Photonic quantum simulations of coupled PT -symmetric Hamiltonians

May combine two claims.

At the exceptional point, the probabilities evolve towards a never-reached turning point, and the system effectively tends to a steady state with coefficients that follow a polynomial power law.

“The probabilities evolve towards a never-reached turning point, and the systems effectively tends to a steady state with coefficients that follows a polynomial power law.”

May combine two claims.

In the broken symmetry phase, the system approaches its steady state with an exponential power law whose characteristic time τ=π/√(γ^2-1) decreases for increasing γ.

“In contrast, in the broken symmetry phase, the system approaches its steady state with an exponential power law whose characteristic time, τ = π/ γ 2 -1, decreases for increasing γ .”

May combine two claims.

The no-signaling condition is violated for all γ=0 and is maximal for γ=γ_c.

“The no-signaling condition is violated for all γ = 0 and is maximal for γ = γ c .”

May combine two claims.

The authors reproduce dynamics in the PT-symmetric regime and across the exceptional point into the PT-symmetry-broken regime, including effects of mutual coherence and interference between forward-in-time and backward-in-time subsystems.

“We reproduce the dynamics in the PT -symmetric regime (defined by real eigenvalues) and across the EP into the PT -symmetry-broken regime (defined by complex-conjugate eigenvalues), including the effects of mutual coherence and interference between the forward-in-time and backward-in-time subsystems.”

May combine two claims.

In the broken symmetry phase with γ=1.1, the system rapidly tends to a steady state with no oscillations and with a nonzero probability of finding the excitation in either the forward or reverse systems.

“Figure 2(b) shows, instead, the evolution in the broken symmetry phase with γ = 1.1.The system rapidly tends to a steady state with no oscillations and with a nonzero probability of finding the excitation in either the forward or reverse systems.”

Classified as probably not a substantial claim.

The data in Figs. 4(d)-4(f) show the evolution of the probability of different photon bunching events in the forward system modes.

“The data in Figs.4(d)-4(f) show the evolution of the probability of different photon bunching events in the forward system modes.”

May combine two claims.

The amplitude of these oscillations is reduced if the coherence between |1_F and |1_R decreases, and it vanishes if the input is a statistical mixture of |1_F and |1_R.

“The amplitude of these oscillations is reduced if the coherence between the states |1 F and |1 R decreases, and it vanishes if the input is a statistical mixture of |1 F and |1 R [Fig.3(a)].”

May combine two claims.

By increasing γ, the system undergoes the transition from the symmetric to the broken symmetry phase characterized by an increase of the period of evolution for γ<√2 and by a nonperiodic evolution for γ>√2.

“By increasing the value of γ , the system undergoes the transition from the symmetric to the broken symmetry phase characterized by an increase of the period of the evolution, for γ < √ 2, and by a nonperiodic evolution, for γ > √ 2.”

May combine two claims.

In the broken symmetry phase, S_F(R) no longer exhibits oscillatory behavior because the system loses its periodicity, but the final configuration changes greatly depending on the initial coherence between the two subsystems.

“We finally note that in the broken symmetry phase [Fig.3(b)], S F(R) no longer exhibits oscillatory behavior since the system loses its periodicity, but the final configuration of the system changes greatly depending on the initial coherence between the two subsystems.”

May combine two claims.

When γ>0, the exchange symmetry between modes 1 and 3 is broken; the weight at smaller times t<T(γ)/2 shifts to the gain mode, and the weight at times t>T(γ)/2 shifts to the loss mode.

“When γ > 0, the exchange symmetry between modes 1 and 3 is broken; the weight at smaller times t < T (γ )/2 shifts to the gain mode, and the weight at times t > T (γ )/2 shifts to the loss mode.”

May combine two claims.

The entropy of the system is periodic for γ<1 and goes asymptotically to 0 for γ>1.

“(c) The entropy of the system is periodic for γ < 1 and goes asymptotically to 0 for γ > 1.”

May combine two claims.

The data for |ψ2 follow the opposite temporal evolution as |ψ1, while the probabilities of observing |1 and |3 are exchanged.

“Conversely, the data for |ψ 2 follow the opposite temporal evolution as |ψ 1 , while the probabilities of observing |1 and |3 are exchanged.”

• AssumptionThe authors state this as an assumption: The production of the desired input state is assumed every time a coincidence detection happens at two detectors, neglecting higher-order pair generation occurring with a two-order-of-magnitude-lower rate.
• AssumptionRests on an assumption: For N=3, the third-order exceptional point occurs at γ_c=√2.

Probably not a substantial claim: may not pass S3 (load-bearing).

The Hong-Ou-Mandel visibility of the interference of two indistinguishable single photons is 0.97.

“The Hong-Ou-Mandel visibility of their interference is measured to be 0.97.”

May combine two claims.

In the Hermitian regime (γ=0), the vacuum probability is constant at 0 and there are sinusoidal oscillations between the two populated levels.

“the inset shows the Hermitian regime (γ = 0) with a constant vacuum probability of 0 and sinusoidal oscillations between the two populated levels.”

May combine two claims.

The partial trace over the reverse (forward) system sums the diagonal elements of the bottom (top) N modes of ρ(t) and places this value in the vacuum state of the forward (reverse) system.

“Similar results are obtained when the partial trace is performed over the forward system, showing that the partial trace operation over the reverse (forward) system sums the diagonal elements of the bottom (top) N modes of ρ(t ) and places this value in the vacuum state of the forward (reverse) system.”

May combine two claims.

In the broken symmetry phase (γ=1.1), the evolution is not periodic, and the system tends rapidly to a steady state.

“(b) In the broken symmetry phase (γ = 1.1), the evolution is not periodic, and the system tends rapidly to a steady state; the inset shows the evolution of the system at the exceptional point.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The spikes appearing in H_1,3 and H_2,4 represent asymptotes.

“The spikes appearing in H 1,3 and H 2,4 represent asymptotes.”

Classified as probably not a substantial claim.

The Fock spaces of the forward and reverse systems are decomposed into particle number subspaces as F(F/R)=H(F/R)_0 ⊕ H(F/R)_1.

“That is, the Fock spaces of the systems are decomposed into particle number subspaces as F (F/R) = H (F/R) 0 ⊕ H (F/R) 1 .”

May combine two claims.

When H_N is Hermitian, G_N(t) is unitary and the singular values are 1, so D_N=0, giving two independently evolving systems.

“In the case when H N is Hermitian, G N (t ) is unitary and the singular values are 1, so D N = 0, giving two independently evolving systems.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The unplotted elements in Fig. 5 are all numerically 0, including all diagonal elements H_k,k for 1≤k≤4.

“The unplotted elements in Fig. 5 are all (numerically) 0 including all diagonal elements H k,k for 1 k 4.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In classical wave systems, PT-symmetric systems have shown unidirectional invisibility, single-mode and topological lasing, topological energy transfer, mode switching, and enhanced sensitivity.

“In the classical domain, PT -symmetric systems have shown remarkable properties such as unidirectional invisibility [5], single-mode [6,7] and topological [8,9] lasing, topological energy transfer [10], mode switching [11], and enhanced sensitivity [12,13].”

• cites [5]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [6]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [7]Not matched

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• cites [8]Not matched

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• cites [9]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The role of thermal and quantum noise on exceptional points in semiclassical PT-symmetric systems is not yet understood.

“For all these advances, the role of thermal and quantum noise on EPs in these semiclassical systems is not yet understood [14,15].”

• cites [14]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [15]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

PT-symmetric Hamiltonians are generally non-Hermitian and exhibit exceptional points at which two or more eigenvalues become degenerate and the corresponding eigenstates coalesce.

“While generally non-Hermitian, PT -symmetric Hamiltonians exhibit exceptional points (EPs) at which two (or more) eigenvalues become degenerate and the corresponding eigenstates coalesce.”

• cites [1]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [2]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Due to PT symmetry between the input states |ψ1 and |ψ2, their evolution proceeds in opposite directions in time and presents an exchange of probabilities between the patterns |1p1p0p and |0p1p1p.

“We note that, due to the PT symmetry between these input states, we expect their evolution to proceed in opposite directions in time and to present an exchange of probabilities between the observation of the patterns |1 p 1 p 0 p and |0 p 1 p 1 p .”

Classified as probably not a substantial claim.

Passive PT symmetry is based on the equivalence, after normalization, of a PT-symmetric Hamiltonian and a dissipative Hamiltonian with mode-selective losses.

“The first approach, known as passive PT symmetry, is based on the equivalence (after normalization) of a PT -symmetric Hamiltonian and a dissipative Hamiltonian with mode-selective losses [17,18].”

• cites [17]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [18]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Whenever γ=0, the two systems are coupled by the defect matrix D_N, and excitations can tunnel between these two systems.

“Whenever γ = 0, the two systems are coupled by the defect matrix D N , and excitations can tunnel between these two systems.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Calibration of the on-chip thermo-optic phase shifters allows mapping a desired phase shift to a voltage bias to be set.

“Calibration of the on-chip thermo-optic phase shifters allows us to map a desired phase shift to a voltage bias to be set.”

Probably not a substantial claim: may not pass S3 (load-bearing).

With N=2, the eigenvalues of the Hamiltonian are ±√(1-γ^2).

“With N = 2, the eigenvalues of the Hamiltonian are = ± 1 -γ 2 .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

As γ increases, the coupling between the systems iD(γ,t) increases, correspondingly decreasing the probability of a photon remaining in the desired subsystem.

“As we increase γ , we also increase the coupling between the systems, iD(γ , t ), which correspondingly decreases the probability of a photon remaining in the desired subsystem.”

May combine two claims.

G†_N(γ,t)=G*_N(γ,t)=T G_N(γ,t) T^{-1}, meaning a system initialized in the last N modes transforms with a backwards-in-time evolution.

“Indeed, G † N (γ , t ) = G * N (γ , t ) = T GN (γ , t )T -1 , meaning that a system initialized in the last N modes of the interferometer transforms with a backwards-in-time (reverse) evolution.”

May combine two claims.

The spectrum of H_N(γ) is purely real for γ<γ_c, and the corresponding eigenvectors are simultaneous eigenvectors of the antilinear PT operator.

“The spectrum of the Hamiltonian H N (γ ) is purely real for γ < γ c , and the corresponding eigenvectors are simultaneous eigenvectors of the antilinear PT operator (unbroken symmetry phase).”

• cites [28]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [29]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Hamiltonian dilation places the PT-symmetric Hamiltonian in a larger Hilbert space by introducing an ancillary two-level system, and the coherent nonunitary dynamics are recovered from the unitary dynamics in the larger Hilbert space.

“The second method uses Hamiltonian dilation that places the PT -symmetric Hamiltonian in a larger Hilbert space by introducing an ancillary two-level system [20][21][22], and the coherent, nonunitary dynamics of interest are recovered from the unitary dynamics in the larger Hilbert space.”

• cites [20]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [21]Not matched

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• cites [22]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

All values of γ can be implemented in the same way, allowing investigation of system dynamics in four regimes: uncoupled Hermitian evolution, coupled PT-symmetric evolution, evolution at the EP, and evolution in the broken symmetry phase.

“In this way, all values of γ can be implemented in the same way, allowing us to investigate the system dynamics in four regimes: uncoupled Hermitian evolution, coupled PT -symmetric evolution, evolution at the EP, and evolution in the broken symmetry phase.”

May combine two claims.

The photonic chip comprises a triangular mesh of 15 interconnected Mach-Zehnder interferometers, controlled by 30 metallic thermo-optic phase shifters, and is capable of implementing any unitary linear optical transformation over its six waveguides.

“Our photonic chip comprises a triangular mesh of 15 interconnected Mach-Zehnder interferometers, controlled by 30 metallic thermo-optic phase shifters, and is capable of implementing any unitary linear optical transformation over its six waveguides (see Ref. [27] for additional device details).”

• cites [27]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

In passive PT symmetry, evolution time is limited by exponentially decaying signal or postselection success probability, while Hamiltonian dilation requires a time-dependent Hermitian Hamiltonian for the larger Hilbert space.

“In the first approach, the evolution time of the system is limited by the exponentially decaying signal or postselection success probability, while the second method requires a time-dependent Hermitian Hamiltonian for the larger Hilbert space.”

Classified as probably not a substantial claim. May combine two claims.

The first section of the chip implements the state evolution described by {U_4(γ,t_i)}_i, and the remaining part performs projective measurements allowing reconstruction of density matrices for evolved single-photon states.

“The first section of the chip implements the state evolution described by {U 4 (γ , t i )} i .The remaining part of the chip is used to perform projective measurements allowing the reconstruction of the density matrices for evolved single-photon states.”

May combine two claims.

γ_c/J equals 1 whenever N is even and equals √((N+1)/(N-1)) when N is odd.

“γ c /J = 1 whenever N is even and γ c /J = √ (N + 1)/(N -1) when N is odd.”

• cites [28]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

A dissipative Hamiltonian arises naturally from a Lindblad approach for open quantum systems when certain quantum jumps are ignored and allows coherent nonunitary dynamics in a minimal quantum system.

“Such a dissipative Hamiltonian arises naturally from a Lindblad approach for open quantum systems when certain quantum jumps are ignored and allows the implementation of coherent, nonunitary dynamics in a minimal quantum system [19].”

• cites [19]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

The dilation process has already been used to simulate Lindbladian dynamics.

“This process has already been used to simulate Lindbladian dynamics [32] and could be extended to simulate anti-PT systems [33].”

• cites [32]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S2 (checkable).

These demonstrations point towards the inclusion of non-Hermitian simulations in the realm of quantum technologies.

“Combined with the current advances in photonic technologies, these demonstrations point towards the inclusion of non-Hermitian simulations in the realm of quantum technologies.”

Classified as probably not a substantial claim.

The relative detector efficiency at different output modes for single-photon events is obtained by testing 500 random phase configurations and comparing experimental statistics with expected photon statistics.

“The relative detector efficiency at the different output modes, for single-photon events, is obtained by testing 500 random phase configurations of our interferometer and comparing the experimental statistic with the expected photon statistics due to the phase settings.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The final probabilities are normalized over all output states characterized by three photons in the forward system modes but with less than three photons bunched in the same mode.

“The final probabilities are normalized over all the output states characterized by three photons in the forward system modes but with less than three photons bunched in the same mode.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In boson sampling, classical intractability arises because the detection statistics are given by permanents of complex matrices.

“In the case of boson sampling [46], in which photons propagate in a circuit that is ideally described by a unitary matrix, classical intractability arises because the detection statistics are given by permanents of complex matrices.”

• cites [46]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The efficiency correction of each mode that maximizes statistical fidelity between experimental statistics and theoretical distribution is chosen as numerical correction for single-photon data.

“The efficiency correction of each mode that maximizes the statistical fidelity between the experimental statistics and the theoretical distribution is chosen as a numerical correction for the data collected during single-photon experiments.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Photon loss reduces the computational complexity in experimental implementations of boson sampling.

“Yet photon loss reduces the computational complexity in experimental implementations of boson sampling [47].”

• cites [47]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

For two-photon antibunched events, the occurrence of each pattern is multiplied by the product of the correction factor of the individual output modes populated with photons.

“For two-photon antibunched events, the occurrence of each pattern is multiplied by the product of the correction factor of the individual output modes populated with photons.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Error bars are obtained via bootstrapping methods.

“Error bars are obtained via bootstrapping methods [45].”

• cites [45]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

The same device can be used as a cascade of 2×2 interferometers to sequentially implement state preparation, evolution, or projection in physically different parts of the chip.

“Alternatively, the same device can be used as a cascade of 2 × 2 interferometers in order to sequentially implement state preparation, evolution, or projection in physically different parts of the chip.”

• cites [27]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The number of events associated with an output state |N1p N2p N3p with N_kp = n_k,1 + n_k,2 is given as the average of the occurrences of equivalent detection patterns.

“The number of events associated with an output state |N 1p N 2p N 3p with N k p = n k,1 + n k,2 is given as the average of the occurrences of equivalent detection patterns.”

Probably not a substantial claim: may not pass S3 (load-bearing).

It is computationally complex to simulate the evolution of quantum Hamiltonians that produce particle statistics governed by intractable matrix functions.

“It is computationally complex to simulate the evolution of quantum Hamiltonians [32] that produce particle statistics which are governed by intractable matrix functions.”

• cites [32]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

A photonic simulation of a traditional model for an open system, such as the Lindblad approach, could become classically tractable via the loss that is necessarily built into the model.

“Similarly, a photonic simulation of a traditional model for an open system, such as the Lindblad approach, could become classically tractable via the loss that is necessarily built into the model.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the pseudo-number-resolving configuration, the detection calibration factor is obtained by using the single-photon detector efficiency ν_i recorded at each output of the FBS.

“In the pseudo-number-resolving configuration, the detection calibration factor is obtained by using the single-photon detector efficiency ν i recorded at each output of the FBS.”

May combine two claims.

In the authors' simulation model for twin PT-symmetric Hamiltonians, while each subsystem is open and lossy, the overall model is unitary and an ideal implementation is lossless; generally, statistics provide information about both the forward and backward subsystems and the relation between them.

“However, in our simulation model for twin PT -symmetric Hamiltonians, while each subsystem is open and lossy, the overall model is unitary and an ideal implementation is lossless; generally, statistics provide information about both the forward and backward subsystems, and the relation between them.”