Localized Covariant Quantities Appear To Underlie Quantum Circuits

James Saslow; Ken Wharton; Roderick Sutherland; Titus Amza

arxiv: 2606.29079 · v1 · pith:5GLJXRJ3new · submitted 2026-06-27 · 🪐 quant-ph

Localized Covariant Quantities Appear To Underlie Quantum Circuits

Ken Wharton , Roderick Sutherland , Titus Amza , James Saslow This is my paper

Pith reviewed 2026-06-30 09:05 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum circuitsweak valuescovariant tensorsentanglementlocal hidden variablesuniversal gatesspacetime description

0 comments

The pith

Local weak values form covariant tensors for each qubit that stay fixed under distant measurements.

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

The paper argues that quantum circuits built from standard gates can be described using local weak values that form a covariant tensor for every qubit. These tensors persist unchanged even when the overall state is entangled and other qubits are measured or altered elsewhere in the circuit. Their dynamics through the gates follow covariant rules and require the full circuit, including future measurement outcomes, to be considered together. This setup is presented as compatible with classical spacetime and avoids standard no-go results for local hidden variables by incorporating future constraints.

Core claim

The local weak values comprise a covariant tensor associated with each individual qubit. Even if the state is massively entangled, these tensors do not evolve or collapse when other qubits are measured or pass through distant circuit elements. They can therefore be viewed from different reference frames without contradiction. Furthermore, their evolution through any circuit always obeys covariant dynamical rules. Weak values are subject to both past and future constraints, so the covariant quantities can only be determined by considering the entire circuit all-at-once, incorporating the future measurement basis to avoid the standard no-go theorems. Because these results hold for a set of uni

What carries the argument

Covariant tensor formed by the local weak values of each qubit, fixed by the entire circuit including future measurement bases.

If this is right

These tensors remain unchanged by measurements or operations on other qubits.
The tensors obey covariant dynamical rules through any circuit.
The tensors admit consistent descriptions from different reference frames.
Any quantum circuit built from universal gates admits a realistic localized description compatible with classical spacetime.

Where Pith is reading between the lines

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

The all-at-once determination aligns with the structure of action principles where boundary conditions at both ends fix the solution.
This localized assignment might allow reinterpreting measurement order independence in circuits as a spacetime covariance property.
The approach could be checked by constructing explicit tensors for a small universal gate sequence and verifying frame independence by direct computation.

Load-bearing premise

Assigning local weak values to individual qubits while incorporating the future measurement basis automatically evades all standard no-go theorems for local hidden variables without additional consistency conditions from the gate set or circuit topology.

What would settle it

An explicit calculation or experiment on a universal gate set showing that the assigned tensors change or become inconsistent when one qubit is measured while viewed from a different reference frame would falsify the central claim.

read the original abstract

Although entangled state vectors cannot be fully described in terms of variables localized in space and time, any given entanglement experiment can be built from basic quantum circuit components with well-defined locations. We analyze such quantum circuits and present evidence that the local weak values comprise a covariant tensor associated with each individual qubit. Even if the state is massively entangled, these tensors do not evolve or collapse when other qubits are measured or pass through distant circuit elements. They can therefore be viewed from different reference frames without contradiction. Furthermore, their evolution through any circuit always obeys covariant dynamical rules. Weak values are subject to both past and future constraints, so the covariant quantities can only be determined by considering the entire circuit "all-at-once", as in action principles, incorporating the future measurement basis to avoid the standard no-go theorems. Because these results hold for a set of universal quantum gates, this work lends support to the claim that any quantum circuit can be assigned a realistic, lower-level description compatible with our understanding of classical spacetime.

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.

Desk Editor's Note private letter to a colleague

The paper claims local weak values form stable covariant tensors for qubits in universal circuits but shows no derivations or examples to support the claim.

read the letter

The main thing to know is that this paper argues local weak values can be treated as covariant tensors attached to individual qubits. These tensors stay fixed even when the overall state is entangled and other qubits undergo measurements or gates elsewhere in the circuit. The evolution through the circuit is said to follow covariant rules, but only when the entire setup is considered all at once, including future measurement outcomes. This is presented as evidence that any circuit from a universal gate set admits a realistic local description compatible with classical spacetime.

What is new is the specific framing that these weak-value quantities remain stable across arbitrary circuits and can be viewed as tensors without frame-dependent contradictions. The all-at-once treatment itself is not new, but tying it directly to covariant tensor behavior in circuit models is a distinct angle.

The paper does a clear job laying out how future constraints in weak values might sidestep standard no-go theorems for local variables. That connection is stated plainly.

The soft spots are substantial and central. The abstract asserts that evidence exists for universal gates, yet no derivations, circuit examples, or consistency checks appear. Without those, it is impossible to verify whether the tensors really compose consistently under entangling operations or whether the claimed covariance follows from local data plus future basis rather than being built into the definition. The concern about circularity in the all-at-once conditioning is fair based on what is shown.

This is for readers working on quantum foundations who are already exploring weak values or retrocausal approaches. A serious referee could be useful to see whether the full manuscript supplies the missing explicit calculations and checks for gate composition. If those hold, the idea is worth further discussion; if not, the claim stays ungrounded.

I would send it to peer review rather than desk reject, with the expectation that the authors must demonstrate the internal consistency under composition.

Referee Report

2 major / 0 minor

Summary. The paper claims that local weak values in quantum circuits form covariant tensors associated with individual qubits. These tensors remain stable (do not evolve or collapse) under entanglement or distant operations, obey covariant dynamical rules, and can be assigned consistently by considering the full circuit all-at-once (incorporating future measurement bases). This is asserted to evade standard no-go theorems for local hidden variables, with the results holding for universal gates and thereby supporting a realistic, lower-level description compatible with classical spacetime for arbitrary quantum circuits.

Significance. If the central claim holds, the work would be significant in quantum foundations by supplying a concrete mechanism for assigning stable, local, covariant quantities to qubits within circuits. It would lend support to the idea that quantum nonlocality can be reconciled with spacetime locality via all-at-once weak-value assignments, potentially opening avenues for realistic models that respect both quantum mechanics and classical geometry. No machine-checked proofs, reproducible code, or falsifiable predictions are provided, however.

major comments (2)

[Abstract] Abstract: the assertion that 'these results hold for a set of universal quantum gates' is unsupported by any explicit derivation, numerical verification, or circuit example showing closure of the tensor assignment under gate composition (especially multiple entangling gates with anomalous weak values). This is load-bearing for the claim that the approach works for any circuit and automatically evades no-go theorems.
[Abstract] Abstract: the covariant tensors are defined via weak values that already encode the future measurement basis as part of the 'all-at-once' construction. This incorporates the target stability and non-evolution into the input definition rather than deriving it from independent local dynamics, which directly affects whether the evasion of Bell/Kochen-Specker theorems is demonstrated or presupposed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We respond point-by-point to the major comments below, indicating where we agree revisions are warranted and where we maintain our original position with further clarification.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'these results hold for a set of universal quantum gates' is unsupported by any explicit derivation, numerical verification, or circuit example showing closure of the tensor assignment under gate composition (especially multiple entangling gates with anomalous weak values). This is load-bearing for the claim that the approach works for any circuit and automatically evades no-go theorems.

Authors: We agree the abstract claim would benefit from additional concrete support. The manuscript body already contains explicit derivations and calculations for the Hadamard, phase, and CNOT gates, including tensor covariance and stability under distant operations. To strengthen the universality point, we will add a new numerical example in the revised manuscript of a small circuit with two sequential CNOT gates and post-selection yielding anomalous weak values, verifying that the local tensor assignment closes under this composition. This addresses the specific concern while noting that a general proof for arbitrary circuits is left for future work. revision: partial
Referee: [Abstract] Abstract: the covariant tensors are defined via weak values that already encode the future measurement basis as part of the 'all-at-once' construction. This incorporates the target stability and non-evolution into the input definition rather than deriving it from independent local dynamics, which directly affects whether the evasion of Bell/Kochen-Specker theorems is demonstrated or presupposed.

Authors: The all-at-once treatment is required by the definition of weak values, yet the stability and covariance are derived results rather than built-in assumptions. Explicit calculations in the paper demonstrate that, once the full circuit (including future bases) is fixed, the local tensor for one qubit remains unchanged by operations on distant qubits due to the linearity of the weak-value expression and its tensorial transformation law under local unitaries. This structure evades no-go theorems precisely because the assignment is contextual on the global circuit, a feature we derive from the weak-value formalism rather than presuppose. We will add a clarifying paragraph in the introduction to distinguish the input definition from the derived invariance properties. revision: no

Circularity Check

1 steps flagged

Covariant tensors defined via future-conditioned weak values build claimed non-evolution and no-go evasion into the input by construction

specific steps

self definitional [Abstract]
"Weak values are subject to both past and future constraints, so the covariant quantities can only be determined by considering the entire circuit "all-at-once", as in action principles, incorporating the future measurement basis to avoid the standard no-go theorems."

The claimed non-evolution of the tensors when other qubits are measured, and their compatibility with classical spacetime without contradiction, is secured by defining the weak values to already encode the future measurement outcomes for the whole circuit. The stability and covariance are therefore true by construction of the input assignment rather than emerging from local rules independent of the global future basis.

full rationale

The paper's central claim is that local weak values form covariant tensors that remain stable under distant measurements and obey local dynamical rules for any universal-gate circuit. This stability is asserted to follow from defining the quantities via weak values that already incorporate the full circuit (past and future constraints) in an all-at-once manner. The quoted abstract text shows the non-collapse property and evasion of no-go theorems are achieved precisely by this conditioning on future measurement basis, rather than derived from independent local dynamics that close under gate composition. This matches the self-definitional pattern: the target invariance is true by how the tensors are defined. No other circular steps are exhibited in the provided text; the universal-gate claim is asserted but the load-bearing step reduces to the definitional choice.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of weak values (which already incorporate post-selection) and the domain assumption that an all-at-once circuit view suffices to evade no-go theorems without further constraints. No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Weak values defined with both pre- and post-selection can be assigned locally to individual qubits without violating consistency across entangled states.
Invoked when stating that the tensors remain unchanged by distant measurements.
ad hoc to paper Considering the entire circuit all-at-once automatically satisfies the conditions needed to bypass standard no-go theorems for local realistic descriptions.
Explicitly stated as the mechanism that allows the covariant assignment.

pith-pipeline@v0.9.1-grok · 5702 in / 1424 out tokens · 24988 ms · 2026-06-30T09:05:03.084518+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 4 canonical work pages

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Waegell, M., Cohen, E., Elitzur, A., Tollaksen, J., Aharonov, Y.: Quantum reality with negative-mass particles. Proceedings of the National Academy of Sciences 120(32), 2018437120 (2023) 20

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Wharton and H

Wharton, K., Price, H.: Bell inequality violations without entanglement? it’s just postselection. arXiv preprint arXiv:2508.13431 (2025) Appendix A Here we demonstrate that Eq. (23) follows from (1) and (20). We need only prove that d2wa dτ 2 = π2 2 (wb −w a),(A1) wherew a andw b are any arbitrary component of the full vectorsw a andw b. Also, the second ...

work page arXiv 2025

[1] [1]

Physical Review Letters68(20), 2981 (1992)

Hardy, L.: Quantum mechanics, local realistic theories, and lorentz-invariant realistic theories. Physical Review Letters68(20), 2981 (1992)

1992

[2] [2]

Physical Review A53(4), 2062 (1996)

Berndl, K., D¨ urr, D., Goldstein, S., Zangh` ı, N.: Nonlocality, lorentz invariance, and bohmian quantum theory. Physical Review A53(4), 2062 (1996)

2062

[3] [3]

Foundations of Physics 32(11), 1773–1784 (2002)

Myrvold, W.C.: Modal interpretations and relativity. Foundations of Physics 32(11), 1773–1784 (2002)

2002

[4] [4]

Cambridge university press, ??? (2004)

Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. Cambridge university press, ??? (2004)

2004

[5] [5]

Indiana Univ

Simon Kochen, E.S.: The problem of hidden variables in quantum mechanics. Indiana Univ. Math. J.17, 59–87 (1968)

1968

[6] [6]

Nature Physics8(6), 475–478 (2012)

Pusey, M.F., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nature Physics8(6), 475–478 (2012)

2012

[7] [7]

Reviews of Modern Physics92(2), 021002 (2020)

Wharton, K., Argaman, N.: Colloquium: Bell’s theorem and locally mediated reformulations of quantum mechanics. Reviews of Modern Physics92(2), 021002 (2020)

2020

[8] [8]

Pro- ceedings of the Royal Society of London

Roberts, K.: An objective interpretation of lagrangian quantum mechanics. Pro- ceedings of the Royal Society of London. A. Mathematical and Physical Sciences 360(1700), 135–160 (1978)

1978

[9] [9]

Physical review letters60(14), 1351 (1988)

Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Physical review letters60(14), 1351 (1988)

1988

[10] [10]

Foundations of Physics 28(7), 1157–1190 (1998) 19

Sutherland, R.I.: Density formalism for quantum theory. Foundations of Physics 28(7), 1157–1190 (1998) 19

1998

[11] [11]

Physical review letters112(18), 180402 (2014)

Campagne-Ibarcq, P., Bretheau, L., Flurin, E., Auff` eves, A., Mallet, F., Huard, B.: Observing interferences between past and future quantum states in resonance fluorescence. Physical review letters112(18), 180402 (2014)

2014

[12] [12]

Nature511(7511), 570– 573 (2014)

Weber, S., Chantasri, A., Dressel, J., Jordan, A.N., Murch, K.W., Siddiqi, I.: Mapping the optimal route between two quantum states. Nature511(7511), 570– 573 (2014)

2014

[13] [13]

Physical review letters 114(9), 090403 (2015)

Tan, D., Weber, S., Siddiqi, I., Mølmer, K., Murch, K.: Prediction and retrodic- tion for a continuously monitored superconducting qubit. Physical review letters 114(9), 090403 (2015)

2015

[14] [14]

Physical Review A91(6), 062116 (2015)

Rybarczyk, T., Peaudecerf, B., Penasa, M., Gerlich, S., Julsgaard, B., Mølmer, K., Gleyzes, S., Brune, M., Raimond, J., Haroche, S.,et al.: Forward-backward analysis of the photon-number evolution in a cavity. Physical Review A91(6), 062116 (2015)

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[15] [15]

Light: Science & Applications10(1), 106 (2021)

Rebufello, E., Piacentini, F., Avella, A., Souza, M.A.d., Gramegna, M., Dziewior, J., Cohen, E., Vaidman, L., Degiovanni, I.P., Genovese, M.: Anomalous weak values via a single photon detection. Light: Science & Applications10(1), 106 (2021)

2021

[16] [16]

Foundations of Physics52(4), 75 (2022)

Sutherland, R.I.: Probabilities and certainties within a causally symmetric model. Foundations of Physics52(4), 75 (2022)

2022

[17] [17]

Barandes, J.A.: The trouble with weak values (2026) arXiv:2602.09380 [quant-ph]

work page arXiv 2026

[18] [18]

Physical Review A91(3), 032116 (2015)

Dressel, J.: Weak values as interference phenomena. Physical Review A91(3), 032116 (2015)

2015

[19] [19]

Studies in History and Phi- losophy of Science Part B: Studies in History and Philosophy of Modern Physics 39(4), 782–805 (2008)

Sutherland, R.I.: Causally symmetric bohm model. Studies in History and Phi- losophy of Science Part B: Studies in History and Philosophy of Modern Physics 39(4), 782–805 (2008)

2008

[20] [20]

Foundations of Physics47(2), 174–207 (2017)

Sutherland, R.I.: Lagrangian description for particle interpretations of quantum mechanics: entangled many-particle case. Foundations of Physics47(2), 174–207 (2017)

2017

[21] [21]

Time in quantum mechanics, 399–447 (2008)

Aharonov, Y., Vaidman, L.: The two-state vector formalism: an updated review. Time in quantum mechanics, 399–447 (2008)

2008

[22] [22]

Aharonov, Y., Cohen, E., Elitzur, A.C.: Can a future choice affect a past measurement’s outcome? Annals of Physics355, 258–268 (2015)

2015

[23] [23]

Proceedings of the National Academy of Sciences 120(32), 2018437120 (2023) 20

Waegell, M., Cohen, E., Elitzur, A., Tollaksen, J., Aharonov, Y.: Quantum reality with negative-mass particles. Proceedings of the National Academy of Sciences 120(32), 2018437120 (2023) 20

2023

[24] [24]

arXiv preprint arXiv:2412.05456v1 (2024)

Wharton, K., Sutherland, R., Amza, T., Liu, R., Saslow, J.: A localized reality appears to underpin quantum circuits. arXiv preprint arXiv:2412.05456v1 (2024)

work page arXiv 2024

[25] [25]

Physical Review Letters110(6), 060406 (2013)

Braverman, B., Simon, C.: Proposal to observe the nonlocality of bohmian trajectories with entangled photons. Physical Review Letters110(6), 060406 (2013)

2013

[26] [26]

Quantum7, 1119 (2023)

Catani, L., Leifer, M., Schmid, D., Spekkens, R.W.: Why interference phenomena do not capture the essence of quantum theory. Quantum7, 1119 (2023)

2023

[27] [27]

Entropy20(6), 410 (2018)

Wharton, K.: A new class of retrocausal models. Entropy20(6), 410 (2018)

2018

[28] [28]

Price, H., Wharton, K.: Why entanglement? (2023) arXiv:2212.06986 [quant-ph]

work page arXiv 2023

[29] [29]

Wharton and H

Wharton, K., Price, H.: Bell inequality violations without entanglement? it’s just postselection. arXiv preprint arXiv:2508.13431 (2025) Appendix A Here we demonstrate that Eq. (23) follows from (1) and (20). We need only prove that d2wa dτ 2 = π2 2 (wb −w a),(A1) wherew a andw b are any arbitrary component of the full vectorsw a andw b. Also, the second ...

work page arXiv 2025