Emergent dynamics from entangled mixed states

A. P. Majtey; A. R. Plastino; A. Vald\'es-Hern\'andez; C. G. Maglione

Emergent dynamics from entangled mixed states

T0 review · 0 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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

Pith's one-line read Under the zero-total-energy constraint, the only source of time evolution is clock-system entanglement.

desk verdict A clean, honest extension of Page-Wootters to mixed states; the math is correct and the novelty is real, though the ideal-clock assumption is the load-bearing prerequisite. read the letter →

arxiv 2009.12451 v2 pith:W7BTHFXQ submitted 2020-09-25 quant-ph

A. Valdés-Hernández , C. G. Maglione , A. P. Majtey , A. R. Plastino This is my paper · ORCID

classification quant-ph

keywords timelessquantumdynamicsemergenttimemixedstatesclock-systementanglementlinearentropyenergyuncertaintydiscordstationaryuniversestate

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 extends the timeless picture of quantum dynamics—where the whole universe is a stationary bipartite state of a clock $C$ and the rest $R$—from pure states to mixed states. It establishes a quantitative link between $R$'s time evolution and $R$–$C$ entanglement: whenever the relative state of $R$ changes with the clock's reading, the joint state must be entangled. The proof runs through an entropy inequality, Eq. (23), stating that the linear entropy of the reduced state $\rho_R$ is never smaller than that of the global state $\rho$, with equality meaning no evolution. This matters because it shows that weaker non-classical correlations, such as quantum discord, cannot by themselves generate dynamics; entanglement is the essential ingredient. The paper also bounds the entanglement indicator in terms of the spread of $R$'s energy distribution.

What carries the argument

The central object is the linear-entropy entanglement indicator $\Delta S = S_L[\rho_R] - S_L[\rho]$, built from $S_L[\varrho] = 1 - \mathrm{Tr}(\varrho^2)$. It compares the mixedness of the reduced state of $R$ after tracing out the clock with the mixedness of the global state; $\Delta S>0$ is an entropic criterion for entanglement. Under the zero-total-energy constraint, concavity of the linear entropy yields $S_L[\rho_R] \ge S_L[\rho]$ along the trajectory of $\sigma_{R,t}$, with equality tied to stationarity. The clock is assumed ideal and isolated, with clock observable $T$ and energy operator $\hat H_C$ satisfying $[T,\hat H_C]=i\hbar$, so that the conditional state obeys the unitary evolution equation and the $R|C$ split is unambiguous. The same machinery produces the upper bound $\Delta S_{\max}$ and the short-time expansion in terms of the commutator $[\hat H_R,\sigma_{R,t}]$.

What would settle it

Look for a zero-total-energy stationary state of $R+C$ whose conditional state of $R$ changes with the clock reading but for which the linear-entropy indicator $S_L[\rho_R]-S_L[\rho]$ vanishes; Eq. (23) says no such state exists, so exhibiting one would settle the claim false.

Watch

Extended reading notes

Core claim

Within the mixed-state timeless picture, the global state $\rho$ of $R+C$ is stationary with definite total energy zero. The paper proves that if the conditional state $\sigma_{R,t}$ of $R$, read off when the clock's hands show $t$, changes with $t$, then $\rho$ is necessarily entangled across the $R|C$ split. The central quantitative result is $S_L[\rho_R] \ge S_L[\rho]$, where $S_L$ is the linear entropy, with equality only when there is no time evolution; hence the indicator $\Delta S = S_L[\rho_R] - S_L[\rho]$ is positive exactly when dynamics occurs. The same indicator has an upper bound and an asymptotic limit, given for pure states by the linear entropy of the energy probability distribution of $R$, and its short-time expansion is proportional to $-\mathrm{Tr}([\hat H_R,\sigma_{R,t}]^2)$, a quantum component of $R$'s energy dispersion. Thus, for mixed states, only the quantum part of the energy spread contributes to the clock-system entanglement bound.

Load-bearing premise

The whole derivation rests on assuming an ideal clock that is dynamically isolated from $R$ and whose reading operator $T$ and energy operator $\hat H_C$ satisfy $[T,\hat H_C]=i\hbar$, since this is what lets the stationary global state produce the unitary evolution of $R$'s conditional state.

Editorial extensions

If this is right

A timeless-picture scenario works for mixed states, not just pure ones: the system's conditional state evolves unitarily while the total state stays static.
Entanglement is necessary for evolution: if $R$ is only classically correlated or discord-correlated with $C$, no ticking of the clock produces dynamics.
For any finite interval, the entanglement indicator $\Delta S$ is bounded above by the quantum spread of $R$'s energy distribution, and in the pure-state case this bound equals the linear entropy of the energy probabilities.
To lowest order in the interval length $T$, $\Delta S$ grows as $T^2$ times the quantum contribution to the energy dispersion of $R$, giving a quantitative time-energy-like relation.
A zero indicator means no evolution, although whether a zero indicator also forces separability remains open.

Reading between the lines

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

The quantity $D = -\mathrm{Tr}([\hat H_R,\sigma_{R,t}]^2)$ could be treated as a resource measure for the "quantum" share of energy uncertainty that powers emergent time, a resource-theoretic reading the paper does not develop.
Testing the mixed-state scenario with a finite-dimensional clock could make the entanglement-necessity claim directly verifiable in current ion-trap or optical experiments.
Because diagonal mixed states have zero indicator, the results suggest that classical energy fluctuations in a mixed universe state do not contribute to emergent dynamics; only coherence in the energy basis does.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's my honest read. The paper is a solid, self-contained extension of the Page-Wootters timeless picture to mixed global states. It proves that if the conditional state of R evolves in time, the global state of R+C must be entangled. The proof uses a linear-entropy inequality that is both simple and rigorous. The authors also derive an upper bound on their entanglement indicator (expressed as the difference between the linear entropy after and before a non-selective energy measurement) and a short-time expansion that connects ΔS to the energy dispersion. These are genuine new results for mixed states; the pure-state versions were known, but the mixed-state statements are not.

What the paper does well: it is transparent about assumptions. The ideal clock ([T,H_C]=iℏ, no interaction with R) is stated explicitly, and the main theorem is conditional on it. The equality conditions in the concavity argument are handled correctly. The authors also flag the open question whether entangled but non-evolving mixed states exist, which keeps the claim honest.

Soft spots: The ideal-clock assumption is the obvious load-bearing piece. It's standard in the PW literature, but it's still an idealization, and the paper doesn't extend to realistic clocks or interactions. That's okay—it's a proof of principle. Minor: there is a sign typo in the phase of Eq. (25); it cancels in the entropy calculations. Also, the paper's headline "entanglement is necessary for evolution" is a theorem under the PW constraints, not an unconditional foundational result; the authors do not overclaim, but a careless reader might.

Overall: the math is correct, the novelty is real, and the citations are appropriate (including their own earlier work, which is the right thing to cite here). It's a good paper for a specialized quantum-foundations or quantum-information journal. It deserves peer review; I would send it out and expect minor revisions. If I were working on time-in-quantum-mechanics, I would cite it.

Recommendation: engage with it; referee it if asked.

Referee Report

0 major / 5 minor

Summary. The paper extends the Page-Wootters (PW) timeless formulation of quantum dynamics to mixed states of a composite system consisting of a clock C and the rest of the Universe R. Under the assumptions that the joint state ρ is stationary with zero total energy and that the clock is an ideal, dynamically isolated clock satisfying [T,H_C]=iℏ, the conditional state σR,t of R obeys the von Neumann equation. The authors introduce the linear-entropy entanglement indicator ΔS = SL[ρR] − SL[ρ] and prove, using a concavity inequality, that ΔS ≥ 0 with equality if and only if σR,t is time-independent. Consequently, any nontrivial dynamical evolution of R forces R and C to be entangled, so correlations weaker than entanglement (e.g., discord) are insufficient for emergent time. The paper further derives an upper bound and asymptotic limit for ΔS in terms of the linear entropy of the energy probability distribution, and a short-time expansion relating ΔS to the quantum contribution to the energy dispersion. Illustrative qubit and noisy-state examples are provided.

Significance. The paper is a clean and rigorous mixed-state generalization of the PW theorem. The central inequality is derived self-contained from standard concavity arguments, with no adjustable parameters, and the assumptions are stated explicitly. The proof that evolution implies entanglement, the upper bound and asymptotic value of the entanglement indicator, and the short-time connection to energy dispersion are valuable quantitative contributions. The authors also honestly flag the open question of whether entangled non-evolving states exist. This is a worthwhile contribution to the quantum-foundations literature on time and clocks.

minor comments (5)

[Section III, Eq. (25)] The phase factor in Eq. (25) should be e^{-i(En−Em)T/2ℏ} rather than the printed e^{+i(...)}; the sign cancels in the products |⟨n|ρR|m⟩|² used for the entropies, so the error does not affect the theorem, but it should be corrected.
[Section III, Eq. (20)] In Eq. (20), the notation is ambiguous: it should read SL[ρR] = SL[σR,t] with the time average placed on the state inside the entropy, to distinguish this from the time-independent quantity SL[σR,t] appearing in Eq. (19).
[Section II, around Eq. (6)] The assumption [T,H_C]=iℏ, together with the claim that the zero-energy constraint effectively discretizes the spectrum of H_C, deserves a brief comment on its mathematical status: the commutation relation cannot hold exactly on a discrete-spectrum subspace, and the finite-box analogy is not exact because [x,p]=iℏ also fails on the boxed Hilbert space. The derivation is best understood as formal in the sense of Dirac kets.
[Section IV, Eq. (36)] The interpretation of ΔSmax as the difference between the linear entropies of the state after and before a non-selective energy measurement is a nice physical characterization; a sentence clarifying that the measurement is performed on the conditional state σR,t, rather than on the global state ρ, would help the reader.
[Section VI and Introduction] There are several small typos: 'In would be interesting' in Section VI should read 'It would be interesting', 'Deutch' in the Introduction should be 'Deutsch', and 'Sciencies' in the Acknowledgments should be 'Sciences'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mixed-state Page-Wootters theorem is derived self-contained from explicit ideal-clock assumptions and standard entropy inequalities; the cited prior work is a post-hoc consistency check, not a load-bearing input.

full rationale

The paper's central claim, that under a zero-energy stationary global state of R+C the evolution of R forces entanglement between R and C, is derived directly from the stated PW assumptions rather than from any fitted parameter or from the conclusion placed among the premises. The key inequality S_L[rho_R] >= S_L[rho] follows from Eq. (19), which uses only the unitary invariance of the overlaps of the conditional states, Eq. (20), which is the definition of rho_R as a time average of sigma_{R,t}, and the standard concavity inequality (21) applied to f(x)=x-x^2. The equality condition is supplied by the strict concavity of f, so nonzero evolution gives the strict inequality Delta S > 0, and the entropic entanglement criterion (14)-(17) is an external, previously established separator result, not something derived from the present conclusion. The clock assumptions [T,H_C]=i hbar and absence of R-C interaction are explicitly stated rather than hidden, and the derivation is conditional on them. The recovery of the pure-state bound from Ref. [23] in Eq. (39) is a consistency check after an independent derivation, not a premise; no step in the derivation requires accepting that prior result. The open question about whether entangled states with no evolution exist is explicitly acknowledged and does not affect the sufficiency direction that the paper proves. There are no fitted parameters, no renaming of known results as new predictions, and no self-citation chain used to forbid alternatives. Therefore the derivation is self-contained and exhibits no significant circularity.

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

The central theorem relies on the standard PW ideal-clock setup plus known entropy inequalities; no ad hoc parameters or invented entities. The qubit figures use illustrative parameters only.

assumptions (6)

domain assumption The total Hamiltonian of R+C is H_U = H_R⊗I + I⊗H_C and the global state ρ satisfies ⟨H_U⟩=0 and ⟨H_U^2⟩-⟨H_U⟩^2=0.
This is the PW constraint that the universe has definite zero total energy; used in Section III before Eq. (8).
domain assumption The clock observable T and Hamiltonian H_C obey [T,H_C]=iℏ, with T having a continuous spectrum restricted to [0,T].
Ideal clock assumption in Section II, Eq. (6), needed for relative states to obey the Schrödinger equation.
domain assumption The clock C and system R do not interact.
Section II, stating a good clock is dynamically isolated; guarantees uniqueness of bipartition and that conditional states evolve unitarily.
standard math For any concave function f, Tr[f(ϱ_t)] ≥ Tr[f(ϱ_t)] (Jensen's inequality).
Used in Eq. (21) to prove ΔS ≥ 0.
standard math The entropic criterion S_L[ρ_R] > S_L[ρ] implies ρ is entangled; equivalent to separable states satisfying S_L[ρ_R] ≤ S_L[ρ].
Cited from [32-36], used to infer entanglement from ΔS > 0.
domain assumption Each component pure state |Ψ_j⟩ of the mixture is stationary with zero energy.
Eq. (8): ρ = ∑ p_j |Ψ_j⟩⟨Ψ_j| with H_U|Ψ_j⟩=0; follows from vanishing energy dispersion.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Emergent dynamics from entangled mixed states." pith.science (2026). https://pith.science/paper/W7BTHFXQ

@misc{pith2026200912451,
  author       = {Pith},
  title        = {Pith review of: Emergent dynamics from entangled mixed states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7BTHFXQ}},
  note         = {Machine review of arXiv:2009.12451}
}

abstract

Entanglement is at the core of quantum physics, playing a central role in quantum phenomena involving composite systems. According to the timeless picture of quantum dynamics, entanglement may also be essential for understanding the very origins of dynamical evolution and the flow of time. Within this point of view, the Universe is regarded as a bipartite entity comprising a clock $C$ and a system $R$ (or "rest of the Universe") jointly described by a global stationary state, and the dynamical evolution of $R$ is construed as an emergent phenomena arising from the entanglement between $C$ and $R$. In spite of substantial recent efforts, many aspects of this approach remain unexplored, particularly those involving mixed states. In the present contribution we investigate the timeless picture of quantum dynamics for mixed states of the clock-system composite, focusing on quantitative relations linking the clock-system entanglement with the emerging dynamical evolution experienced by the system.

Figures

Figures reproduced from arXiv: 2009.12451 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

[1]

Goldstein, J

S. Goldstein, J. L. Lebowitz, R. Tumulka and N. Zanghi, Canonical typicality, Phys. Rev. Lett. 96, 050403 (2006)

work page 2006
[2]

Deﬀner and W

S. Deﬀner and W. Zurek, Foundations of statistical me- chanics from symmetries of entanglement , New J. Phys. 18 063013 (2016)

work page 2016
[3]

D. N. Page and W. K. Wootters, Evolution without Evo- lution: Dynamics Described by Stationary Observables , Phys. Rev. D 27, 2885 (1983)

work page 1983
[4]

W. K. Wootters, “Time” replaced by quantum correla- tions, Int. J. Theor. Phys. 23, 701 (1984)

work page 1984
[5]

B. S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory, Phys. Rev. 160, 1113 (1967)

work page 1967
[6]

J. B. Hartle and S. W. Hawking, Wave function of the Universe, Phys. Rev. D 28, 2960 (1983)

work page 1983
[7]

Rovelli, Quantum mechanics without time: A model , Phy

C. Rovelli, Quantum mechanics without time: A model , Phy. Rev. D 42, 8, 2638, (1990)

work page 1990
[8]

Gambini, R

R. Gambini, R. A. Porto, J. Pullin, and S. Torterolo, Conditional probabilities with Dirac observables and the problem of time in quantum gravity , Phys. Rev. D 79, 041501(R) (2009)

work page 2009

Show all 36 references

[9]

Albrecht and A

A. Albrecht and A. Iglesias, Clock Ambiguity and the Emergence of Physical Laws , Phys. Rev. D 77, 063506 (2008)

2008
[10]

Vedral, Time, (Inverse) Temperature and Cosmo- logical Inﬂation as Entanglement , in Time in Physics , edited by R

V. Vedral, Time, (Inverse) Temperature and Cosmo- logical Inﬂation as Entanglement , in Time in Physics , edited by R. Renner and S. Stupar, Tutorials, Schools, and Workshops in the Mathematical Siences (Birkh¨ auser, Cham, 2017)

2017
[11]

Marletto and V

C. Marletto and V. Vedral, Evolution Without Evolu- tion and Without Ambiguities, Phys. Rev. D 95, 0435101 (2017)

2017
[12]

Giovannetti, S

V. Giovannetti, S. Lloyd and L. Maccone, Quantum time, Phys. Rev. D 92, 0450331 (2015)

2015
[13]

Leon and L

J. Leon and L. Maccone, The Pauli Objection , Found. Phys. 47, 1597 (2017)

2017
[14]

L. R. Mendes and D. O. Soares-Pinto, Time as a con- sequence of internal coherence , Proc. R. Soc. A 475, 20190470 (2019). 10

2019
[15]

N. L. Diaz and R. Rossignoli, History state formalism for Dirac’s theory, Phys. Rev. D 99, 045008 (2019)

2019
[16]

N. L. Diaz, J. M. Matera, and R. Rossignoli, History state formalism for scalar particles, Phys. Rev. D 100, 125020 (2019)

2019
[17]

McClean, J.A

J.R. McClean, J.A. Parkhill, and A. Aspuru-Guzik,Feyn- man’s clock, a new variational principle, and parallel-in- time quantum dynamics , PNAS 110, E3901 (2013)

2013
[18]

Moreva, G

E. Moreva, G. Brida, M. Gramegna, V. Giovannetti, L. Maccone, and M. Genovese, Time from quantum entan- glement: An experimental illustration , Phys. Rev. A 89, 052122 (2014)

2014
[19]

Massar, P

S. Massar, P. Spindel, A. F. Varon and C. Wunderlich, Investigating the emergence of time in stationary states with trapped ions, Phys. Rev. A 92 030102(R)1 (2015)

2015
[20]

Pab´ on, L Reb´ on, S

D. Pab´ on, L Reb´ on, S. Bordakevich, N. Gigena, A. Boette, C. Iemmi, R. Rossignoli, and S. Ledesma, Parallel-in-time optical simulation of history states , Phys. Rev. A 99, 062333 (2019)

2019
[21]

Boette, R

A. Boette, R. Rossignoli, N. Gigena and M. Cerezo, System-time entanglement in a discrete-time model , Phys. Rev. A 93 0621271 (2016)

2016
[22]

Boette and R

A. Boette and R. Rossignoli, History states of systems and operators, Phys. Rev. A 98 032108 (2018)

2018
[23]

A. P. Majtey, A. Vald´ es-Hern´ andez, C. G. Maglione and A. R. Plastino, Entropic characterization of quantum states with maximal evolution under given energy con- straints, Entropy 21, 770 (2019)

2019
[24]

Schild, Time in quantum mechanics: A fresh look at the continuity equation, Phys

A. Schild, Time in quantum mechanics: A fresh look at the continuity equation, Phys. Rev. A, 98, 052113 (2018)

2018
[25]

R. V. Buniy and S. D. H. Hsu, Everything is entangled , Phys. Lett. B 718, 233 (2012)

2012
[26]

D. N. Page, Density matrix of the Universe , Phys. Rev. D 34 (1986) 2267

1986
[27]

V. G. Gurzadyan and A. A. Kocharyan, On pure and mixed quantum states of the Universe , Mod. Phys. Lett. A 4 (1989) 507

1989
[28]

Deutsch, Quantum mechanics near closed timelike lines, Phys

D. Deutsch, Quantum mechanics near closed timelike lines, Phys. Rev. D 44 (1991) 3197

1991
[29]

Anandan and Y

J. Anandan and Y. Aharonov, Meaning of the Density Matrix, Found. Phys. Lett. 12, 571 (1999)

1999
[30]

Relative State

H. Everett III, “Relative State” Formulation of Quantum Mechanics, Rev. Mod. Phys. 29, 454 (1957)

1957
[31]

Wehrl, General properties of entropy, Rev

A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221 (1978)

1978
[32]

M. A. Nielsen and J. Kempe, Separable States Are More Disordered Globally than Locally , Phys. Rev. Lett. 86, 5184 (2001)

2001
[33]

Batle, A

J. Batle, A. R. Plastino, M. Casas and A. Plastino, Con- ditional q-entropies and quantum separability: a numeri- cal exploration, J. Phys. A: Math. Gen. 35, 10311 (2002)

2002
[34]

Horodecki and M

R. Horodecki and M. Horodecki, Information-theoretic aspects of inseparability of mixed states , Phys. Rev. A 54, 1838 (1996)

1996
[35]

Tsallis, S

C. Tsallis, S. Lloyd, and M. Baranger Peres criterion for separability through nonextensive entropy , Phys. Rev. A 63, 042104 (2001)

2001
[36]

Rossignoli and N

R. Rossignoli and N. Canosa, Generalized entropic crite- rion for separability, Phys. Rev. A 66, 042306 (2002)

2002

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Goldstein, J

S. Goldstein, J. L. Lebowitz, R. Tumulka and N. Zanghi, Canonical typicality, Phys. Rev. Lett. 96, 050403 (2006)

work page 2006

[2] [2]

Deﬀner and W

S. Deﬀner and W. Zurek, Foundations of statistical me- chanics from symmetries of entanglement , New J. Phys. 18 063013 (2016)

work page 2016

[3] [3]

D. N. Page and W. K. Wootters, Evolution without Evo- lution: Dynamics Described by Stationary Observables , Phys. Rev. D 27, 2885 (1983)

work page 1983

[4] [4]

W. K. Wootters, “Time” replaced by quantum correla- tions, Int. J. Theor. Phys. 23, 701 (1984)

work page 1984

[5] [5]

B. S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory, Phys. Rev. 160, 1113 (1967)

work page 1967

[6] [6]

J. B. Hartle and S. W. Hawking, Wave function of the Universe, Phys. Rev. D 28, 2960 (1983)

work page 1983

[7] [7]

Rovelli, Quantum mechanics without time: A model , Phy

C. Rovelli, Quantum mechanics without time: A model , Phy. Rev. D 42, 8, 2638, (1990)

work page 1990

[8] [8]

Gambini, R

R. Gambini, R. A. Porto, J. Pullin, and S. Torterolo, Conditional probabilities with Dirac observables and the problem of time in quantum gravity , Phys. Rev. D 79, 041501(R) (2009)

work page 2009