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REVIEW 3 major objections 5 minor 21 references

Discrete-Time Modelling of Quantum Evolutions, the Energy-Time Uncertainty Relation and General Extensions in the Entangled History Formalism

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Pith's one-line read A single vector in the tensor product of instantaneous Hilbert spaces can represent a quantum system's discretized evolution, including its temporal correlations.

desk verdict A useful but overclaimed paper: the entangled-history answer to Aharonov is basis-relative, and the energy-time 'relation' is only a pair of informal examples. read the letter →

arxiv 1908.02935 v1 pith:EPXK5X4X submitted 2019-08-08 quant-ph

Hai Wang , Ray-Kuang Lee , Junde Wu This is my paper · ORCID
classification quant-ph
keywords entangledhistoriestemporalcorrelationsdiscrete-timequantumevolutionenergy-timeuncertaintychannelsChoimatrixmonitorsystemstwo-vectorformalism
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

The paper addresses a challenge raised in the two-vector formalism: whether any vector in a tensor product of Hilbert spaces can faithfully represent a quantum system's state at several chosen instants. It answers yes, using the entangled history formalism, in which the evolution is encoded by a history state built from a preferred orthonormal basis at each instant. The key example is a qubit undergoing trivial evolution, represented by $\alpha_0 \odot_{i=0}^N |0\rangle_i + \alpha_1 \odot_{i=0}^N |1\rangle_i$, which reproduces the correct temporal correlations without revealing the full unknown state from a single subsystem. The construction is then used to give a qualitative account of the energy-time uncertainty relation from two extreme cases, and to extend the formalism from pure states and unitary evolutions to density matrices and general quantum operations. A sympathetic reader would care because the result turns a supposed limitation into a concrete representational scheme for temporal quantum correlations.

What carries the argument

The central object is the entangled history state, a vector in the tensor product $\odot_{i=0}^N H_i$ of Hilbert spaces assigned to successive instants, where $\odot$ behaves like $\otimes$ but marks the temporal character of the factor spaces. Its work is to store information about the evolution path rather than about a single instant. The paper pairs this with the monitor-system construction, which couples the evolving system to auxiliary qubits at each instant and projects the main system out, so that temporal correlations among the history labels become ordinary spatial correlations among the monitor qubits. For the general case of a density matrix $\rho$ and a quantum operation $\Lambda$, the construction uses the operation's matrix representation with elements $\Lambda_{kl,ij} = \operatorname{tr}(F_{kl}^{\dagger}\Lambda(E_{ij}))$ and writes the history state as $\sum_{ij,kl} \Lambda_{kl,ij}\,\rho_{ij}\, F_{kl}\odot E_{ij}$.

What would settle it

Prepare a qubit in an arbitrary superposition, let it evolve trivially, and use the monitor-system procedure to build the history state over three instants; then measure the same spin direction on two different monitor qubits. The paper's claim predicts perfectly correlated outcomes for every direction; observing any direction for which those outcomes are not perfectly correlated would falsify the representational claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a vector in $\odot_{i=0}^N H_i$ can represent the discretization of a quantum system's evolution, giving a definite affirmative answer to the question posed in the two-vector formalism. For a qubit in state $\alpha_0|0\rangle + \alpha_1|1\rangle$ that evolves trivially, choosing the spin basis $\{|0\rangle_{\vec r}, |1\rangle_{\vec r}\}$ at every instant yields the entangled history state $\alpha_0 \odot_{i=0}^N |0\rangle_{i,\vec r} + \alpha_1 \odot_{i=0}^N |1\rangle_{i,\vec r}$. This state has the two desired properties: any spin measurement at an earlier instant is perfectly correlated with the same measurement at a later instant, and no measurement on a single subsystem leaks the identity of the full unknown state. The paper further claims that when energy is fixed, time uncertainty is maximal, and when time is fixed (in the sense that the history label is perfectly distinguishable), energy uncertainty is maximal; and it generalizes the history construction to mixed states and quantum channels by replacing unitary matrix elements with the elements of the operation's matrix representation.

Load-bearing premise

The construction depends on choosing an orthonormal basis at every instant; if a meaningful temporal correlation involves incompatible bases at different times, this history vector cannot represent it, and the claim would collapse.

Editorial extensions

If this is right

  • The affirmative answer to the question means that a tensor-product vector can faithfully encode sequential-measurement statistics, so the entangled history formalism is a viable alternative to the earlier two-vector treatment of temporal correlations.
  • For a trivial evolution, the history state predicts that repeated measurements of any fixed spin observable at different instants always agree, a temporal correlation that the naive product state $|\phi\rangle^{\otimes N}$ fails to capture.
  • The two extreme cases of the energy-time relation imply that, in this formalism, energy certainty and time certainty cannot both be achieved; the relation is represented as a complementarity between the energy eigenbasis and the distinguishability of time labels in the history.
  • The matrix-representation generalization shows that mixed states and general quantum operations can be handled directly, without enlarging the Hilbert space or struggling with partial traces in the two-vector formalism.

Reading between the lines

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

  • Editorial inference: because the history state is defined relative to a chosen orthonormal basis at each instant, the construction suggests a natural test: the representational claim should be evaluated separately for each choice of basis, and correlations involving incompatible bases at different times are not part of the representation.
  • Editorial inference: the matrix-representation form of the history state offers a way to connect temporal correlations to quantum process reconstruction; one could, in principle, recover the process from the history state's correlations, which the paper does not explicitly develop.
  • Editorial inference: the qualitative energy-time extremes invite a quantitative extension in which time uncertainty is measured by an information-theoretic distinguishability of history labels; whether such a measure yields the standard inequality is not settled by the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper claims three results: (i) an affirmative answer to Aharonov's question whether a vector in the tensor-product Hilbert space ⊙_{i=0}^N H_i can represent the discretization of a quantum evolution, via the entangled-history state α0 ⊙ |0>_{i,σ} + α1 ⊙ |1>_{i,σ} for trivial evolution; (ii) an energy-time uncertainty relation read off from two extreme cases in the entangled-history formalism; and (iii) a generalization of the formalism to quantum channels and mixed states using Choi-matrix coefficients. The constructions follow the monitor-system approach of Cotler and Wilczek [8]. The paper is written as an early-stage research note, and the strength of its claims is not always matched by the supporting arguments.

Significance. If the central claims held as stated, the paper would provide a simple tensor-product answer to a long-standing question about representing temporal evolution, and it would extend the entangled-history formalism to general quantum operations. The explicit construction for trivial evolution is a correct and instructive illustration of how temporal correlations in a fixed basis can be stored without enabling tomography of an unknown state, and the Choi-matrix extension is a natural and potentially useful step beyond unitary evolutions and pure states. However, the affirmative answer to Aharonov is basis-relative and does not cover noncommuting observables, and the energy-time uncertainty section contains no derivation. The paper would be a useful contribution if reframed as a basis-dependent history-state construction, with the energy-time material presented as a conjecture or motivation for future work.

major comments (3)
  1. [III.B (history-state construction and the 'any spin' sentence)] The central claim that a vector in ⊙ H_i gives a definite affirmative answer to Aharonov's question is overstated. The history state α0 ⊙ |0>_{i,σ} + α1 ⊙ |1>_{i,σ} is defined only after fixing an orthonormal basis for each H_i, and the paper's own caveat in the same section says that only observables compatible with the chosen basis can be accessed. For the trivial evolution of |0_z>, the z-basis history state is |0_z>⊙|0_z>; expressed in the x-basis it is (|0_x>+|1_x>)⊙(|0_x>+|1_x>)/2, so measuring both factors in σ_x yields different outcomes with probability 1/2, whereas sequential σ_x measurements on the actual system agree with probability 1. Hence the sentence 'the spin σ_r can be any spin' holds only if a separate history vector is constructed for each spin axis; no single vector encodes temporal correlations for noncommuting axes. The abstract and the conclusion should be revised to state this basis-dependence explicitly rather than presenting an unqualified affirmative answer.
  2. [III.C (energy-time uncertainty)] The energy-time uncertainty relation is not derived. The section only observes two extreme cases: for a fixed-energy state the history state is |φ>⊙...⊙|φ>, making subsystem discrimination impossible, and for the history state |1>⊙|0> subsystem discrimination is perfect while the energy spread in the {|+>,|->} basis is maximal. No quantitative definitions of time uncertainty or energy uncertainty are provided, and no inequality or general argument connects the two extremes. The authors themselves describe the reasoning as 'quite immature' at the end of Section III.C. Since the abstract claims the paper 'reveals the energy-time uncertainty relationship,' this claim is unsupported as it stands; it should either be replaced by a rigorous derivation or explicitly labeled as a conjecture motivating future work.
  3. [III.D (Choi-matrix generalization)] The expression Σ_{ij,kl} Λ_{kl,ij} ρ_{ij} F_{kl} ⊙ E_{ij} is presented as the generalization of the history-state construction to general quantum operations, but the presentation is incomplete. The paper does not explain how temporal correlations of measurements are recovered for non-unitary evolution, how the operator-space tensor product is physically interpreted, or how such an object would be prepared or measured. The basis-dependence already noted in Section III.B applies here as well and is not stated. This weakens the third claimed achievement, although the formula itself is a natural formal extension of the unitary case.
minor comments (5)
  1. [Abstract] The abstract contains the typo 'Firstyl' for 'Firstly' and the grammatical error 'we reveals' for 'we reveal'.
  2. [III.A] Several spelling errors appear in this section: 'unitory' should be 'unitary', 'Feymann' should be 'Feynman', and 'minitor' should be 'monitor'.
  3. [III.C] In the 'time is fixed' example, the Hamiltonian whose energy uncertainty is being discussed is never specified; for the qubit with evolution X, the eigenstates {|+>,|->} of some Hermitian operator H should be stated explicitly.
  4. [III.D] The basis-dependence of the Choi-matrix history expression should be stated in this section, in line with the caveat already present in Section III.B.
  5. [References] Reference [1] lists the author as 'Tollaken'; the correct spelling is 'Tollaksen'. The title also appears without the comma present in the original publication.

Circularity Check

2 steps flagged · score 3.0 of 10

Affirmative answer to Aharonov and the energy-time extremes are definitional encodings of the history state, not independent derivations.

  1. self definitional [Section III.B ('Answer to problems of Aharonov'), paragraph beginning 'Now, back to our initial problem.']
    "By analog, in the spin σ− →r view, we will immediately see that under the above setting, once we measure σ− →r at some earlier instant, we will get the same value absolutely about the same spin for later measurements."

    The history state is defined as a coherent superposition of paths in which every time-slice has the same σ_r eigenvalue (all 0 or all 1). The 'same value for earlier and later measurements' is therefore a literal property of the chosen state, not a consequence derived from independent dynamics or a measurement model. The paper itself qualifies the construction in the following paragraph: 'once orthonormal basis are fixed, we can only get information of those observables compatible with our orthonormal basis.' Thus the affirmative answer to Aharonov is basis-relative: for each spin axis one may construct a different history vector, but a single vector does not encode temporal correlations for noncommuting observables. The claimed result reduces to the definition of the history vector.

  2. self definitional [Section III.C.1 ('Energy is fixed'), paragraph after the heading.]
    "It is clear that for states like |φ⟩ ⊙ |φ⟩ ⊙... ⊙ |φ⟩, it is impossible to find a measurement to tell which subsystem it is from measurement results. That means that in this case, we can’t make any inference about time, just can do random guessing. So in this case. the time uncertainty is biggest."

    The section defines 'time uncertainty' as the inability to identify which subsystem a measurement outcome came from. A product history state with identical factors has exactly that property by construction, so the conclusion 'time uncertainty is biggest' is read off from the definition rather than derived from an independent uncertainty measure. Similarly, in the opposite extreme (III.C.2), 'time is fixed' is defined as being able to tell which subsystem is measured perfectly, and 'energy uncertainty' is read off from a basis choice. The two extreme cases are therefore chosen to match the stipulated definitions; they illustrate an interpretation rather than provide an independent derivation of an energy-time uncertainty relation.

full rationale

This paper contains no fitted parameters, no data fitting, and no self-citations by the authors. Its central constructive claim—that the entangled-history vector α0⊙|0>+α1⊙|1> represents trivial evolution—is not an empirical prediction but a definitional encoding: the temporal correlation 'same value' is built into the support of the state by the choice of basis and amplitudes. The paper is explicit about the basis-dependence ('once orthonormal basis are fixed, we can only get information of those observables compatible with our orthonormal basis'), so the broad 'any spin' sentence is an equivocation rather than a derived result. The energy-time section is similarly definitional: 'time uncertainty' is equated with subsystem indistinguishability, so the two extreme cases follow by construction. These steps are self-definitional but not loaded with fitted parameters or self-citation chains, and the paper honestly calls the reasoning 'quite immature.' The generalization to Choi matrices is a straightforward transcription of quantum operations and is not circular. Overall, the paper is an application of an existing formalism with definitional illustrations; the circularity is mild.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The paper's central construction is inherited from the entangled history formalism of Cotler and Wilczek; the only new conceptual step is the proposed definitional reading of energy-time uncertainty.

assumptions (3)
  • domain assumption The entangled history formalism, including the monitor-system representation, correctly encodes temporal correlations of quantum evolutions.
    The paper relies on refs [4,8] for the validity of representing evolution paths as tensor-product states; this is the core assumption.
  • ad hoc to paper Time uncertainty can be defined via state discrimination of the history subsystems, following the approach of ref [17].
    The energy-time discussion in Section III.C assumes this definition to map time uncertainty to distinguishability of the N instant states.
  • domain assumption Only observables compatible with the chosen orthonormal basis at each instant can be extracted from the history state.
    The paper states this in Section III.B and it restricts the claimed representation, but it is necessary for their answer to Aharonov.

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Cite this review

Pith. "Pith review of Discrete-Time Modelling of Quantum Evolutions, the Energy-Time Uncertainty Relation and General Extensions in the Entangled History Formalism." pith.science (2026). https://pith.science/paper/EPXK5X4X

@misc{pith2026190802935,
  author       = {Pith},
  title        = {Pith review of: Discrete-Time Modelling of Quantum Evolutions, the Energy-Time Uncertainty Relation and General Extensions in the Entangled History Formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EPXK5X4X}},
  note         = {Machine review of arXiv:1908.02935}
}
read the original abstract

Time evolution is an indivisible part in any physics theory. Usually, people are accustomed to think that the universe is a fixed background and the system itself evolves step by step in time. However, Yakir Aharonov challenges this view using his two-vector formalism. In this paper, using the entangled history formalism, we attain three achievements.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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    It tells too much. From this model, we can use quantum tomography to know what the unknown state it is. However, this is not allowed in the original experiment setting

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    From this model, we cannot get the internal correlation between different instants

    It tells too little. From this model, we cannot get the internal correlation between different instants. Based on these two considerations, [ 1] concludes that it is impossible to find a simple vector in ⊗N i=0Hi to represent a state’s evolution at N different instants. However, later we will show this strategy actually works. Using the entangled history m...

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    Then if the initial state of the system, |φ⟩ has a fixed energy, an eigenvector of our hermitian H

    Energy is fixed Firstly, assume that our hermitian operator is non- degenerated. Then if the initial state of the system, |φ⟩ has a fixed energy, an eigenvector of our hermitian H. Now sup- pose that evolution time under this hermitian operator H isT . ChooseN different instants t1,...t N from (0,T ] with the initial time t0 = 0 . Then the history space for...

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    Time is fixed As done in [ 17], time uncertainty is transformed into dis- crimination of states. In the entangled history formalism, we think that the sentence, time is fixed, means that we can find a measurement and from its outcomes, we can tell which sub- system it is perfectly. Let us use qubit system to show our idea. And to make it easier, the evolutio...

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    Y . Aharonov, S. Popescu and J. Tollaken Each Instant of Time a New Universe , in Quantum Theory: A Two-Time Success Story, Yakir Aharonov Festschrift, edited by D. C. Struppa a nd J. M. Tollaksen (Springer, New Y ork, 2013), pp. 21-36

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