If this is right

  • Nearly pure coarse-grained states become exponentially rarer with growing qubit number.
  • The average preimage of the maximally mixed state under the two-qubit inverse channel retains a finite singlet fraction.
  • Monte-Carlo sampling of coarse-grained statistics reproduces the analytically derived probability densities.
  • Detection of quantum many-body properties with imperfect devices grows harder as system size increases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The concentration effect implies that entanglement witnesses or purity estimates extracted from coarse-grained data will be biased toward classical mixtures for large systems.
  • Designing addressing protocols that reduce the effective mixture weight could restore access to purer coarse-grained states.
  • The persistent singlet component in the averaged preimage may allow partial recovery of two-qubit correlations even after coarse-graining.

Load-bearing premise

The coarse-graining is accurately captured by a convex mixture of the two partial traces for two qubits and extends to larger systems via random-matrix methods without further assumptions on the distribution of addressing errors.

What would settle it

Compute or measure the histogram of coarse-grained states obtained from many random pure inputs on systems of 8–12 qubits and check whether the mass lies overwhelmingly near the maximally mixed state rather than spreading toward purer states.

Figures

Figures reproduced from arXiv: 2512.08150 by C. Chryssomalakos, C. Pineda, I. V\'azquez Mota, K. Uriostegui, V. Rasc\'on Barajas.

Figure 1
Figure 1. Figure 1: FIG. 1. Visualization of a pure state formed by two qubits [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Green and red subspheres corresponding to the locus [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A couple of remarks are due at this point. The first one, is on Eq. (23), given that as h goes to zero, it ap￾pears that VΩϵ increases boundlessly as rts < h. This would be problematic, as VΩϵ is a subset of P, and the volume of the latter is normalized to 1. However, 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 FIG. 3. Probability density P2(h; rts) vs. rts, for various values of h. In all case… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The simplex [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. PDF of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. PDF for N=2, 3, 4, and 5. Each of the functions has [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The top plot shows the behavior of purity of the [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Plot of the PDF function [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Plot of the PDF, [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Approximation to [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Subspheres corresponding to the reduced states that [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Components of the average state [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
read the original abstract

We study a coarse-graining map arising from incomplete and imperfect addressing of particles in a multipartite quantum system. In its simplest form, corresponding to a two-qubit state, the resulting channel produces a convex mixture of the two partial traces. We derive the probability density of obtaining a given coarse-grained state, using geometric arguments for two qubits coarse-grained to one, and random-matrix methods for larger systems. As the number of qubits increases, the probability density sharply concentrates around the maximally mixed state, making nearly pure coarse-grained states increasingly unlikely. For two qubits, we also compute the inverse state needed to characterize the effective dynamics under coarse-graining and find that the average preimage of the maximally mixed state contains a finite singlet component. Finally, we validate the analytical predictions by inferring the underlying probabilities from Monte-Carlo-generated coarse-grained statistics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a coarse-graining map for multipartite quantum states arising from incomplete and imperfect particle addressing. For two qubits this reduces to a convex mixture of the two partial traces; the authors derive the probability density of the output state via geometric arguments in the two-qubit case and random-matrix techniques for n>2. They conclude that the density concentrates sharply around the maximally mixed state with increasing n, rendering nearly pure coarse-grained states unlikely. For two qubits they also compute the inverse map and show that the average preimage of the maximally mixed state contains a finite singlet fraction. Analytical predictions are checked against Monte-Carlo sampling of coarse-grained statistics.

Significance. If the modeling of the coarse-graining map is physically justified, the concentration result would be a useful quantitative statement about the bias introduced by imperfect addressing when attempting to detect many-body quantum states. The combination of exact two-qubit geometry, random-matrix asymptotics, and numerical validation provides a concrete handle on how measurement imperfections affect state purity at large n.

major comments (1)
  1. [derivation of probability density for larger systems] The central large-n concentration claim rests on extending the exact two-qubit convex-mixture channel to n>2 via random-matrix methods. This extension implicitly assumes that addressing errors are distributed according to an ensemble compatible with the random-matrix treatment, yet no microscopic model of the physical error process (e.g., possible spatial correlations in misaddressing) is supplied to justify the ensemble choice. If the actual error statistics deviate, the claimed concentration need not follow. This assumption is load-bearing for the main result.
minor comments (2)
  1. [abstract] The abstract states that Monte-Carlo validation was performed but supplies neither error bars on the inferred probabilities nor explicit statements of the assumed error distribution or number of samples.
  2. [introduction / model definition] Notation for the coarse-graining map and its action on density operators would benefit from an explicit equation label when first introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [derivation of probability density for larger systems] The central large-n concentration claim rests on extending the exact two-qubit convex-mixture channel to n>2 via random-matrix methods. This extension implicitly assumes that addressing errors are distributed according to an ensemble compatible with the random-matrix treatment, yet no microscopic model of the physical error process (e.g., possible spatial correlations in misaddressing) is supplied to justify the ensemble choice. If the actual error statistics deviate, the claimed concentration need not follow. This assumption is load-bearing for the main result.

    Authors: We agree that a microscopic model would provide stronger physical grounding for the ensemble choice. In the manuscript, the coarse-graining for n>2 is defined by randomly selecting a subset of particles to address, with the selection drawn from a uniform distribution over possible misaddressings, which is modeled using random matrix techniques to compute the induced distribution on the coarse-grained state. This corresponds to an i.i.d. error model without spatial correlations. We acknowledge that if addressing errors exhibit strong correlations (e.g., due to global laser misalignment affecting multiple qubits similarly), the concentration around the maximally mixed state might be altered. However, for the phenomenological model of independent imperfect addressing considered here, the random-matrix approach is appropriate and the concentration holds. We will revise the manuscript to explicitly state the i.i.d. assumption and discuss its implications, including the potential effects of correlations, in a new subsection on model assumptions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations use standard techniques on defined map

full rationale

The paper defines a coarse-graining map from imperfect addressing, derives the probability density of coarse-grained states via geometric arguments for two qubits and random-matrix methods for n>2, and validates predictions via Monte-Carlo sampling of the map. No equation reduces by construction to a fitted parameter, self-citation, or renamed input; the concentration around the maximally mixed state is a direct consequence of applying these external methods to the explicitly constructed channel. The derivation is self-contained against standard quantum information and random-matrix benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the modeling choice that imperfect addressing produces a convex mixture of partial traces; random-matrix methods are imported from prior literature without new axioms stated in the abstract.

axioms (1)
  • domain assumption Imperfect particle addressing in a multipartite quantum system produces a coarse-graining channel that is a convex mixture of partial traces.
    This modeling assumption defines the channel whose statistics are then derived.

pith-pipeline@v0.9.0 · 5462 in / 1187 out tokens · 48749 ms · 2026-05-17T01:03:08.202362+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    the authors propose choosing the fine-grained state of maximum entropy compatible with the given target state. Motivated by quantum state tomography proce- dures—where reconstructed states naturally correspond to averages over multiple experimental realizations—we assign, as the preimage of a target state, the average of all fine-grained states that the c...

    work page

  2. [2]

    Parametrization of pure two-qubit states We now turn to a convenient parametrization of two- qubit pure states. This parametrization is defined in terms of (i) the coordinates of the reduced density ma- trices of each qubit, and (ii) two nonlocal parameters: one quantifying the degree of entanglement between the qubits, and another representing a relative...

    work page

  3. [3]

    (6) treats all subsystems equally, in the caseN= 2 we assume thatp 1 < p2 with- out loss of generality

    Coarse-graining statistics via preimage volume Since the CG map in Eq. (6) treats all subsystems equally, in the caseN= 2 we assume thatp 1 < p2 with- out loss of generality. In this way, we write the coarse- grained state asρ=pρ 1 + (1−p)ρ 2 with 0≤p≤ 1

    work page

  4. [4]

    In terms of Bloch vectors, this is expressed as rts = 1−h 2 r1 + 1 +h 2 r2 ,(19) Qubit 1 Entanglement Qubit 2 hemisphere FIG. 1. Visualization of a pure state formed by two qubits in the parameterization (16). The state shown corresponds to the valuesη= π 6 ,γ= π 3 ,θ 1 = 3π 4 ,ϕ 1 = π 2 ,θ 2 = π 4 andϕ 2 = 7π 4 . The orange and purple spheres correspond ...

    work page

  5. [5]

    Coarse-graining statistics via RMT methods The methods developed in the previous sections al- lowed us to gain deeper physical insight into the problem. However, when considering systems composed of a larger number of particles, the number of parameters involved in the parametrization grows exponentially, which in- creases computational complexity. For th...

    work page

  6. [6]

    thickness

    A geometric interlude The techniques used in the previous section are both powerful and intuitively opaque (at least to the authors). We try here to gain an intuitive, geometric understanding of some of the previous results. ConsiderR 4 with coordinates{x 00, x01, x10, x11}, then the fact thex ij’s are non-negative and sum to one means that the locus ofxi...

    work page 2000

  7. [7]

    Heisenberg,Physics and Beyond: Encounters and Conversations, World Perspectives, Vol

    W. Heisenberg,Physics and Beyond: Encounters and Conversations, World Perspectives, Vol. 42 (Harper & Row, Publishers, New York, Evanston, and London, 1971)

    work page 1971

  8. [8]

    Heisenberg,Physics and Philosophy: The Revolution in Modern Science, World Perspectives, Vol

    W. Heisenberg,Physics and Philosophy: The Revolution in Modern Science, World Perspectives, Vol. 19 (Harper & Brothers Publishers, New York, 1958)

    work page 1958

  9. [9]

    Eisert, D

    J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, U. Parekh, R. Chabaud, and E. Kashefi, Quantum cer- tification and benchmarking, Nature Rev. Phys.2, 382 (2020)

    work page 2020

  10. [10]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, and B. DeMarco, Quantum simulators: Ar- chitectures and opportunities, PRX Quantum2, 017003 (2021)

    work page 2021

  11. [11]

    Pelucchi, G

    E. Pelucchi, G. Fagas, I. Aharonovich, D. Englund, E. Figueroa, Q. Gong, H. Hannes, J. Liu, C. Y. Lu, and N. Matsuda, The potential and global outlook of inte- grated photonics for quantum technologies, Nature Rev. Phys.4, 194 (2022)

    work page 2022

  12. [12]

    Rosset, R

    D. Rosset, R. Ferretti-Sch¨ obitz, J. D. Bancal, N. Gisin, and Y. C. Liang, Imperfect measurement settings: Impli- cations for quantum state tomography and entanglement witnesses, Phys. Rev. A86, 062325 (2012)

    work page 2012

  13. [13]

    Naikoo, S

    J. Naikoo, S. Banerjee, A. K. Pan, and S. Ghosh, Projec- tive measurements under qubit quantum channels, Phys. Rev. A104, 042608 (2021)

    work page 2021

  14. [14]

    Z. Wu, S. Zhang, and J. Wu, Non-disturbance criteria of quantum measurements, Int. J. Theor. Phys.50, 1325 (2011)

    work page 2011

  15. [15]

    J. Zhu, Z. Liu, Z. Liao, and S. Wu, Learning informa- tive latent representation for quantum state tomography, Phys. Rev. A107, 032412 (2023)

    work page 2023

  16. [16]

    X. Gao, M. Sun, F. Zhang, K. Xu, S. Gu, X. Deng, Y. Zhang, and H. Wang, Deep learning-based quantum state tomography with imperfect measurement, Interna- tional Journal of Theoretical Physics61, 134 (2022)

    work page 2022

  17. [17]

    Lange, M

    H. Lange, M. B. Matjaˇ z Kebriˇ z and, U. Schollw¨ ock, F. Grusdt, and A. Bohrdt, Adaptive quantum state to- mography with active learning, Quantum7, 1129 (2023)

    work page 2023

  18. [18]

    Pineda, D

    C. Pineda, D. D´ avalos, C. Viviescas, and A. Rosado, Fuzzy measurements and coarse graining in quantum many-body systems, Phys. Rev. A104, 042218 (2021)

    work page 2021

  19. [19]

    Busch and R

    P. Busch and R. Quadt, Concepts of coarse graining in quantum mechanics, Int. J. Theor. Phys.32, 2261 (1993)

    work page 1993

  20. [20]

    Quadt and P

    R. Quadt and P. Busch, Coarse graining and the quan- tum—classical connection, Open Systems & Information Dynamics2, 129 (1994)

    work page 1994

  21. [21]

    Duarte, G

    C. Duarte, G. D. Carvalho, N. K. Bernandes, and F. De melo, Emerging dynamics arising from coarse- grained quantum systems, Phys. Rev. A96, 032113 (2017)

    work page 2017

  22. [22]

    Castillo, C

    A. Castillo, C. Pineda, E. S. Navarrete, and D. Davalos, Coarse-grained dynamics in quantum many-body sys- tems using the maximum entropy principle, Phys. Rev. A112, 032204 (2025)

    work page 2025

  23. [23]

    J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature467, 68 (2010)

    work page 2010

  24. [24]

    J. Yang, L. Liu, J. Mongkolkiattichai, and P. Schauss, Site-resolved imaging of ultracold fermions in a triangu- lar-lattice quantum gas microscope, PRX Quantum2, 020344 (2021)

    work page 2021

  25. [25]

    K. Kwon, K. Kim, J. Hur, S. Huh, and J. yoon Choi, Site-resolved imaging of a bosonic Mott insulator of 7li 21 atoms, Phys. Rev. A105, 033323 (2022)

    work page 2022

  26. [26]

    Saideh, A

    I. Saideh, A. D. Ribeiro, G. Ferrini, T. Coudreau, P. Mil- man, and A. Keller, General dichotomization procedure to provide qudit entanglement criteria, Phys. Rev. A92, 052334 (2015)

    work page 2015

  27. [27]

    Bengtsson and K

    I. Bengtsson and K. ˙Zyczkowski,Geometry of quan- tum states: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2008) p. 107

    work page 2008

  28. [28]

    Ekert and P

    A. Ekert and P. L. Knight, Entangled quantum systems and the Schmidt decomposition, Am. J. Phys.63, 415 (1995)

    work page 1995

  29. [29]

    P. K. Aravind, Geometry of the Schmidt decomposition and Hardy’s theorem, Am. J. Phys.64, 1143 (1996)

    work page 1996

  30. [30]

    Hill and W

    S. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett.78, 5022 (1997)

    work page 1997

  31. [31]

    Christandl, B

    M. Christandl, B. Doran, S. Kousidis, and M. Wal- ter, Eigenvalue distributions of reduced density matrices, Commun. Math. Phys.332, 1 (2014)

    work page 2014

  32. [32]

    Mej´ ıa, C

    J. Mej´ ıa, C. Zapata, and A. Botero, The difference be- tween two random mixed quantum states: exact and asymptotic spectral analysis, J. Phys. A: Math. Theor. 50, 025301 (2017)

    work page 2017

  33. [33]

    Kieburg and J

    M. Kieburg and J. Zhang, Derivative principles for in- variant ensembles, Adv. Math.413, 108833 (2023)

    work page 2023

  34. [34]

    ˙Zyczkowski, Volume of the set of separable states

    K. ˙Zyczkowski, Volume of the set of separable states. II, Phys. Rev. A60, 3496 (1999)

    work page 1999

  35. [35]

    R. F. Werner, Quantum states with Einstein-Podolsky- Rosen correlations admitting a hidden-variable model, Phys. Rev. A40, 4277 (1989)

    work page 1989