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REVIEW 4 major objections 7 minor 30 references

Quantum simulacra

T0 review · 4 major / 7 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Changing a quantum system’s metric redefines its observables so measurements can emulate entanglement, squeezing, and Bell violations the Hamiltonian itself does not produce.

desk verdict Solid metric algebra packaged as a strong interpretive claim; the Bell example is a reconstruction, not a nonlocality result. read the letter →

arxiv 2607.24739 v1 pith:ARJKAY3Q submitted 2026-07-27 quant-ph

L. F. Alves da Silva , M. H. Y. Moussa This is my paper · ORCID
classification quant-ph
keywords quantumsimulacrapseudo-HermitianmetricsPOVMpostselectionmetric-dependententanglementBellinequalityDysonmapcontextualitysuperradiantphasetransition
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 argues that the inner-product metric used to describe a quantum system is not fixed scenery but an operational choice. By adopting a metric other than the usual L² product, one redefines which operators count as observables and therefore which experimental arrangements measure them. The resulting statistics can display squeezing of a free field, entanglement and teleportation of uncoupled qubits, a superradiant transition without light–matter coupling, and Bell-violating correlations from product states—phenomena the authors call quantum simulacra. The cost is that the new observables must be measured; the authors propose doing so with positive-operator-valued measures followed by postselection of a subensemble. A sympathetic reader cares because the construction relocates part of the burden of engineering quantum effects from Hamiltonian design to the choice of metric and measurement context, and because it supplies a metric-based reading of a recent experiment that reported Bell violation with unentangled photons.

What carries the argument

The positive-definite metric operator Θ = η†η built from a Dyson map η, together with a POVM-plus-postselection protocol: POVM elements are built from the biorthogonal (left) eigenvectors associated with Θ, the failure outcome E× is discarded, and the accepted subensemble is renormalized so that Born-rule probabilities and expectation values are those of the Θ-metric.

What would settle it

Prepare two uncoupled qubits in a product state, implement the paper’s POVM elements for the metric-induced spin observable without any entangling gate in the measurement stage, postselect, and check whether the concurrence or Bell correlator still reaches the values predicted for the Θ-metric; if the signals vanish once those gates are removed, the simulacrum collapses to ordinary postselection.

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Extended reading notes

Core claim

Quantum simulacra are phenomena that appear when a Hermitian or non-Hermitian system is treated in a metric other than standard L². The metric redefines the set of observables and the measurement context, so that postselected statistics can emulate a microscopic reality—squeezing, entanglement, teleportation, Bell violation, superradiance—distinct from the dynamics prescribed by the Hamiltonian alone. Photons separable in L² become entangled under the new metric, accounting for reported Bell inequality violation with unentangled photons.

Load-bearing premise

That measuring the redefined observables by POVM elements from the biorthogonal basis, then discarding failures and renormalizing, produces a genuinely new metric-dependent reality rather than ordinary L² statistics on a postselected subensemble whose discrimination already uses the global gates the simulacrum claims to avoid in the Hamiltonian.

Editorial extensions

If this is right

  • A free bosonic mode can exhibit unbounded quadrature squeezing solely by choice of metric and the associated quadrature measurement.
  • Uncoupled qubits can display metric-induced concurrence up to 1 and support a teleportation protocol once the corresponding POVM is postselected.
  • Product-state fourfold coincidence rates, normalized under the new metric, reproduce E(φA, φB) = cos(φA + φB) and thereby Bell violation.
  • A Tavis–Cummings-type coupling strong enough for an equilibrium superradiant transition can be realized by metric choice without leaving the rotating-wave regime.
  • Part of the engineering of effective interactions can be moved from Hamiltonian design onto the design of the measurement metric.

Reading between the lines

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

  • If the measurement stage must itself contain a CNOT or parity-controlled NOT to resolve the biorthogonal basis, simulacra may be re-describable as ordinary circuits whose entangling resources have simply been relocated from evolution to readout.
  • The same metric-plus-postselection pattern could be tried on continuous-variable cluster states or measurement-based computation to ask whether graph-state entanglement can be “metric-induced” from product inputs.
  • An experiment that reports both the accepted-subensemble correlators and the discarded E× rates would let one test whether the effective metric is uniquely fixed or under-determined by the postselection.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The manuscript extends Mostafazadeh's pseudo-Hermitian metric framework to Hermitian Hamiltonians: for Hermitian H, any positive metric Θ with [Θ,H]=0 defines, via the Dyson map η (Θ=η†η), a Hermitian partner h=ηHη⁻¹ and redefined observables O=η⁻¹oη whose expectation values are taken in the Θ-inner product. Since h generically contains interaction terms absent from H (by construction of η), measuring the pseudo-Hermitian observables reproduces statistics of a system with interactions the original Hamiltonian lacks — the authors call such phenomena "quantum simulacra." A measurement protocol is proposed (Sec. III): a POVM with elements En=q|χ̃n⟩⟨χ̃n| built from the biorthogonal basis, plus postselection discarding the failure outcome E×=1−qΘ, which yields the Born rule in the Θ-metric on the accepted subensemble. Worked examples include squeezing from a free/driven mode (Sec. IV), entanglement of uncoupled qubits (Sec. V), reproduction of the Wang et al. Bell-type correlator E(φA,φB)=cos(φA+φB) for product states (Sec. VI), teleportation (Sec. VII), and a Tavis–Cummings superradiant transition (Sec. VIII). The algebraic framework is standard and internally consistent.

Significance. If the claims are appropriately scoped, the paper offers a coherent unifying observation — postselection plus renormalized POVM statistics define an effective non-L² metric — and applies it to several nontrivial worked examples (squeezing, entanglement, teleportation, superradiance), with a concrete, in-principle implementable measurement protocol. The identification of the Wójcik & Wójcik mechanism with an effective metric is a genuinely neat reframing. However, the present draft overstates what is achieved: the 'simulacra' require, at the measurement stage, precisely the resources (entangling gates, per-setting preparation, strong couplings) whose absence from the Hamiltonian motivates the construction, and the Bell section cannot support any nonlocality claim. With honest resource accounting and toned-down interpretation, this would be a worthwhile contribution to the pseudo-Hermitian/QI literature; as written, its most quotable claims are not supported.

major comments (4)
  1. [VI, Eqs. (19)–(21)] The claimed correspondence with Wang et al. is achieved by choosing the product-state amplitudes a1,a2,b1,b2 as explicit functions of the phase settings (Eqs. (20a–d) contain cos²((φA+φB)/2) and Φ(φA+φB)). The text is candid about this ('we choose the coefficients of the initial product state to reproduce the phase-dependent fourfold-coincidence probabilities'), but the consequence is not drawn: for each setting pair a different state is prepared, so E(φA,φB)=cos(φA+φB) is inserted at preparation rather than extracted by the metric. In any Bell analysis, settings must be chosen independently of and after state preparation; a per-setting preparation trivially reproduces any correlator. The abstract's statement that 'photons that are separable in L² become entangled when analyzed within a new metric framework' should be qualified accordingly: what is reproduced is the numerical value of a
  2. [V, VI, Eqs. (17), (22)–(23); VII, Eq. (26)] The POVM elements En = q|χn⟩⟨χn| with |χn⟩ = η†|n⟩ (Eq. (22)) require, by the authors' own construction, a global entangling gate (CNOT in Sec. V, PCNOT in Sec. VII) to discriminate the biorthogonal basis. A joint entangling measurement on both 'parties' is precisely the resource a Bell test excludes, so the 'entanglement in metric Θ' carries no nonlocality content, and the teleportation of Sec. VII uses the full standard teleportation resource (Bell measurement plus two-control gate) dressed by η. More broadly, the claim (abstract, Sec. X) that simulacra 'provide a route to implementing quantum operations that lie beyond the reach of the L² metric' is undermined by the authors' own resource accounting: nothing beyond L² is implemented; an interaction is moved from the Hamiltonian to the measurement apparatus, at the cost of postselection. Please restate the claim as a relocation of reso
  3. [IV–VIII (choice of Dyson map)] In each example the Dyson map is chosen inside the algebra that already contains the desired coupling: su(1,1) quadratics for squeezing (Sec. IV), the bilinear σ⁺₁σ⁻₂ − σ⁻₁σ⁺₂ for entanglement (Eq. (8)), collective aS⁺ − a†S⁻ for Tavis–Cummings (Sec. VIII). The partner h then 'exhibits' the coupling by construction. The manuscript acknowledges this in passing (Sec. V: η 'carries the bilinear interaction... required for the metric-induced construction'), but the recurring framing that 'phenomena absent from the Hamiltonian emerge from the choice of metric' (Secs. I, IV-A, X) overstates what is shown. The honest statement is: given a target interaction, one may hide it in η and recover its statistics on a postselected subensemble. This is still a coherent and possibly useful observation, but the manuscript should state it as such, ideally with a short general proposition characterizing whi
  4. [III; IV, Fig. 1(c); VIII, Eqs. (29)–(31)] No success-probability analysis is given for the POVM+postselection scheme, yet it is load-bearing for the practical claims. Positivity of E× = 1 − qΘ requires q ≤ 1/λmax(Θ). The most striking results occur at the boundary of the metric domain — unbounded squeezing as ζ→±1 (Sec. IV), maximal concurrence at ζ=±1 (Eq. (16)), critical coupling at ζ± (Eq. (31)) — precisely where ‖Θ‖ diverges or becomes singular, forcing q and hence the acceptance probability toward zero. There appears to be an unavoidable tradeoff between the strength of the simulacrum and the postselection rate; without quantifying it (e.g., acceptance probability vs. ζ for the Sec. V example), claims of feasibility — in particular the Sec. VIII assertion that the simulacrum makes the superradiant transition 'feasible' where the RWA blocks it — are unsupported. Note also that measuring n̂ = η⁻¹(a†a)η in Sec. VIII requires i
minor comments (7)
  1. [I, final paragraph] Typo: 'and thus thequantum contextuality' (missing space; also 'redefined' reads awkwardly).
  2. [V, Eq. (12)] The right-hand side e^{+2ϵ(z)} of the normalization condition is surprising given that the left side is a product of state-amplitude factors; please verify the exponent and state the convention for |a_i|²+|b_i|² ≠ 1 that permits complex amplitudes here (the a_i, b_i are written with modulus/phase factored, which is nonstandard notation).
  3. [III] The POVM scale q is introduced with only 'q > 0 chosen such that E× ≥ 0'; state the explicit bound q ≤ 1/λmax(Θ) and note that the choice of q does not affect the accepted-subensemble statistics.
  4. [IV-B (U× postselection gate)] The continuous-spectrum extension assumes the existence of a unitary U× mapping E× onto |0⟩⟨0|; since E× = 1 − qΘ is generally full-rank, such a unitary does not exist in the single-mode Hilbert space and an ancillary system is required. Please make the ancilla explicit.
  5. [Fig. 1(c)] Fig. 1(c) would benefit from marking the divergent-squeezing limits and stating the quadrature uncertainty relation for the pseudo-Hermitian X1, X2 (do ΔX1·ΔX2 saturate a Θ-modified bound?).
  6. [VI, final paragraph] Reference 12 is described as having 'correctly identified' the postselection/renormalization mechanism; given that this paper's added contribution is the metric reinterpretation, a crisper delineation of what is new relative to Ref. 12 would help the reader.
  7. [References] Several references have formatting inconsistencies (e.g., Ref. [14] 'J.Math.Phys.45, 932-946'; Ref. [26] journal name abbreviated differently).

Circularity Check

5 steps flagged · score 6.0 of 10

Desired couplings are inserted into the Dyson map η by ansatz, so the Hermitian partner h and the Bell correlator are recovered by construction; the Sec. VI amplitudes are explicitly tuned to the target cos(φ_A+φ_B).

  1. self definitional [Sec. V, Eqs. (8)–(10)]
    "We introduce a Dyson map η=exp[ϵσz_1+χ(ασ+_1σ-_2−βσ-_1σ+_2)] which carries the bilinear interaction that, although absent from the Hamiltonian, is required for the metric-induced construction of entanglement between the two subsystems. ... The Hermitian counterpart of H then reads h=...−(2Δζ)/(1+ζ²)(σ+_1σ-_2+σ-_1σ+_2)"

    η is defined to contain exactly the bilinear coupling whose absence from H is being advertised. The similarity transform then puts that coupling into h by algebra, so the subsequent concurrence C(t)∝|ζ|/(1+ζ²) and the claim that ‘separability depends on the choice of metric’ are not derived phenomena but the direct image of the interaction terms written into η.

  2. self definitional [Sec. IV, text before Eq. (6) and Eqs. (5a)–(6)]
    "The Dyson map η is chosen within su(1,1) algebra, rather than within the Heisenberg–Weyl algebra associated with the linear terms in H, so that the metric itself induces the parametric amplification underlying the squeezing mechanism. The resulting Hermitian counterpart of H is given by h=...+(ωζ)/(1−ζ²)(a†²+a²)."

    The algebra of η is selected expressly to generate the quadratic squeezing term. Once that choice is made, h necessarily contains parametric amplification and the quadrature uncertainties ΔX_ℓ=e^{±r}/√2 follow by construction; the ‘simulacrum of the squeezing mechanism’ is the input ansatz renamed as an emergent metric effect.

3 more flagged steps
  1. fitted input called prediction [Sec. VI, Eqs. (20a)–(21) and surrounding text]
    "In order to establish the correspondence with the experiment, we choose the coefficients of the initial product state to reproduce the phase-dependent fourfold-coincidence probabilities. For simplicity, we evaluate the state at t=0, obtaining a_1=...[...(α/2β)e^{-2ϵ}cos²((φ_A+φ_B)/2)]^{1/4}, ... With this choice, the corresponding detection probabilities are p_↑↑=p_↓↓=¼[1+cos(φ_A+φ_B)], ... Consequently, the correlation function becomes E(φ_A,φ_B)=cos(φ_A+φ_B), thereby reproducing Eq. (18)"

    The product amplitudes are fitted—made explicit functions of φ_A+φ_B and of the metric parameters—so that the Θ-metric Born probabilities equal the target fourfold rates. The correlator identity E=cos(φ_A+φ_B) is then an arithmetic consequence of that fit, not an independent prediction or an explanation of the Wang et al. data from a metric principle alone.

  2. self definitional [Sec. VIII, Eqs. (29)–(30)]
    "We then define the Dyson map η=exp[ϵS_z+χ(α a S_+−β a† S_-)] ... For each choice of z ... the Hermitian counterpart of H takes the Tavis–Cummings model form, h=[ω_0−2Δζ²/(1+ζ²)]S_z+ω(a†a+1/2)+g(a S_++a† S_-), where ... g=2Δζ/√N(1+ζ²)."

    Collective raising/lowering factors are built into η; the light–matter coupling g in h is therefore present by the similarity transform. Declaring a ‘simulacrum of the Tavis–Cummings interaction’ and a metric-induced superradiant transition simply restates the coupling strength that was parametrized into η (via ζ).

  3. renaming known result [Sec. IX, opening paragraph]
    "In addition to the examples discussed above, several results already reported in the literature [24–27] also admit a natural interpretation as quantum simulacra. ... These examples are as genuine quantum simulacra because the corresponding amplification or correlation channels are absent in the standard-metric description."

    References [24–27] are prior works by the same group on PT-symmetric/bosonic models. Relabeling those constructions as ‘quantum simulacra’ imports the authors’ earlier ansatz-driven metrics under a new name without an independent derivation; the section is rebranding, not new evidence.

full rationale

The paper’s central constructive move is not an independent derivation of new dynamics from a fixed Hamiltonian, but a deliberate choice of metric generator η that already carries the interaction or amplification one wishes to see. Once η is so chosen, the similarity h=ηHη^{-1} exhibits those terms automatically, and expectation values/concurrence computed in the Θ-metric reproduce the engineered phenomenology. This pattern repeats for squeezing (η in su(1,1)), bipartite entanglement and teleportation (bilinear spin factors in η), and the Tavis–Cummings/superradiant case (collective J± factors in η). The load-bearing Bell ‘explanation’ goes further: product-state coefficients a_i,b_i are written as explicit functions of (φ_A+φ_B) chosen so that the postselected probabilities equal the Wang et al. fourfold rates; the correlator E=cos(φ_A+φ_B) then follows by arithmetic identity, not as a prediction. Self-citations to the authors’ prior PT-symmetric models (Sec. IX) are present but secondary—they rebrand earlier constructions as ‘simulacra’ rather than underwrite uniqueness. The framework therefore has real technical content (POVM+postselection protocol, extension of pseudo-Hermiticity to Hermitian H), yet the headline claim that metric choice makes entanglement/Bell violation ‘emerge’ for separable L² states reduces, in the paper’s own equations, to inputs placed in η and in the prepared amplitudes. Score 6 reflects partial, constructional circularity on the central examples without total vacuity of the surrounding formalism.

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

The central claim rests on standard pseudo-Hermitian algebra plus three paper-specific moves: (i) nontrivial metrics commuting with Hermitian H still redefine “reality” via observables; (ii) POVM failure discard plus renormalization is identified with the Θ-metric Born rule; (iii) metric parameters and, in the Bell example, state amplitudes may be chosen freely to produce the target phenomenology. Invented baggage is mostly the “simulacrum” entity as an interpretive category; independent evidence outside this framing is limited to existing postselected experiments reinterpreted after the fact.

free parameters (5)
  • metric parameter z (or ζ) = free in open interval; special values ζ→±1, ζ=1 used for maximal effects
    Continuous knob in (-z0,z0) or (-1,1) that sets the strength of squeezing, effective coupling, or concurrence prefactor; chosen to obtain the desired simulacrum, including maximal entanglement at |ζ|=1.
  • deformation parameters α, β
    Enter every Dyson map and the metric; constrained only by signs/products (e.g. αβ>0) and used to shape η and the biorthogonal basis.
  • POVM scale q = any q>0 with E_×≥0
    Must satisfy E_×=1−qΘ≥0; sets acceptance rate of the postselected subensemble but is not fixed by the theory.
  • Bell product-state amplitudes a1,a2,b1,b2 = phase- and Φ-dependent closed forms fixed to target correlator
    Sec. VI chooses these explicitly (Eqs. 20a–d) so that p↑↑,p↓↓,p↑↓,p↓↑ match the Wang fourfold pattern and E=cos(φ_A+φ_B).
  • drive/detuning and frequency scales (μ, ω, ωi, Δ, N)
    Physical scales in example Hamiltonians; not fitted to external datasets here, but together with ζ they set effective g and critical-metric windows (e.g. ζ± for superradiance).
assumptions (5)
  • domain assumption Mostafazadeh-style pseudo-Hermiticity: a positive metric Θ=η†η exists such that ΘH=H†Θ, partner h=ηHη⁻¹ is Hermitian in L², and expectation values match via ⟨ψ|O|ψ⟩_Θ=⟨ϕ|o|ϕ⟩.
    Taken from the cited pseudo-Hermitian framework (Sec. II, Eq. 1–3); standard in that literature, assumed without re-proof.
  • ad hoc to paper For Hermitian H, any Θ with [Θ,H]=0 is a legitimate alternative metric that can redefine the physically relevant observables O=η⁻¹oη and thereby the experimental context.
    Sec. II extension beyond the usual non-Hermitian motivation; the paper treats this commuting metric as an extra instrument for shaping reality, which is a foundational choice not forced by the algebra alone.
  • ad hoc to paper Discarding POVM failure outcomes E_× and renormalizing accepted counts implements the Born rule in the Θ-metric and defines a physical subensemble described by that metric.
    Sec. III equates postselected frequencies Nn/N with |⟨ψ̃_n|ψ⟩_Θ|²; this identification is the operational core of every simulacrum and is stronger than the bare POVM postulate.
  • standard math Standard POVM and postselection measurement postulates in quantum information (Nielsen & Chuang; discrete and continuous cases).
    Used to justify En=q|χ_n⟩⟨χ_n| and continuous analogues (Secs. III–IV).
  • domain assumption Concurrence and Bell correlators computed in the Hermitian partner representation witness entanglement/Bell nonlocality of the simulacrum in the same operational sense as in L².
    Secs. V–VI import C(t) and E(φ_A,φ_B) as if metric-redefined statistics inherit the usual foundational meaning; contested when postselection is essential.
invented entities (2)
  • quantum simulacrum
    purpose: Name phenomena that appear only after metric redefinition and metric-adapted measurement, allegedly distinct from Hamiltonian-prescribed reality.
    Umbrella concept introduced in the abstract and Sec. I; groups squeezing, entanglement, Bell, teleportation, and superradiance examples. Falsifiable content reduces to ordinary postselected statistics under specified POVMs, not a new physical field or particle.
  • POVM + postselection scheme for pseudo-Hermitian observables
    purpose: Provide an experimental route to measure O=η⁻¹oη by accepting only En outcomes and discarding E_×.
    Proposed in Sec. III and reused throughout; no demonstrated laboratory realization in the paper, though built from standard POVM parts.

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Pith. "Pith review of Quantum simulacra." pith.science (2026). https://pith.science/paper/ARJKAY3Q

@misc{pith2026260724739,
  author       = {Pith},
  title        = {Pith review of: Quantum simulacra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARJKAY3Q}},
  note         = {Machine review of arXiv:2607.24739}
}
abstract

Here we analyze the creation of quantum simulacra: phenomena that emerge from treating a Hermitian or non-Hermitian quantum system in metrics other than the standard $L^{2}$. Changing the metric redefines the set of system observables and thus the experimental arrangement for their measurement, making quantum contextuality and microscopic reality metric-dependent. The simulacra therefore consist, on the one hand, of a resizing of the status of quantum measurement, which has always occupied a central role in quantum mechanics: beyond the connection between quantum and classical dynamics, measurements performed in an appropriate metric can emulate a microscopic reality distinct from that prescribed by the Hamiltonian. On the other hand, simulacra provide a route to implementing quantum operations that lie beyond the reach of the $L^2$ metric. Quantum simulacra offer, as an example, an explanation for the recent observation of the violation of Bell inequalities with unentangled photons [Sci. Adv. \textbf{11}, eadr1794 (2025)]: photons that are separable in $L^2$ metric, become entangled when analyzed within a new metric framework. Simulacrum comes at the cost of implementing measurements of the metric-redefined observables; to address this challenge, we propose a scheme combining positive operator-valued measures with postselected subensembles.

Figures

Figures reproduced from arXiv: 2607.24739 by the authors.

Figure 1
Figure 1. (a) Success-failure mechanism implementing the postselection required by the con [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

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Works this paper leans on

30 extracted references · 2 linked inside Pith

  1. [1]

    C. M. Bender and S. Boettcher, Phys. Rev. Lett.80,5243 (1998)

    1998

  2. [2]

    This is the standard teleportation decomposition in the Hermitian representation, in which Alice performs a Bell’s measurement on qubits 2 and 3, while Bob holds qubit 1

    In this basis, the state|ϕ 123⟩can be written as |ϕ123⟩= X B∈{ψ±,ϕ±} (aB |↑1⟩+b B| ↓1⟩)⊗ |B23⟩/2,(24) with coefficients ( aψ±, bψ±) = ( ±a, b) and ( aϕ±, bϕ±) = ( ∓b,±a ). This is the standard teleportation decomposition in the Hermitian representation, in which Alice performs a Bell’s measurement on qubits 2 and 3, while Bob holds qubit 1. Given the meas...

  3. [3]

    POVM + postselection

    presented a framework that, in addition to the real spectrum, ensures norm conservation for pseudo-Hermitian dynamics. The key ingredient is a positive-definite metric operator that defines a modified Hilbert-space inner product, distinct from the standard L2, with respect to which the non-Hermitian Hamiltonian becomes self-adjoint. Since then, many theor...

  4. [4]

    POVM + post-selection

    In the pseudo-Hermitian representation, with H and |ψ123⟩ under the metric Θ, the corresponding observable is O123 = η−1 (I1 ⊗o 23)η , which describes how the Bell’s measurement can be performed taking into account the Θ-pseudo-Hermiticity. Expanding |ψ123⟩ in the biorthogonal basis defined by right and left eigenvector of O123, given by |ψs1,B⟩ = η−1 (|s...

  5. [5]

    C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys.40, 2201 (1999)

    1999

  6. [6]

    Mostafazadeh, J

    A. Mostafazadeh, J. Math. Phys.43, 205 (2002)

    2002

  7. [7]

    Klaiman, U

    S. Klaiman, U. G¨ unther, and N. Moiseyev, Phys. Rev. Lett.101, 080402 (2008); A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett.103, 093902 (2009); C. T. West, T. Kottos, and T. Prosen, Phys. Rev. Lett.104, 054102 (2010); Z. Lin, H. Ramezani, T. Eichelkraut, T. Kott...

    2008

  8. [8]

    Znojil, Phys

    M. Znojil, Phys. Rev. D78, 085003 (2008)

    2008

Show all 30 references
  1. [9]

    Gong and Q.-H

    J. Gong and Q.-H. Wang, J. Phys. A: Math. Theor.46, 485302 (2013)

    2013

  2. [10]

    Fring and M

    A. Fring and M. H. Y. Moussa, Phys. Rev. A93, 042114 (2016)

    2016

  3. [11]

    L. F. Alves da Silva and M. H. Y. Moussa, arXiv:2504.02050 [quant-ph] (2025)

    2025 arXiv

  4. [12]

    L. F. A. da Silva, R. A. Dourado, and M. H. Y. Moussa, SciPost Phys. Core5, 012 (2022)

    2022

  5. [13]

    Budroni, A

    C. Budroni, A. Cabello, O. G¨ uhne, M. Kleinmann, and J.-˚A. Larsson, Rev. Mod. Phys.94, 045007 (2022)

    2022

  6. [14]

    K. Wang, Z. Hou, K. Qian, L. Chen, M. Krenn, M. Aspelmeyer,A. Zeilinger, S. Zhu, and X.-S. Ma, Sci. Adv.11, eadr1794 (2025). 19

    2025

  7. [15]

    W´ ojcik and J

    A. W´ ojcik and J. W´ ojcik, arXiv:2509.03127 [quant-ph] (2025)

    2025 arXiv

  8. [16]

    T. J. Herzog, J. G. Rarity, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett.72, 629 (1994)

    1994

  9. [17]

    Mostafazadeh, J.Math.Phys.45, 932-946 (2004)

    A. Mostafazadeh, J.Math.Phys.45, 932-946 (2004)

    2004

  10. [18]

    D. C. Brody, J. Phys. A47, 035305 (2014)

    2014

  11. [19]

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

    2010

  12. [20]

    Leonhardt, Measuring the quantum state of light (Cambridge university press, 1997)

    U. Leonhardt, Measuring the quantum state of light (Cambridge university press, 1997)

    1997

  13. [21]

    W. K. Wootters, Phys. Rev. Lett.80, 2245 (1998)

    1998

  14. [22]

    C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and William K. Wootters, Phys. Rev. Lett.70, 1895 (1993)

    1993

  15. [23]

    Y. K. Wang and F. T. Hioe, Phys. Rev. A7, 831 (1973)

    1973

  16. [24]

    Dimer, B

    F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael Phys. Rev. A75, 013804 (2007)

    2007

  17. [25]

    Comer Duncan, Phys

    G. Comer Duncan, Phys. Rev. A9, 418 (1974)

    1974

  18. [26]

    Kirton, M

    P. Kirton, M. M. Roses, J. Keeling, E. G. Dalla Torre, Adv. Quantum Technol.2, 1800043 (2018)

    2018

  19. [27]

    M. A. de Ponte, F. S. Luiz, O. S. Duarte, and M. H. Y. Moussa, Phys. Rev. A100, 012128 (2019);ErratumPhys. Rev. A102, 049903 (2020)

    2019

  20. [28]

    M. A. de Ponte, F. Oliveira Neto, P. M. Soares, M. H. Y. Moussa, SciPost Phys.15, 091 (2023)

    2023

  21. [29]

    D. Cius, F. M. Andrade, A. S. M. de Castro, M. H. Y. Moussa, Phys. A: Stat. Mech. Appl. 593, 126945 (2022)

    2022

  22. [30]

    D. Cius, G. M. Uhdre, A. S. M. de Castro, and F. M. Andrade, Phys. Rev. A107, 022403 (2023). 20

    2023

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.