Quantum Dynamics and Information Measures in PT and Anti-PT-Symmetric Systems

Amir Ahmadi; Javad T. Firouzjaee; Roozbeh H. Asgari

REVIEW 4 major objections 5 minor 50 references

Quantum Dynamics and Information Measures in PT and Anti-PT-Symmetric Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Anti-PT-symmetric qubits in a bosonic bath lose coherence and grow entropy more slowly than PT-symmetric qubits.

desk verdict The anti-PT robustness claim is an artifact of an unequal parameter point; the QSL and Rényi plots are new but derivative. read the letter →

arxiv 2508.03169 v2 pith:W4KBJLW6 submitted 2025-08-05 quant-ph

Amir Ahmadi , Roozbeh H. Asgari , Javad T. Firouzjaee This is my paper · ORCID

classification quant-ph

keywords PTsymmetryanti-PTnon-HermitianHamiltoniandecoherenceRényientanglemententropyquantumspeedlimitspin-bosonmodelqubitdynamics

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

A two-level system governed by a non-Hermitian Hamiltonian with anti-PT symmetry is argued to resist decoherence better than its PT-symmetric counterpart when both are weakly coupled to a bosonic bath. The paper maps the non-Hermitian dynamics to an equivalent Hermitian one and isolates the decoherence factor $D(t)=e^{-\omega_0^2\gamma(t)}$, whose anti-PT effective frequency is much smaller at the parameters studied. As a result the off-diagonal coherences live longer, the Rényi entropies of orders 1, 2, and $\infty$ grow more slowly, and the quantum speed limit shows a rapid early rise followed by gradual decay. If correct, the claim gives a symmetry-based route to longer-lived quantum memories and more secure quantum key distribution.

What carries the argument

The load-bearing object is a time-dependent Dyson map $\eta(t)=e^{-\theta t(1+V_B)}$, which turns the diagonalized non-Hermitian anti-PT Hamiltonian into a Hermitian qubit-plus-bath Hamiltonian. In that rotated frame the reduced density matrix keeps its diagonal populations and multiplies its off-diagonal coherences by $D(t)=e^{-\omega_0^2\gamma(t)}$, where $\gamma(t)$ is the finite-temperature bath dephasing integral. The PT-symmetric case is instead diagonalized by a time-independent similarity transformation, giving the same coherence factor with its own frequency. The comparison of the two symmetry-dependent frequencies $\omega_{\mathrm{PT},0}$ and $\omega_{\mathrm{APT},0}$ is what turns a symmetry classification into a quantitative prediction about coherence time, entropy growth, and quantum speed-limit velocity.

What would settle it

A decisive check is to retune the parameters so the two effective frequencies coincide ($\xi^2+\delta^2-\theta^2=\alpha^2-\xi^2-\delta^2$) and then compare the times at which the coherences $D_{\mathrm{PT}}(t)=e^{-\omega_{\mathrm{PT},0}^2\gamma(t)}$ and $D_{\mathrm{APT}}(t)=e^{-\omega_{\mathrm{APT},0}^2\gamma(t)}$ drop to $1/e$; if the PT qubit then lasts as long or longer, the robustness ordering is parameter-selected rather than symmetry-derived. A second check is to compute the Rényi entropies from the unmapped density matrix and see whether the slower-entropy-growth ordering survives.

Watch

Extended reading notes

Core claim

The central claim is that moving from a PT-symmetric non-Hermitian Hamiltonian $H_{\mathrm{PT}}$, which commutes with $PT$, to an anti-PT-symmetric Hamiltonian $H_{\mathrm{APT}}$, which anticommutes with $PT$, changes the dephasing rate while keeping the qubit in a two-dimensional Hilbert space. In the diagonal frame the coherence decays as $D(t)=e^{-\omega_0^2\gamma(t)}$, with $\omega_{\mathrm{PT},0}^2=\xi^2+\delta^2-\theta^2$ and $\omega_{\mathrm{APT},0}^2=\alpha^2-\xi^2-\delta^2$. At the parameter set used in the figures the anti-PT frequency is much smaller, so its coherences and all finite Rényi entropies evolve more slowly. The paper also reports that increasing the non-Hermitian parameters slows decoherence in both systems, and it concludes that anti-PT-symmetric qubits are better suited than PT-symmetric ones for information-preserving open-system tasks.

Load-bearing premise

The comparison that produces the headline ordering is made at one parameter point, where the anti-PT decay coefficient is roughly seven times smaller than the PT one, and the entropy claims are taken from the Dyson-rotated density matrix; if either choice is changed, the claimed anti-PT advantage may weaken or reverse.

Editorial extensions

If this is right

If the central claim holds, an anti-PT qubit at the parameter set taken from the comparison source retains coherence and low Rényi entropy for longer than the PT qubit at the same raw couplings, making it a candidate for quantum memory.
Raising the non-Hermitian parameters ($\theta$ for PT; $\xi,\delta$ for anti-PT) slows decoherence and entropy growth, so non-Hermiticity functions as a coherence-preserving control knob in both symmetry classes.
The quantum speed-limit velocity rises at early times and then decays, and larger non-Hermitian couplings make that rise-and-fall profile smoother.
The zero-order Rényi entropy stays at $\log 2$ throughout, so neither system leaves the two-dimensional qubit subspace while decohering.
The min-entropy behavior implies that anti-PT qubits leak less information to the environment, which the paper connects to improved quantum-key-distribution security.

Reading between the lines

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

Editorial inference: the quantitative anti-PT advantage is evaluated at a single point where $\omega_{\mathrm{APT},0}^2=0.03$ and $\omega_{\mathrm{PT},0}^2=0.23$; matching the two effective frequencies by retuning $\alpha$ or $\theta$ would provide a sharper test of whether robustness is a property of the symmetry itself.
Editorial inference: entropies are computed on the Dyson-rotated density matrix; computing the same Rényi entropies from the original non-Hermitian density matrix could give different curves, so reporting both versions is a natural extension.
Editorial inference: the Dyson-map construction should generalize to multi-qubit registers, qutrits, or different bath spectral densities, and the predicted ordering is testable in current quantum simulators.
Editorial inference: because the model is pure dephasing, the advantage concerns phase coherence only; adding energy-relaxation channels could alter the comparison, a possibility the paper does not address.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Short version: this is a legitimate but narrow extension of Cen and Saxena (PR A 105, 022404) and Gardas et al., and its headline claim — anti-PT qubits are more robust to dephasing than PT qubits — does not survive a fair parameter comparison. The new content (QSL curves, q=2 and q=∞ Rényi entropies) is real but is a direct consequence of the same decoherence factor D(t)=e^{-ω_0^2 γ(t)}.

What is actually new: Figures 3, 6, and 7 do not appear in [5] or [8]. The authors compute QSL velocity curves and higher-order Rényi entropies for both symmetries, and the algebra connecting the Dyson map to the decoherence function is mostly self-consistent. They also correctly note that S0=log 2 is an identity for a rank-2 state and that the q→1 limit reproduces von Neumann entropy; those checks are honest.

The soft spots are the usual ones, but one is load-bearing. The central comparison is made at a single point (ξ=0.81, δ=0.56, θ=0.86, α=1) imported from [5], where ω_PT0^2=0.230 and ω_APT0^2=0.030. Since D(t)=e^{-ω_0^2 γ(t)} for both systems with the same γ(t), the slower decoherence and slower entropy growth of the anti-PT qubit follow entirely from the smaller ω_APT0^2. No fair mapping between the two symmetry classes is defined, so the abstract's general claim that anti-PT systems show enhanced robustness is not supported. The second concern is that the entropies are computed on the Dyson-rotated density matrix ρ_D^h; the authors never show that its von Neumann or Rényi spectrum matches the physical qubit's reduced state. That may be fixable, but it is currently an assumption.

There are also mechanical defects: Eqs. (38)-(39) for the PT case use ω_APT0 instead of ω_PT0; Eq. (54) claims h_D is Hermitian despite containing an anti-Hermitian term θ t \tilde{V}_B; the QSL formula (74) is quoted without derivation or citation; and the min-entropy remark 'S∞=n' contradicts Eq. (83) (it should be log n for a maximally mixed state in n dimensions). No spectral density is specified beyond the figure parameters, and no code or data is provided, which limits reproducibility.

Bottom line: the paper is not a waste of time — it is a workmanlike extension of known results, and the QSL/entropy plots are new. But the comparative robustness claim needs to be redone under an equal-ω_0 mapping, or dropped. With that fix and the typo cleanup, it could become a modest but useful contribution. I would send it to a competent referee, but I would not cite it in its present form.

Referee Report

4 major / 5 minor

Summary. The manuscript studies a spin-boson model of a qubit whose system Hamiltonian is either PT-symmetric or anti-PT-symmetric. For the PT case the authors use a similarity transformation to a Hermitian diagonal form, while for the anti-PT case they use a time-dependent Dyson map. They derive phase-evolution and decoherence functions, a Mandelstam-Tamm-type quantum speed limit, and Rényi entropies of orders 0, 1, 2, and infinity, and they conclude that anti-PT-symmetric systems show enhanced robustness against decoherence, slower entropy growth, and longer coherence times than PT-symmetric systems.

Significance. If the central comparative claim were established, the paper would be of interest to the community working on non-Hermitian quantum systems and open-system quantum information. The analytic machinery—similarity transformation for the PT qubit and time-dependent Dyson map for the anti-PT qubit—is a useful formal exercise, and the explicit expressions for the decoherence factor and the Rényi entropies are clearly presented. However, the paper's advertised headline result is not supported by the evidence as presented, because the comparison is made at a single parameter point where the effective coupling in the decoherence factor is roughly seven times larger for the PT system than for the anti-PT system; moreover, the entropy quantities are computed on a Dyson-rotated density matrix without proving that this rotation preserves the physical qubit's entropy spectrum. These issues are load-bearing for the abstract and conclusion.

major comments (4)

[Sec. III.B, Eqs. (38)-(39)] The decoherence factor and phase factor for the PT-symmetric qubit are written with omega_APT_0 rather than omega_PT_0 defined in Eq. (23). This is not a harmless notational slip: the central comparison in Figures 1, 2, 5, 6, and 7 depends on which frequency enters D(t)=exp[-omega_0^2 gamma(t)]. Either Eq. (38) should contain omega_PT_0, or the authors must explain why the anti-PT frequency controls PT dynamics.
[Sec. III.C and Sec. V, Figs. 1-7] The claim that anti-PT systems are more robust against decoherence is evaluated at the single point (xi=0.81, delta=0.56, theta=0.86, alpha=1) taken from Ref. [5]. At this point omega_PT_0^2 = delta^2+xi^2-theta^2 = 0.230, while omega_APT_0^2 = alpha^2-xi^2-delta^2 = 0.030. Since gamma(t) in Eq. (41) is identical for both systems, the slower decay, longer coherence time, and slower Rényi entropy growth follow directly from the smaller omega_APT_0^2, not from the anticommutation symmetry. A fair comparison requires a parameter mapping that equalizes omega_0^2; under that mapping the decoherence factors coincide exactly. The authors should either provide such a mapping and revisit the claims, or restrict all comparative conclusions to the specific parameter regime and state clearly that no symmetry-based advantage is claimed at equal effective coupling.
[Sec. III.B, Eqs. (45)-(46), and Sec. V] The Rényi and von Neumann entropies are computed from the Dyson-rotated density matrix rho_D^h = eta rho_D eta^dagger (Eq. (46); used in Eq. (69)). Since eta is not unitary, this congruence does not in general preserve the eigenvalue spectrum, and therefore does not in general preserve the von Neumann or Rényi entropies. The manuscript gives no proof that rho_D^h and the physical qubit density matrix are related by a unitary or isospectral transformation. Without such a proof, the entropy curves in Section V characterize an auxiliary Hermitian picture, not necessarily the entropy of the original PT or anti-PT qubit. This point must be resolved before the information-theoretic claims can be accepted.
[Sec. IV, Eqs. (74)-(77)] The quantum speed limit results are presented as plots, but the manuscript does not specify the Liouville superoperator L(rho(t)), the fidelity F, or the Bures angle for the specific PT and anti-PT density matrices. The reader cannot verify that the QSL curves in Fig. 3 follow from the model equations. Please provide the explicit expressions used in the computation, or cite the exact formula applied to rho_S(t) and rho_D^h(t).

minor comments (5)

[Sec. V.B.1] The statement that S_0 remains at the constant value 0.693 is simply log(rank(rho)) = log 2 for any rank-2 density matrix; it is an identity, not a dynamical result, and should be presented as such.
[Sec. V.B.2] The statement that the first-order Rényi entropy reproduces the von Neumann entropy in the limit q -> 1 is true by definition and does not serve as a numerical check of the model; the perfect agreement in Fig. 5 is therefore tautological.
[Throughout] There are several typographical and formatting errors, including 'thier' in the Acknowledgments, 'the the' in Sec. V.B.2, 'Stmmetric' in Fig. 5a, and 'asumming' in Sec. V.B. These should be corrected.
[Fig. 4] The caption refers to blue and orange curves but the figure appears to have no legend; please add a legend or describe the curves unambiguously.
[Sec. III.C, Figs. 1 and 2] The parameter lists in the captions are inconsistent: Fig. 1 includes theta=0.86 but not alpha, while Fig. 2 includes alpha but not theta. Please list the full parameter set for each panel.

Circularity Check

2 steps flagged · score 2.0 of 10

Derivation is self-contained; only the presented S0 and Rényi-1/von Neumann 'checks' are definitional tautologies, and the anti-PT advantage remains parameter-dependent rather than a fitted or circular result.

self definitional [Section V.B.1, 'Zero order Rényi Entropy for PT and Anti-PT Symmetric Systems']
"For both the PT and Anti-PT-symmetric systems, we can see from Fig.4 that this maximum entropy remains at the constant value 0.693 in time and independent of the non-Hermitian terms θ (for PT) and ξ, δ (for Anti-PT). Using Eq.80 for S0, we can easily see that log(2) = 0.693 and thus the rank of the density matrix, and the dimension of the Hilbert space remain 2."

S0 is defined in Eq. (80) as log rank(ρ). For any 2×2 qubit density matrix, the maximum possible S0 is log 2, so the reported 0.693 is just the definitional upper bound rather than a symmetry-dependent dynamical result. The statement that S0 remains at 0.693 is equivalent to asserting that the reduced state keeps rank 2; it is not an independent entropy calculation and does not involve the PT or anti-PT structure beyond the qubit dimension.
self definitional [Section V.B.2, 'First-order Rényi and the Von Neumann Entropy in PT-Symmetric and Anti-PT-Symmetric Qubits']
"As shown in section (VB2), the 1st Reniy entropy reaches the Von Neumann entropy upon taking the limit q→ 1. This can be verified for the PT-symmetric and Anti-PT-symmetric systems, as we have shown in figure 5 .The solid curves, which are the the Reniy entropies of order 1, perfectly fit the dotted curves, which are the Von Neumann entropies."

By definition, the Rényi entropy in Eq. (78) is S_q = (1/(1−q)) log Tr[ρ^q], and its q→1 limit is the von Neumann entropy −Tr[ρ log ρ] in Eq. (79), for every density matrix. The perfect fit between the first-order Rényi entropy and the von Neumann entropy is therefore an identity built into the definitions, not a numerical or dynamical validation of the PT or anti-PT models.

full rationale

The main derivation chain is a direct calculation: the decoherence factors in Eqs. (38)/(73), the Dyson-map construction in Eqs. (45)-(54), and the entropy curves all follow from the stated Hamiltonians and bath model rather than from fitting or from self-citation. I found no parameter that is fitted to a target and then 'predicted,' and the cited prior work (ref. [5]) is external, not authored by the present authors. The central comparative claim that anti-PT systems are more robust is a direct evaluation of the same decoherence envelope D(t)=exp(−ω0^2 γ(t)) at the imported parameter point (ξ=0.81, δ=0.56, θ=0.86, α=1): since ω_PT0^2=0.230 and ω_APT0^2=0.030, the slower anti-PT dephasing is fixed by these parameter values rather than by the anti-commutation symmetry alone. That is a parameter-dependence/fairness concern, not a circularity. The genuinely circular items are the two entropy 'results' flagged above: they restate definitions of S0 and of the q→1 Rényi limit. These are minor and do not affect the core dynamical calculation, so the overall circularity score is low.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The ledger records everything the central claim rests on that is not derived in the paper: the pseudo-Hermitian framework (standard math), the pure-dephasing spin-boson model and thermal initial state (domain assumption), the specific Dyson map ansatz, the ungrounded QSL velocity formula, and the assertion that the rotated-picture entropy equals the physical entropy. Free parameters are the hand-chosen comparison point from [5] and the spectral-density parameters that appear only in figure captions. No invented physical entities are introduced.

free parameters (2)

Comparison parameter point (α, θ, ξ, δ) = α=1, θ=0.86, ξ=0.81, δ=0.56
Chosen from [5] and used in all comparison plots; it fixes ω_PT_0^2=0.230 vs ω_APT_0^2=0.030, so the central 'anti-PT is more robust' ordering is guaranteed by this hand-picked point.
Spectral density parameters J0, ωc, μ, β = J0=1, ωc=1, μ=-0.5, β=0.5
Given only in figure captions; the explicit J(ω) form is never written in the text, so the plotted integrals are not reproducible from the prose alone.

assumptions (5)

standard math Unbroken PT symmetry implies a real spectrum and a positive Hermitian metric η with H† = ηHη^{-1} (Mostafazadeh pseudo-Hermitian framework).
Invoked in Sec II.C to legitimize unitary time evolution in the η-inner product after rotation; used for the PT part of the derivation.
domain assumption Pure-dephasing spin-boson model, H = H_S + H_B + H_S ⊗ V_B, with the bath initially in a thermal state and only information flow (no heat exchange).
Adopted in Sec II.D and used throughout; the entire result D(t)=e^{-ω_0^2 γ(t)} follows from this model and its initial conditions.
ad hoc to paper Dyson map ansatz η(t) = exp[-θt(1+V_B)] with ∂tφ = -θ (Eqs. 51-53).
This specific time-dependent ansatz, inherited from [5], is imposed to make h_D Hermitian; the anti-PT dynamics and its phase and decoherence functions depend on it.
domain assumption Quantum speed limit velocity formula V_QSL = ||L(ρ)||_op/(2 sin L cos L) of Eq. (74).
Stated without derivation and attributed loosely to [21]; the form is not found there and diverges at L=0 and L=π/2.
domain assumption The von Neumann and Rényi entropies of the Dyson-rotated density matrix equal the physical qubit's entropies under the non-unitary maps.
Used throughout Sec V; eigenvalues are not invariant under non-unitary similarity, so this equality is assumed but never shown.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantum Dynamics and Information Measures in PT and Anti-PT-Symmetric Systems." pith.science (2026). https://pith.science/paper/W4KBJLW6

@misc{pith2026250803169,
  author       = {Pith},
  title        = {Pith review of: Quantum Dynamics and Information Measures in PT and Anti-PT-Symmetric Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4KBJLW6}},
  note         = {Machine review of arXiv:2508.03169}
}

read the original abstract

In this study, we investigate qubit dynamics under PT and Anti-PT-symmetric non-Hermitian Hamiltonians, focusing on phase evolution, decoherence, quantum speed limits (QSL), and R\'enyi entanglement entropies. Using similarity transformations and Dyson maps, we analyze the reduced density matrix evolution in bosonic environments. Anti-PT-symmetric systems show enhanced robustness against decoherence, with slower entropy growth and longer coherence times compared to PT-symmetric counterparts. QSL behavior is non-monotonic, reflecting rapid initial evolution followed by a gradual decrease. Higher-order R\'enyi entropies reveal that Anti-PT-symmetric qubits preserve quantum information more effectively, offering advantages for memory and cryptographic applications.

Figures

Figures reproduced from arXiv: 2508.03169 by the authors.

**Figure 2.** FIG. 2: Parameter values for the Anti - [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3: System parameters used for the simulations, for [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: Zero-order Rényi Entropy: Blue represents Anti - [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5: Solid curves represent the first order Rényi entropy and dotted curves exhibit Von Neumann entropy. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6: The 2nd Rényi entropy [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7: The [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 40 canonical work pages

[5]

H isPT -symmetric: [PT ,H ] = 0 (see Eq. (8)),

work page
[8]

(81) This measures the average uncertainty or entanglement entropy and is additive for independent systems, weighting all of their corresponding eigenstates linearly

q = 1 : Von Neumann Entropy The standard quantum entropy: S1(ρ) =−Tr[ρ logρ]. (81) This measures the average uncertainty or entanglement entropy and is additive for independent systems, weighting all of their corresponding eigenstates linearly. It has deep connections to thermodynamics, quantum channel capacities, and entanglement theory [2, 9]

work page
[1]

q = 0: Max-Entropy 13

work page
[2]

q = 1: Von Neumann Entropy 13

work page
[3]

q = 2: Collision Entropy 13

work page
[4]

q → ∞: Min-Entropy 13 C. Summary 13 Zero order Rényi Entropy forPT and Anti-PT Symmetric Systems 14 First-order Rényi and the Von Neumann Entropy inPT -Symmetric and Anti-PT -Symmetric Qubits 14 Second-order (collision) Rényi Entropy inPT -Symmetric and Anti-PT -Symmetric Qubits 15 Infinite-order (min) Rényi Entropy inPT -Symmetric and Anti-PT -Symmetric ...

work page Pith review arXiv 2025
[6]

ThePT -symmetry is unbroken: each eigenfunctionψ satisfiesPT ψ =ψ (cf. Eq. (10)). Then, according to [8], if the spectrum of the Hamiltonian is real, there exists a Hermitian, positive-definite operator η such thatH† =ηHη−1 (Eq. (13)). AlthoughPT and Anti-PT systems have different physical properties and definitions, their dynamics are governed by the sam...

work page 2013
[7]

max-entropy

q = 0 : Max-Entropy The Rényi entropy atq = 0 captures the logarithmic rank of the density matrix: S0(ρ) = log rank(ρ). (80) This quantity is insensitive to the actual eigenvalues and simply counts the number of non-zero ones, weighting all of their corresponding eigenstates with equal probabilities. It reflects the maximum possible entropy or largest pos...

work page

Show all 50 references

[9]

q = 2 : Collision Entropy Atq = 2, we find: S2(ρ) =− log Tr[ρ2]. (82) This entropy is directly related to the purity and discrimination between quantum states, and is frequently used to detect decoherence, quantify entanglement, and study complexity in many-body systems [39]
[10]

q → ∞ : Min-Entropy Taking the limitq→∞ , the Rényi entropy becomes: S∞(ρ) =− logλmax, (83) where λmax is the largest eigenvalue ofρ. This form, called the min-entropy, has operational meaning in single-shot information theory and bounds the optimal probability of correctly gu...
[11]

C. M. Bender and S. Boettcher, Physical Review Letters80, 5243 (1998)

1998
[12]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems (Oxford University Press, New York, 2002)

2002
[13]

Mostafazadeh, Journal of Physics A: Mathematical and General38, 6557 (2005)

A. Mostafazadeh, Journal of Physics A: Mathematical and General38, 6557 (2005)

2005
[14]

Mostafazadeh, Physics Letters B650, 208 (2007)

A. Mostafazadeh, Physics Letters B650, 208 (2007)

2007
[15]

Cen and A

J. Cen and A. Saxena, Physical Review A105, 022404 (2022)

2022
[16]

Y. Wang, S. Mukhopadhyay, and G. S. Agarwal, Physical Review Research4, 013131 (2022)

2022
[17]

W. Ma, H. Zhang, and J. Liu, Phys. Rev. A112, 012204 (2025)

2025
[18]

Gardas, S

B. Gardas, S. Deffner, and A. Saxena, Physical Review A94, 040101 (2016)

2016
[19]

M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information (Cambridge University Press, 2010)

2010
[20]

X. Xu, D. Zhang, D. Yu, Y. He, and Y. Liu, Preprint (2024), arXiv:2401.00549

2024 arXiv
[21]

El-Ganainy, M

R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and S. K. Ozdemir, Physical Review A97, 022109 (2018)

2018
[22]

Miri and A

M.-A. Miri and A. Alù, Science363
[23]

Zhang and X

S. Zhang and X. Lu, Sensors22, 3977 (2023)

2023
[24]

Hodaei, A

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017)

2017
[25]

Zhang, C

B.-W. Zhang, C. Shang, J. Y. Sun, Z.-C. Gu, and X. X. Yi, Phys. Rev. A111, 063702 (2025)

2025
[26]

S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, Nature Materials18, 783 (2019)

2019
[27]

Y. Choi, C. Hahn, J. W. Yoon, S. H. Song, P. Berini, and H.-G. Lee, Nature Communications9, 2182 (2018)

2018
[28]

P. Miao, J. Li, Z. Zhang, J. Zuo, and S. Fan, Light: Science and Applications13, 91 (2024)

2024
[29]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Physical Review Letters103, 093902 (2009)

2009
[30]

Deffner and E

S. Deffner and E. Lutz, Physical Review Letters111, 010402 (2013)

2013
[31]

Deffner and S

S. Deffner and S. Campbell, Journal of Physics A: Mathematical and Theoretical50, 453001 (2017)

2017
[32]

Funo and T

K. Funo and T. Sagawa, Entropy25, 254 (2023)

2023
[33]

Y. Li, Y. Luo, Z. Wang, X. Chen, and H. Zhai, Physical Review Letters126, 083604 (2021)

2021
[34]

Bhattacharya, A

U. Bhattacharya, A. Lakshminarayan, and S. Ghosh, Physical Review A101, 042319 (2020)

2020
[35]

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Physical Review X8, 031079 (2018)

2018
[36]

Yao and Z

S. Yao and Z. Wang, Physical Review Letters121, 086803 (2018)

2018
[37]

J. Bhat, M. Lone, S. Datta, G. Dar, and A. Farouk, Ukranian journal of physics68, 101 (2023)

2023
[38]

J. J. Sakurai and J. Napolitano,Modern Quantum Mechanics, 2nd ed. (Addison-Wesley, 1995). 18

1995
[39]

E. P. Wigner,Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959)

1959
[40]

Viola and E

L. Viola and E. Knill, Reports on Progress in Physics85, 066001 (2022)

2022
[41]

C. M. Bender, Reports on Progress in Physics70, 947 (2007)

2007
[42]

Lindblad, Communications in Mathematical Physics48, 119 (1976)

G. Lindblad, Communications in Mathematical Physics48, 119 (1976)

1976
[43]

Mostafazadeh, Journal of Mathematical Physics43, 205 (2002)

A. Mostafazadeh, Journal of Mathematical Physics43, 205 (2002)

2002
[44]

Mandelstam and I

L. Mandelstam and I. Tamm, J. Phys. (USSR)9, 249 (1945)

1945
[45]

Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. A67, 052109 (2003)

2003
[46]

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett.72, 3439 (1994)

1994
[47]

Tóth and I

G. Tóth and I. Apellaniz, J. Phys. A: Math. Theor.47, 424006 (2014)

2014
[48]

A. Rényi, inProceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Con- tributions to the Theory of Statistics (University of California Press, 1961) pp. 547–561

1961
[49]

Müller-Lennert, F

M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, Journal of Mathematical Physics54, 122203 (2013)

2013
[50]

Tomamichel,Quantum Information Processing with Finite Resources: Mathematical Foundations (Springer, 2015)

M. Tomamichel,Quantum Information Processing with Finite Resources: Mathematical Foundations (Springer, 2015)

2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [5]

H isPT -symmetric: [PT ,H ] = 0 (see Eq. (8)),

work page

[2] [8]

(81) This measures the average uncertainty or entanglement entropy and is additive for independent systems, weighting all of their corresponding eigenstates linearly

q = 1 : Von Neumann Entropy The standard quantum entropy: S1(ρ) =−Tr[ρ logρ]. (81) This measures the average uncertainty or entanglement entropy and is additive for independent systems, weighting all of their corresponding eigenstates linearly. It has deep connections to thermodynamics, quantum channel capacities, and entanglement theory [2, 9]

work page

[3] [1]

q = 0: Max-Entropy 13

work page

[4] [2]

q = 1: Von Neumann Entropy 13

work page

[5] [3]

q = 2: Collision Entropy 13

work page

[6] [4]

q → ∞: Min-Entropy 13 C. Summary 13 Zero order Rényi Entropy forPT and Anti-PT Symmetric Systems 14 First-order Rényi and the Von Neumann Entropy inPT -Symmetric and Anti-PT -Symmetric Qubits 14 Second-order (collision) Rényi Entropy inPT -Symmetric and Anti-PT -Symmetric Qubits 15 Infinite-order (min) Rényi Entropy inPT -Symmetric and Anti-PT -Symmetric ...

work page Pith review arXiv 2025

[7] [6]

ThePT -symmetry is unbroken: each eigenfunctionψ satisfiesPT ψ =ψ (cf. Eq. (10)). Then, according to [8], if the spectrum of the Hamiltonian is real, there exists a Hermitian, positive-definite operator η such thatH† =ηHη−1 (Eq. (13)). AlthoughPT and Anti-PT systems have different physical properties and definitions, their dynamics are governed by the sam...

work page 2013

[8] [7]

max-entropy

q = 0 : Max-Entropy The Rényi entropy atq = 0 captures the logarithmic rank of the density matrix: S0(ρ) = log rank(ρ). (80) This quantity is insensitive to the actual eigenvalues and simply counts the number of non-zero ones, weighting all of their corresponding eigenstates with equal probabilities. It reflects the maximum possible entropy or largest pos...

work page