Metric Congruence in Finite-Dimensional Non-Hermitian Quantum Mechanics

Ramirez Romina; Reboiro Marta

arxiv: 2606.28868 · v1 · pith:73XUAGBPnew · submitted 2026-06-27 · 🧮 math-ph · math.MP

Metric Congruence in Finite-Dimensional Non-Hermitian Quantum Mechanics

Ramirez Romina , Reboiro Marta This is my paper

Pith reviewed 2026-06-30 08:42 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords non-Hermitian quantum mechanicsKrein spaceHilbert space isomorphismmetric representationsRobertson uncertainty relationfinite-dimensional systemstwo-level spin model

0 comments

The pith

Non-Hermitian systems admit equivalent descriptions in isomorphic Hilbert spaces when states, metrics, and operators are transported through the isomorphism.

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

The paper shows that a non-Hermitian quantum system can be formulated in different but isomorphic Hilbert spaces, with the Krein space carrying an indefinite metric made explicitly isomorphic to the standard Hilbert space. Physical states, metrics, and operators must be mapped through this isomorphism so that equivalent representations carry identical physical content. A sympathetic reader would care because this mapping resolves apparent inconsistencies, such as violations of the Robertson uncertainty relation, by attributing them to mismatched representations rather than intrinsic failures. The claim is tested on a two-level non-Hermitian spin model.

Core claim

Within the Krein-space formalism, the vector space endowed with an indefinite Krein metric can be explicitly related to the standard Hilbert space through a suitable isomorphism. Physical states, metrics, and operators must be transported through the corresponding Hilbert-space isomorphism. In this way, equivalent representations of the same system can be used without changing the physical content of the theory. Apparent violations of the Robertson uncertainty relation arise only when operators and states are kept fixed while the metric is changed, reflecting a mismatch of representations rather than a failure of the principle.

What carries the argument

The Hilbert-space isomorphism that maps the indefinite Krein metric space onto the standard positive-definite Hilbert space, allowing consistent transport of states and operators between representations.

If this is right

Equivalent representations of a non-Hermitian system can be used interchangeably without altering physical predictions.
The Robertson uncertainty relation is preserved when representations are aligned via the isomorphism.
Apparent violations of uncertainty relations indicate only a mismatch between metric, states, and operators.
The same transport procedure applies to any finite-dimensional non-Hermitian Hamiltonian treated in the Krein-space setting.

Where Pith is reading between the lines

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

Choosing a convenient metric representation could simplify numerical work on specific observables while the isomorphism guarantees unchanged physics.
The same congruence argument may clarify apparent inconsistencies reported in other non-Hermitian models once representations are aligned.
The finite-dimensional result supplies a concrete test bed for checking whether infinite-dimensional non-Hermitian systems obey analogous transport rules.

Load-bearing premise

That the Robertson uncertainty relation continues to hold once states and operators have been correctly transported through the isomorphism.

What would settle it

A direct calculation on the two-level non-Hermitian spin model that shows a violation of the Robertson relation after the states and operators have been transported through the isomorphism.

read the original abstract

We study metric representations in finite-dimensional non-Hermitian quantum mechanics. The main purpose of this work is to emphasize that the physical description of a non-Hermitian system may be formulated in different, but isomorphic, Hilbert spaces. In particular, within the Krein-space formalism, we show that the vector space endowed with an indefinite Krein metric can be explicitly related to the standard Hilbert space through a suitable isomorphism. This observation is essential for a consistent description of non-Hermitian Hamiltonians. Physical states, metrics, and operators must be transported through the corresponding Hilbert-space isomorphism. In this way, equivalent representations of the same system can be used without changing the physical content of the theory. We illustrate these theoretical aspects by studying a two-level non-Hermitian spin model. We use the Robertson uncertainty relation as a consistency test. Apparent violations can arise when operators and states are kept fixed while the metric is changed, and therefore reflect a mismatch of representations rather than a failure of the uncertainty principle.

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 notes that different metric representations of finite-dimensional non-Hermitian systems are equivalent only after transporting states, operators, and the metric under the isomorphism, with a two-level model showing apparent uncertainty violations are representation artifacts.

read the letter

The main point is that in finite-dimensional non-Hermitian QM you can switch between isomorphic Hilbert spaces (including via Krein metrics) but must map everything—states, operators, and the metric itself—to keep the physics unchanged. The authors illustrate this with a two-level non-Hermitian spin model and check the Robertson relation as a consistency test, showing mismatches arise only when the mapping is skipped.

The paper does a clean job spelling out the isomorphism and why it matters for avoiding inconsistent descriptions. The example is concrete and directly demonstrates the fix for the apparent violation, which is useful as a reminder.

The limitation is that none of this is new. Finite-dimensional vector spaces of equal dimension are always linearly isomorphic; once the map is applied the uncertainty relation holds by direct substitution. The consistency test is therefore automatic rather than a derivation of new physics. The work stays within standard linear algebra and does not resolve open questions or introduce new tools.

This is aimed at people already working in the Krein-space approach to non-Hermitian QM who might hit representation mismatches in their own calculations. Readers outside that narrow subfield or looking for fresh results will find little to engage with.

I would send it to peer review. It is a modest clarification note that could reduce confusion in the literature, and referees can verify the example is presented without gaps.

Referee Report

0 major / 3 minor

Summary. The paper claims that in finite-dimensional non-Hermitian quantum mechanics, the physical description of a system may be formulated in different but isomorphic Hilbert spaces, including those with indefinite Krein metrics. The vector space with a Krein metric can be related to the standard Hilbert space via a suitable isomorphism, and physical states, metrics, and operators must be transported under this map to preserve physical content. This is illustrated with a two-level non-Hermitian spin model, where the Robertson uncertainty relation is used as a consistency test; apparent violations are attributed to mismatches of representations rather than failures of the principle.

Significance. If correct, the result provides a useful clarification on maintaining consistency when switching metric representations in non-Hermitian QM. It rests on the elementary fact that all finite-dimensional vector spaces of equal dimension are linearly isomorphic, together with standard properties of the Krein-space formalism, and offers an explicit two-level illustration rather than new physical predictions. No free parameters, ad-hoc axioms, or invented entities are introduced.

minor comments (3)

[Abstract] Abstract: the statement that the Krein-space vector space 'can be explicitly related' to the standard Hilbert space would be strengthened by a forward reference to the section or equation where the explicit isomorphism map (and its action on the metric) is constructed.
[Two-level model] Two-level model section: the transport of the metric and operators under the isomorphism should be written out explicitly (e.g., as a matrix equation) before applying the Robertson relation, so that the consistency test can be verified by direct substitution.
Notation: the distinction between the original and transported metric should be made with consistent symbols throughout, to avoid any appearance that the metric is being altered while states and operators are held fixed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and positive recommendation of minor revision. The referee's summary accurately captures the central claim of our work: that equivalent physical descriptions of finite-dimensional non-Hermitian systems can be formulated in isomorphic Hilbert spaces, including those equipped with Krein metrics, with states, operators, and metrics transported accordingly. The two-level illustration via the Robertson relation is correctly noted as a consistency check rather than a source of new predictions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's argument rests on the elementary linear-algebra fact that all finite-dimensional vector spaces of equal dimension are isomorphic, together with the requirement that states, operators and metrics must be transported under the isomorphism to preserve physical content. The Robertson relation is invoked only as a consistency test after the map is applied; apparent violations are attributed to representation mismatch rather than any new derivation. No equations, fitted parameters, or self-citations appear as load-bearing steps that reduce the claimed result to its own inputs by construction. The two-level example is presented purely for illustration. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument relies on the existence of a suitable isomorphism between the indefinite Krein metric space and the standard Hilbert space; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The vector space endowed with an indefinite Krein metric can be explicitly related to the standard Hilbert space through a suitable isomorphism.
Invoked in the abstract as essential for consistent description of non-Hermitian Hamiltonians.

pith-pipeline@v0.9.1-grok · 5701 in / 1209 out tokens · 31267 ms · 2026-06-30T08:42:51.969015+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 32 canonical work pages · 3 internal anchors

[1]

Ashida, Z

author author Y. Ashida , author Z. Gong , \ and\ author M. Ueda ,\ title title Non-hermitian physics , \ 10.1080/00018732.2021.1876991 journal journal Advances in Physics \ volume 69 ,\ pages 249--435 ( year 2020 ) ,\ http://arxiv.org/abs/https://doi.org/10.1080/00018732.2021.1876991 https://doi.org/10.1080/00018732.2021.1876991 NoStop

work page doi:10.1080/00018732.2021.1876991 2021
[2]

Gao , author K

author author H. Gao , author K. Wang , author L. Xiao , author M. Nakagawa , author N. Matsumoto , author D. Qu , author H. Lin , author M. Ueda , \ and\ author P. Xue ,\ title title Experimental observation of the yang-lee quantum criticality in open quantum systems , \ 10.1103/PhysRevLett.132.176601 journal journal Phys. Rev. Lett. \ volume 132 ,\ page...

work page doi:10.1103/physrevlett.132.176601 2024
[3]

Ram \' rez , author M

author author R. Ram \' rez , author M. Reboiro , \ and\ author D. Tielas ,\ title title Exceptional points from the hamiltonian of a hybrid physical system: Squeezing and anti-squeezing , \ https://doi.org/10.1140/epjd/e2020-10218-1 journal journal Eur. Phys. J. D \ volume 74 ,\ pages 193 ( year 2020 ) NoStop

work page doi:10.1140/epjd/e2020-10218-1 2020
[4]

Elganainy , author K

author author R. Elganainy , author K. Makris , author M. Khajavikhan , author Z. Musslimani , author S. Rotter , \ and\ author D. Christodoulides ,\ title title Non-hermitian physics and pt symmetry , \ 10.1038/nphys4323 journal journal Nature Physics \ volume 14 ,\ pages 11--19 ( year 2018 ) NoStop

work page doi:10.1038/nphys4323 2018
[5]

Minganti , author A

author author F. Minganti , author A. Miranowicz , author R. W. \ Chhajlany , author I. I. \ Arkhipov , \ and\ author F. Nori ,\ title title Hybrid-liouvillian formalism connecting exceptional points of non-hermitian hamiltonians and liouvillians via postselection of quantum trajectories , \ 10.1103/PhysRevA.101.062112 journal journal Phys. Rev. A \ volum...

work page doi:10.1103/physreva.101.062112 2020
[6]

Chen , author M

author author W. Chen , author M. Abbasi , author Y. N. \ Joglekar , \ and\ author K. W. \ Murch ,\ title title Quantum jumps in the non-hermitian dynamics of a superconducting qubit , \ https://doi.org/10.1103/PhysRevLett.127.140504 journal journal Phys. Rev. Lett. \ volume 127 ,\ pages 140504 ( year 2021 ) NoStop

work page doi:10.1103/physrevlett.127.140504 2021
[7]

Naghiloo , author M

author author M. Naghiloo , author M. Abbasi , author Y. N. \ Joglekar , \ and\ author K. W. \ Murch ,\ title title Quantum state tomography across the exceptional point in a single dissipative qubit , \ @noop journal journal Nature Physics \ volume 15 ,\ pages 1232--1236 ( year 2019 ) NoStop

2019
[8]

Kato ,\ @noop title Perturbation theory for linear operators ,\ Vol.\ volume 132 \ ( publisher Springer ,\ year 1966 ) NoStop

author author T. Kato ,\ @noop title Perturbation theory for linear operators ,\ Vol.\ volume 132 \ ( publisher Springer ,\ year 1966 ) NoStop

1966
[9]

author author W. D. \ Heiss ,\ title title The physics of exceptional points , \ 10.1088/1751-8113/45/44/444016 journal journal Journal of Physics A: Mathematical and Theoretical \ volume 45 ,\ pages 444016 ( year 2012 ) NoStop

work page doi:10.1088/1751-8113/45/44/444016 2012
[10]

author author G. A. \ Álvarez , author E. P. \ Danieli , author P. R. \ Levstein , \ and\ author H. M. \ Pastawski ,\ title title Environmentally induced quantum dynamical phase transition in the spin swapping operation , \ 10.1063/1.2193518 journal journal J. Chem. Phys. \ volume 124 ,\ pages 194507 ( year 2006 ) NoStop

work page doi:10.1063/1.2193518 2006
[11]

author author E. J. \ Bergholtz , author J. C. \ Budich , \ and\ author F. K. \ Kunst ,\ title title Exceptional topology of non-hermitian systems , \ 10.1103/RevModPhys.93.015005 journal journal Rev. Mod. Phys. \ volume 93 ,\ pages 015005 ( year 2021 ) NoStop

work page doi:10.1103/revmodphys.93.015005 2021
[12]

Rahmani , author A

author author A. Rahmani , author A. Opala , \ and\ author M. Matuszewski ,\ title title Exceptional points and phase transitions in non-hermitian nonlinear binary systems , \ 10.1103/PhysRevB.109.085311 journal journal Phys. Rev. B \ volume 109 ,\ pages 085311 ( year 2024 ) NoStop

work page doi:10.1103/physrevb.109.085311 2024
[13]

Fu \ and\ author G

author author Y. Fu \ and\ author G. Xianlong ,\ title title Anatomy of non-hermitian dynamical quantum phase transitions , \ 10.1103/92w1-71rt journal journal Phys. Rev. B \ volume 112 ,\ pages L140302 ( year 2025 ) NoStop

work page doi:10.1103/92w1-71rt 2025
[14]

Shukla , author R

author author N. Shukla , author R. Modak , \ and\ author B. P. \ Mandal ,\ title title Uncertainty relation for non-hermitian systems , \ 10.1103/PhysRevA.107.042201 journal journal Phys. Rev. A \ volume 107 ,\ pages 042201 ( year 2023 ) NoStop

work page doi:10.1103/physreva.107.042201 2023
[15]

author author D. C. \ Brody ,\ title title Biorthogonal quantum mechanics , \ https://doi.org/10.1088/1751-8113/47/3/035305 journal journal J. Phys. A: Math. Theor. \ volume 47 ,\ pages 035305 ( year 2013 ) NoStop

work page doi:10.1088/1751-8113/47/3/035305 2013
[16]

author author C. M. \ Bender , author D. C. \ Brody , \ and\ author H. F. \ Jones ,\ title title Complex extension of quantum mechanics , \ 10.1103/PhysRevLett.89.270401 journal journal Phys. Rev. Lett. \ volume 89 ,\ pages 270401 ( year 2002 ) ,\ note erratum: 92, 119902 (2004). Stop

work page doi:10.1103/physrevlett.89.270401 2002
[17]

author author A. Mostafazadeh ,\ title title Time-dependent pseudo-hermitian hamiltonians defining a unitary quantum system and uniqueness of the metric operator , \ https://doi.org/10.1016/j.physletb.2007.04.064 journal journal Phys. Lett. B \ volume 650 ,\ pages 208--212 ( year 2007 ) NoStop

work page doi:10.1016/j.physletb.2007.04.064 2007
[18]

Fring \ and\ author M

author author A. Fring \ and\ author M. H. Y. \ Moussa ,\ title title Non-hermitian swanson model with a time-dependent metric , \ 10.1103/PhysRevA.94.042128 journal journal Phys. Rev. A \ volume 94 ,\ pages 042128 ( year 2016 ) NoStop

work page doi:10.1103/physreva.94.042128 2016
[19]

U ber den anschaulichen inhalt der quantentheoretischen kinematik und mechanik , \ @noop journal journal Zeitschrift f \

author author W. Heisenberg ,\ title title \"U ber den anschaulichen inhalt der quantentheoretischen kinematik und mechanik , \ @noop journal journal Zeitschrift f \"u r Physik \ volume 43 ,\ pages 172--198 ( year 1927 ) NoStop

1927
[20]

author author H. P. \ Robertson ,\ title title The uncertainty principle , \ 10.1103/PhysRev.34.163 journal journal Phys. Rev. \ volume 34 ,\ pages 163--164 ( year 1929 ) NoStop

work page doi:10.1103/physrev.34.163 1929
[21]

o dinger ,\ title title Zum heisenbergschen unsch \

author author E. Schr \"o dinger ,\ title title Zum heisenbergschen unsch \"a rfeprinzip , \ @noop journal journal Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse \ volume 14 ,\ pages 296--303 ( year 1930 ) NoStop

1930
[22]

Maccone \ and\ author A

author author L. Maccone \ and\ author A. K. \ Pati ,\ title title Stronger uncertainty relations for all incompatible observables , \ 10.1103/PhysRevLett.113.260401 journal journal Phys. Rev. Lett. \ volume 113 ,\ pages 260401 ( year 2014 ) NoStop

work page doi:10.1103/physrevlett.113.260401 2014
[23]

Zhao , author X

author author X. Zhao , author X. Yu , author W. Zhou , author C. Zhang , author J.-S. \ Xu , author C.-F. \ Li , \ and\ author G.-C. \ Guo ,\ title title Experimental investigation of uncertainty relations for non-hermitian operators , \ 10.1103/PhysRevLett.132.070203 journal journal Phys. Rev. Lett. \ volume 132 ,\ pages 070203 ( year 2024 ) NoStop

work page doi:10.1103/physrevlett.132.070203 2024
[24]

\ Wang , author Z.-A

author author Y.-T. \ Wang , author Z.-A. \ Wang , author Z.-P. \ Li , author X.-D. \ Zeng , author J.-M. \ Ren , author W. Liu , author Y.-Z. \ Yang , author N.-J. \ Guo , author L.-K. \ Xie , author J.-Y. \ Liu , author Y.-H. \ Ma , author J.-S. \ Tang , author C. Zhang , author C.-F. \ Li , \ and\ author G.-C. \ Guo ,\ title title Direct non-hermitian ...

work page doi:10.1103/d7bd-qtpw 2025
[25]

\ Song , author Y.-F

author author X.-F. \ Song , author Y.-F. \ Ren , author S. Liu , author X.-H. \ Chen , \ and\ author Y. Turek ,\ title title Stronger sum uncertainty relations for non-hermitian operators , \ 10.1103/PhysRevA.111.012221 journal journal Phys. Rev. A \ volume 111 ,\ pages 012221 ( year 2025 ) NoStop

work page doi:10.1103/physreva.111.012221 2025
[26]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian , \ 10.1063/1.1418246 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 205--214 ( year 2002 a ) NoStop

work page doi:10.1063/1.1418246 2002
[27]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity versus pt-symmetry ii: A complete characterization of non-hermitian hamiltonians with a real spectrum , \ 10.1063/1.1461427 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 2814--2816 ( year 2002 b ) ,\ http://arxiv.org/abs/math-ph/0110016 arXiv:math-ph/0110016 [math-ph] NoStop

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.1461427 2002
[28]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity versus pt-symmetry iii: Equivalence of pseudo-hermiticity and the presence of antilinear symmetries , \ 10.1063/1.1489072 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 3944--3951 ( year 2002 c ) NoStop

work page doi:10.1063/1.1489072 2002
[29]

Sim , author N

author author K. Sim , author N. Defenu , author P. Molignini , \ and\ author R. Chitra ,\ title title Observables in non-hermitian systems: A methodological comparison , \ 10.1103/PhysRevResearch.7.013325 journal journal Physical Review Research \ volume 7 ( year 2025 ),\ 10.1103/PhysRevResearch.7.013325 NoStop

work page doi:10.1103/physrevresearch.7.013325 2025
[30]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity and generalized pt- and cpt-symmetries , \ 10.1063/1.1539304 journal journal Journal of Mathematical Physics \ volume 44 ,\ pages 974--989 ( year 2003 ) NoStop

work page doi:10.1063/1.1539304 2003
[31]

Ramirez \ and\ author M

author author R. Ramirez \ and\ author M. Reboiro ,\ title title Dynamics of finite dimensional non-hermitian systems with indefinite metric , \ https://doi.org/10.1063/1.5075628 journal journal J. Math. Phys. \ volume 60 ,\ pages 012106 ( year 2019 ) NoStop

work page doi:10.1063/1.5075628 2019
[32]

Kitagawa \ and\ author M

author author M. Kitagawa \ and\ author M. Ueda ,\ title title Squeezed spin states , \ 10.1103/PhysRevA.47.5138 journal journal Phys. Rev. A \ volume 47 ,\ pages 5138--5143 ( year 1993 ) NoStop

work page doi:10.1103/physreva.47.5138 1993
[33]

Decoherence, the measurement problem, and interpretations of quantum mechanics

author author M. Schlosshauer ,\ title title Decoherence, the measurement problem, and interpretations of quantum mechanics , \ 10.1103/RevModPhys.76.1267 journal journal Rev. Mod. Phys. \ volume 76 ,\ pages 1267--1305 ( year 2005 ) NoStop

work page internal anchor Pith review doi:10.1103/revmodphys.76.1267 2005
[34]

Uncertainty inequalities in a non-Hermitian scenario

author author Y. Alvarez , author M. Portesi , author R. Ramirez , \ and\ author M. Reboiro ,\ title title Uncertainty inequalities in a non-hermitian scenario: the problem of the metric , \ @noop journal journal arXiv preprint arXiv:2512.24437 \ ( year 2025 ) NoStop

work page internal anchor Pith review Pith/arXiv arXiv 2025
[35]

author author G. A. \ Starkov , author M. V. \ Fistul , \ and\ author I. M. \ Eremin ,\ title title Formation of exceptional points in pseudo-hermitian systems , \ 10.1103/PhysRevA.108.022206 journal journal Phys. Rev. A \ volume 108 ,\ pages 022206 ( year 2023 ) NoStop

work page doi:10.1103/physreva.108.022206 2023
[36]

Bagarello ,\ title title Uncertainty relation for non-hermitian operators , \ 10.1088/1751-8121/acfbc7 journal journal J

author author F. Bagarello ,\ title title Uncertainty relation for non-hermitian operators , \ 10.1088/1751-8121/acfbc7 journal journal J. Phys. A: Math. Theor. \ volume 56 ,\ pages 425201 ( year 2023 ) NoStop

work page doi:10.1088/1751-8121/acfbc7 2023

[1] [1]

Ashida, Z

author author Y. Ashida , author Z. Gong , \ and\ author M. Ueda ,\ title title Non-hermitian physics , \ 10.1080/00018732.2021.1876991 journal journal Advances in Physics \ volume 69 ,\ pages 249--435 ( year 2020 ) ,\ http://arxiv.org/abs/https://doi.org/10.1080/00018732.2021.1876991 https://doi.org/10.1080/00018732.2021.1876991 NoStop

work page doi:10.1080/00018732.2021.1876991 2021

[2] [2]

Gao , author K

author author H. Gao , author K. Wang , author L. Xiao , author M. Nakagawa , author N. Matsumoto , author D. Qu , author H. Lin , author M. Ueda , \ and\ author P. Xue ,\ title title Experimental observation of the yang-lee quantum criticality in open quantum systems , \ 10.1103/PhysRevLett.132.176601 journal journal Phys. Rev. Lett. \ volume 132 ,\ page...

work page doi:10.1103/physrevlett.132.176601 2024

[3] [3]

Ram \' rez , author M

author author R. Ram \' rez , author M. Reboiro , \ and\ author D. Tielas ,\ title title Exceptional points from the hamiltonian of a hybrid physical system: Squeezing and anti-squeezing , \ https://doi.org/10.1140/epjd/e2020-10218-1 journal journal Eur. Phys. J. D \ volume 74 ,\ pages 193 ( year 2020 ) NoStop

work page doi:10.1140/epjd/e2020-10218-1 2020

[4] [4]

Elganainy , author K

author author R. Elganainy , author K. Makris , author M. Khajavikhan , author Z. Musslimani , author S. Rotter , \ and\ author D. Christodoulides ,\ title title Non-hermitian physics and pt symmetry , \ 10.1038/nphys4323 journal journal Nature Physics \ volume 14 ,\ pages 11--19 ( year 2018 ) NoStop

work page doi:10.1038/nphys4323 2018

[5] [5]

Minganti , author A

author author F. Minganti , author A. Miranowicz , author R. W. \ Chhajlany , author I. I. \ Arkhipov , \ and\ author F. Nori ,\ title title Hybrid-liouvillian formalism connecting exceptional points of non-hermitian hamiltonians and liouvillians via postselection of quantum trajectories , \ 10.1103/PhysRevA.101.062112 journal journal Phys. Rev. A \ volum...

work page doi:10.1103/physreva.101.062112 2020

[6] [6]

Chen , author M

author author W. Chen , author M. Abbasi , author Y. N. \ Joglekar , \ and\ author K. W. \ Murch ,\ title title Quantum jumps in the non-hermitian dynamics of a superconducting qubit , \ https://doi.org/10.1103/PhysRevLett.127.140504 journal journal Phys. Rev. Lett. \ volume 127 ,\ pages 140504 ( year 2021 ) NoStop

work page doi:10.1103/physrevlett.127.140504 2021

[7] [7]

Naghiloo , author M

author author M. Naghiloo , author M. Abbasi , author Y. N. \ Joglekar , \ and\ author K. W. \ Murch ,\ title title Quantum state tomography across the exceptional point in a single dissipative qubit , \ @noop journal journal Nature Physics \ volume 15 ,\ pages 1232--1236 ( year 2019 ) NoStop

2019

[8] [8]

Kato ,\ @noop title Perturbation theory for linear operators ,\ Vol.\ volume 132 \ ( publisher Springer ,\ year 1966 ) NoStop

author author T. Kato ,\ @noop title Perturbation theory for linear operators ,\ Vol.\ volume 132 \ ( publisher Springer ,\ year 1966 ) NoStop

1966

[9] [9]

author author W. D. \ Heiss ,\ title title The physics of exceptional points , \ 10.1088/1751-8113/45/44/444016 journal journal Journal of Physics A: Mathematical and Theoretical \ volume 45 ,\ pages 444016 ( year 2012 ) NoStop

work page doi:10.1088/1751-8113/45/44/444016 2012

[10] [10]

author author G. A. \ Álvarez , author E. P. \ Danieli , author P. R. \ Levstein , \ and\ author H. M. \ Pastawski ,\ title title Environmentally induced quantum dynamical phase transition in the spin swapping operation , \ 10.1063/1.2193518 journal journal J. Chem. Phys. \ volume 124 ,\ pages 194507 ( year 2006 ) NoStop

work page doi:10.1063/1.2193518 2006

[11] [11]

author author E. J. \ Bergholtz , author J. C. \ Budich , \ and\ author F. K. \ Kunst ,\ title title Exceptional topology of non-hermitian systems , \ 10.1103/RevModPhys.93.015005 journal journal Rev. Mod. Phys. \ volume 93 ,\ pages 015005 ( year 2021 ) NoStop

work page doi:10.1103/revmodphys.93.015005 2021

[12] [12]

Rahmani , author A

author author A. Rahmani , author A. Opala , \ and\ author M. Matuszewski ,\ title title Exceptional points and phase transitions in non-hermitian nonlinear binary systems , \ 10.1103/PhysRevB.109.085311 journal journal Phys. Rev. B \ volume 109 ,\ pages 085311 ( year 2024 ) NoStop

work page doi:10.1103/physrevb.109.085311 2024

[13] [13]

Fu \ and\ author G

author author Y. Fu \ and\ author G. Xianlong ,\ title title Anatomy of non-hermitian dynamical quantum phase transitions , \ 10.1103/92w1-71rt journal journal Phys. Rev. B \ volume 112 ,\ pages L140302 ( year 2025 ) NoStop

work page doi:10.1103/92w1-71rt 2025

[14] [14]

Shukla , author R

author author N. Shukla , author R. Modak , \ and\ author B. P. \ Mandal ,\ title title Uncertainty relation for non-hermitian systems , \ 10.1103/PhysRevA.107.042201 journal journal Phys. Rev. A \ volume 107 ,\ pages 042201 ( year 2023 ) NoStop

work page doi:10.1103/physreva.107.042201 2023

[15] [15]

author author D. C. \ Brody ,\ title title Biorthogonal quantum mechanics , \ https://doi.org/10.1088/1751-8113/47/3/035305 journal journal J. Phys. A: Math. Theor. \ volume 47 ,\ pages 035305 ( year 2013 ) NoStop

work page doi:10.1088/1751-8113/47/3/035305 2013

[16] [16]

author author C. M. \ Bender , author D. C. \ Brody , \ and\ author H. F. \ Jones ,\ title title Complex extension of quantum mechanics , \ 10.1103/PhysRevLett.89.270401 journal journal Phys. Rev. Lett. \ volume 89 ,\ pages 270401 ( year 2002 ) ,\ note erratum: 92, 119902 (2004). Stop

work page doi:10.1103/physrevlett.89.270401 2002

[17] [17]

author author A. Mostafazadeh ,\ title title Time-dependent pseudo-hermitian hamiltonians defining a unitary quantum system and uniqueness of the metric operator , \ https://doi.org/10.1016/j.physletb.2007.04.064 journal journal Phys. Lett. B \ volume 650 ,\ pages 208--212 ( year 2007 ) NoStop

work page doi:10.1016/j.physletb.2007.04.064 2007

[18] [18]

Fring \ and\ author M

author author A. Fring \ and\ author M. H. Y. \ Moussa ,\ title title Non-hermitian swanson model with a time-dependent metric , \ 10.1103/PhysRevA.94.042128 journal journal Phys. Rev. A \ volume 94 ,\ pages 042128 ( year 2016 ) NoStop

work page doi:10.1103/physreva.94.042128 2016

[19] [19]

U ber den anschaulichen inhalt der quantentheoretischen kinematik und mechanik , \ @noop journal journal Zeitschrift f \

author author W. Heisenberg ,\ title title \"U ber den anschaulichen inhalt der quantentheoretischen kinematik und mechanik , \ @noop journal journal Zeitschrift f \"u r Physik \ volume 43 ,\ pages 172--198 ( year 1927 ) NoStop

1927

[20] [20]

author author H. P. \ Robertson ,\ title title The uncertainty principle , \ 10.1103/PhysRev.34.163 journal journal Phys. Rev. \ volume 34 ,\ pages 163--164 ( year 1929 ) NoStop

work page doi:10.1103/physrev.34.163 1929

[21] [21]

o dinger ,\ title title Zum heisenbergschen unsch \

author author E. Schr \"o dinger ,\ title title Zum heisenbergschen unsch \"a rfeprinzip , \ @noop journal journal Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse \ volume 14 ,\ pages 296--303 ( year 1930 ) NoStop

1930

[22] [22]

Maccone \ and\ author A

author author L. Maccone \ and\ author A. K. \ Pati ,\ title title Stronger uncertainty relations for all incompatible observables , \ 10.1103/PhysRevLett.113.260401 journal journal Phys. Rev. Lett. \ volume 113 ,\ pages 260401 ( year 2014 ) NoStop

work page doi:10.1103/physrevlett.113.260401 2014

[23] [23]

Zhao , author X

author author X. Zhao , author X. Yu , author W. Zhou , author C. Zhang , author J.-S. \ Xu , author C.-F. \ Li , \ and\ author G.-C. \ Guo ,\ title title Experimental investigation of uncertainty relations for non-hermitian operators , \ 10.1103/PhysRevLett.132.070203 journal journal Phys. Rev. Lett. \ volume 132 ,\ pages 070203 ( year 2024 ) NoStop

work page doi:10.1103/physrevlett.132.070203 2024

[24] [24]

\ Wang , author Z.-A

author author Y.-T. \ Wang , author Z.-A. \ Wang , author Z.-P. \ Li , author X.-D. \ Zeng , author J.-M. \ Ren , author W. Liu , author Y.-Z. \ Yang , author N.-J. \ Guo , author L.-K. \ Xie , author J.-Y. \ Liu , author Y.-H. \ Ma , author J.-S. \ Tang , author C. Zhang , author C.-F. \ Li , \ and\ author G.-C. \ Guo ,\ title title Direct non-hermitian ...

work page doi:10.1103/d7bd-qtpw 2025

[25] [25]

\ Song , author Y.-F

author author X.-F. \ Song , author Y.-F. \ Ren , author S. Liu , author X.-H. \ Chen , \ and\ author Y. Turek ,\ title title Stronger sum uncertainty relations for non-hermitian operators , \ 10.1103/PhysRevA.111.012221 journal journal Phys. Rev. A \ volume 111 ,\ pages 012221 ( year 2025 ) NoStop

work page doi:10.1103/physreva.111.012221 2025

[26] [26]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian , \ 10.1063/1.1418246 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 205--214 ( year 2002 a ) NoStop

work page doi:10.1063/1.1418246 2002

[27] [27]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity versus pt-symmetry ii: A complete characterization of non-hermitian hamiltonians with a real spectrum , \ 10.1063/1.1461427 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 2814--2816 ( year 2002 b ) ,\ http://arxiv.org/abs/math-ph/0110016 arXiv:math-ph/0110016 [math-ph] NoStop

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.1461427 2002

[28] [28]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity versus pt-symmetry iii: Equivalence of pseudo-hermiticity and the presence of antilinear symmetries , \ 10.1063/1.1489072 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 3944--3951 ( year 2002 c ) NoStop

work page doi:10.1063/1.1489072 2002

[29] [29]

Sim , author N

author author K. Sim , author N. Defenu , author P. Molignini , \ and\ author R. Chitra ,\ title title Observables in non-hermitian systems: A methodological comparison , \ 10.1103/PhysRevResearch.7.013325 journal journal Physical Review Research \ volume 7 ( year 2025 ),\ 10.1103/PhysRevResearch.7.013325 NoStop

work page doi:10.1103/physrevresearch.7.013325 2025

[30] [30]

author author A. Mostafazadeh ,\ title title Pseudo-hermiticity and generalized pt- and cpt-symmetries , \ 10.1063/1.1539304 journal journal Journal of Mathematical Physics \ volume 44 ,\ pages 974--989 ( year 2003 ) NoStop

work page doi:10.1063/1.1539304 2003

[31] [31]

Ramirez \ and\ author M

author author R. Ramirez \ and\ author M. Reboiro ,\ title title Dynamics of finite dimensional non-hermitian systems with indefinite metric , \ https://doi.org/10.1063/1.5075628 journal journal J. Math. Phys. \ volume 60 ,\ pages 012106 ( year 2019 ) NoStop

work page doi:10.1063/1.5075628 2019

[32] [32]

Kitagawa \ and\ author M

author author M. Kitagawa \ and\ author M. Ueda ,\ title title Squeezed spin states , \ 10.1103/PhysRevA.47.5138 journal journal Phys. Rev. A \ volume 47 ,\ pages 5138--5143 ( year 1993 ) NoStop

work page doi:10.1103/physreva.47.5138 1993

[33] [33]

Decoherence, the measurement problem, and interpretations of quantum mechanics

author author M. Schlosshauer ,\ title title Decoherence, the measurement problem, and interpretations of quantum mechanics , \ 10.1103/RevModPhys.76.1267 journal journal Rev. Mod. Phys. \ volume 76 ,\ pages 1267--1305 ( year 2005 ) NoStop

work page internal anchor Pith review doi:10.1103/revmodphys.76.1267 2005

[34] [34]

Uncertainty inequalities in a non-Hermitian scenario

author author Y. Alvarez , author M. Portesi , author R. Ramirez , \ and\ author M. Reboiro ,\ title title Uncertainty inequalities in a non-hermitian scenario: the problem of the metric , \ @noop journal journal arXiv preprint arXiv:2512.24437 \ ( year 2025 ) NoStop

work page internal anchor Pith review Pith/arXiv arXiv 2025

[35] [35]

author author G. A. \ Starkov , author M. V. \ Fistul , \ and\ author I. M. \ Eremin ,\ title title Formation of exceptional points in pseudo-hermitian systems , \ 10.1103/PhysRevA.108.022206 journal journal Phys. Rev. A \ volume 108 ,\ pages 022206 ( year 2023 ) NoStop

work page doi:10.1103/physreva.108.022206 2023

[36] [36]

Bagarello ,\ title title Uncertainty relation for non-hermitian operators , \ 10.1088/1751-8121/acfbc7 journal journal J

author author F. Bagarello ,\ title title Uncertainty relation for non-hermitian operators , \ 10.1088/1751-8121/acfbc7 journal journal J. Phys. A: Math. Theor. \ volume 56 ,\ pages 425201 ( year 2023 ) NoStop

work page doi:10.1088/1751-8121/acfbc7 2023