A non-Hermitian matrix can have real eigenvalues and still change the ordinary norm of a state. It can also have two equal eigenvalues but only one eigenvector. Understanding those two possibilities is the beginning of understanding non-Hermitian quantum mathematics.
Non-Hermitian quantum mechanics connects effective loss, gain, PT symmetry, exceptional points, biorthogonal eigenvectors and complex spectra. These ideas appear in several different physical settings: classical wave amplitudes, resonances of an open system, quantum trajectories conditioned on a measurement record, and quantum theories written with a modified inner product. The same-looking matrix can represent different things in those settings, so its probability interpretation must never be guessed from the eigenvalues alone.
This guide develops one coherent rule: solve the matrix, but also specify what its vectors, norms and measurements mean. We will calculate a PT-symmetric two-mode model exactly, derive its exceptional point, construct left and right eigenvectors, recover the missing vector with a Jordan chain, and separate spectral sensitivity from actual measurement precision. We will also examine the non-Hermitian skin effect and explain why an ordinary boundary change can reorganise an extensive set of eigenvectors.
Your 50-second route
Start with meaning: which physical problem does the matrix represent? Start with calculation: the PT dimer and Jordan-chain laboratory. Working with an open quantum system? Read conditional evolution and missing jumps. Reading claims about new devices? Use the sensing test. Studying boundary physics? Go to the asymmetric chain. End with the worked practice set.
Open the reading map and conventions
Part I establishes the probability and linear-algebra distinctions. Part II solves the PT dimer, its metric and its exceptional point. Part III treats conditional quantum evolution, non-normal amplification, boundary sensitivity and sensing. The exercises require only two-by-two matrices before moving to a finite chain. We use ℏ=1, the evolution convention iψ̇=Hψ, and the usual Pauli matrices. A dagger means complex-conjugate transpose; a star means entrywise complex conjugation.
A right eigenvector is a column. A left eigenvector is a row satisfying its own eigenvalue equation. When we discuss a modified metric, we name it explicitly. Otherwise norms and probabilities use the ordinary Hilbert-space inner product. This convention prevents a change of mathematical coordinates from being mistaken for a change in an experimental probability.
A spectrum is not a complete physical model. Ask what was eliminated, what was conditioned on, which norm is used and which measurement will test the prediction.
1. Four interpretations that must not be mixed
In a classical coupled-wave model, a vector may contain optical or mechanical mode amplitudes. Its squared norm can represent an intensity or another amplitude measure. Gain and loss can change that quantity without violating a probability rule because the vector is not necessarily a normalised quantum state.
In an open quantum system, eliminating a continuum or conditioning on no detected emission can produce an effective non-Hermitian Hamiltonian. The remaining vector describes a restricted branch or subspace. Norm lost from that branch must be accounted for through the eliminated degrees of freedom or the probabilities of other measurement records.
In a closed quantum theory written with a nonstandard positive metric, a matrix that is not Hermitian under the ordinary inner product may become self-adjoint under the specified metric. Observables and state normalisation must be transformed consistently. One cannot keep the convenient spectrum from that description while borrowing an incompatible probability rule from the ordinary one.
Finally, a complex eigenvalue problem may describe resonances rather than normalisable stationary states. Boundary conditions and analytic continuation then matter. Ashida, Gong and Ueda discuss these distinct routes to non-Hermitian physics, including effective open-system descriptions. The distinction is foundational, not merely terminological. [1]
2. Separate the Hermitian and anti-Hermitian parts
Every square matrix H can be written uniquely as H=H₀+iK, where H₀=(H+H†)/2 and K=(H−H†)/(2i) are Hermitian. H₀ rotates amplitudes in the familiar way; K controls changes of the ordinary norm.
Differentiate N(t)=⟨ψ|ψ⟩ using ψ̇=−iHψ. The result is Ṅ=i⟨ψ|(H†−H)|ψ⟩=2⟨ψ|K|ψ⟩. If K is negative semidefinite, the norm cannot increase. If K has both signs, growth depends on the state.
This equation is often more informative than the eigenvalues. A real spectrum does not imply K=0 and does not guarantee conservation of the ordinary norm for every superposition. Conversely, a system with only decaying eigenmodes can show transient growth in a chosen amplitude norm when eigenvectors are nonorthogonal.
In a no-jump quantum interpretation, K=−Γ/2 with Γ=ΣLk†Lk⪰0. Then Ṅ=−⟨Γ⟩. The lost norm is not discarded probability; it is the probability weight of trajectories in which a jump occurred.
3. Right eigenvectors and left eigenvectors
A right eigenvector satisfies H|rn⟩=En|rn⟩. A left eigenvector satisfies ⟨ln|H=En⟨ln|. Equivalently, the column associated with the left vector satisfies H†|ln⟩=En*|ln⟩.
For Hermitian matrices, left and right vectors can be identified through the ordinary adjoint, and orthonormal eigenvectors are available. For a general diagonalizable non-Hermitian matrix, they form two dual bases instead. One can choose the normalisation ⟨lm|rn⟩=δmn, called biorthogonality.
The word dual matters. It says how coefficients in a vector expansion are extracted. It does not automatically say that ⟨ln|O|rn⟩ is the outcome average of an ordinary laboratory measurement. That interpretation needs an explicitly defined metric or physical protocol.
When a matrix is complex symmetric, Hᵀ=H, it may be tempting to use a transpose where a dagger belongs everywhere. The left-eigenvector algebra can indeed exploit symmetry, but physical probabilities still require their own consistent inner product. Keep algebraic convenience separate from probability interpretation.
4. Spectral reconstruction and the hidden assumption of diagonalizability
Put the right eigenvectors into an invertible matrix V. Then H=V D V⁻¹, with D diagonal. The rows of V⁻¹ are the left eigenvectors, so I=Σn|rn⟩⟨ln| and H=ΣnEn|rn⟩⟨ln|.
The propagator follows immediately: exp(−iHt)=Σn exp(−iEnt)|rn⟩⟨ln|. This formula is useful only when there is a complete eigenbasis. At an exceptional point V becomes singular and the formula cannot simply be evaluated by substituting equal eigenvalues into a divergent eigenvector normalisation.
The complete matrix exponential remains well defined. It is the diagonal representation that fails. Jordan chains or direct matrix functions provide the correct continuation. Recognising this distinction prevents a mathematical singularity in chosen coordinates from being mistaken for infinite physical evolution at every finite time.
5. A simple triangular example of a biorthogonal basis
Consider H=[[1,q],[0,2]] with real q. Right eigenvectors can be chosen as r₁=(1,0)ᵀ and r₂=(q,1)ᵀ. Therefore V=[[1,q],[0,1]] and V⁻¹=[[1,−q],[0,1]].
The left rows are l₁=(1,−q) and l₂=(0,1). Check the four overlaps: l₁r₁=1, l₁r₂=0, l₂r₁=0 and l₂r₂=1. The right eigenvectors are not orthogonal when q is nonzero, yet the left-right pair gives a perfectly valid resolution of identity.
For a vector ψ=(a,b)ᵀ, the correct expansion coefficients are c₁=a−qb and c₂=b, not simply the ordinary overlaps with unnormalised right eigenvectors. This small example is worth doing by hand because it reveals exactly what a left eigenvector contributes.
6. Degeneracy is not always exceptional
The matrix E₀I has a repeated eigenvalue and two independent eigenvectors in two dimensions. It is diagonalizable. By contrast, J=[[E₀,1],[0,E₀]] has the same repeated eigenvalue but only one eigenvector. The second case is defective.
An exceptional point is a parameter value at which eigenvalues and their eigenvectors coalesce, producing this defective structure. Merely plotting two equal energies is insufficient. One must compare algebraic multiplicity with the dimension of the eigenspace or examine the limiting eigenvectors and matrix rank. Heiss reviews the characteristic singularities and physical consequences of these points. [2]
In numerical work, exact rank is replaced by a tolerance-dependent singular-value calculation. A nearly defective matrix can be very ill-conditioned before a plotted eigenvalue pair visibly merges. Treat that conditioning as part of the result rather than hiding it behind more displayed digits.
7. The PT-symmetric dimer
Our main model is H=gX+iγZ=[[iγ,g],[g,−iγ]], with g>0 and γ≥0. The off-diagonal coupling exchanges amplitudes between two components. The imaginary diagonal entries describe balanced relative gain and loss in the coupled-amplitude interpretation.
Let parity exchange the components: P=X. Let time reversal act by complex conjugation, T=K, for this effective two-mode problem. Then P H* P=H, so H commutes with the combined antiunitary PT operation. This particular T is a model definition; it should not be confused with every physical spin-half time-reversal operator.
The matrix is not Hermitian for γ≠0, because its imaginary diagonal entries change sign under a dagger. Nevertheless, its eigenvalues can be real. PT-symmetric Hamiltonians with real spectra were central to the work of Bender and Boettcher, but PT symmetry alone does not make every spectrum real or provide a complete probability interpretation. [3]
8. Solve the spectrum without a determinant expansion
Use X²=Z²=I and XZ+ZX=0. Squaring the dimer gives H²=(g²−γ²)I. Therefore its eigenvalues obey E²=g²−γ², and
E±=±√(g²−γ²).
When γ<g, the pair is real. At γ=g, both eigenvalues meet at zero. When γ>g, they are purely imaginary and opposite in sign. This is the familiar unbroken-to-broken spectral transition of this dimer.
For g=1 and γ=0.6, the eigenvalues are ±0.8. For g=1 and γ=1.25, they are ±0.75i. The arithmetic is elementary, but the interpretation is not: the real pair does not imply ordinary norm conservation, and the growing imaginary branch is not an allowed no-jump probability amplitude unless the physical model supplies gain or a changed normalisation convention.
9. What antiunitary symmetry really guarantees
Suppose H|r⟩=E|r⟩ and H commutes with PT. Because PT is antiunitary, applying it gives H(PT|r⟩)=E*(PT|r⟩). Thus a nonreal eigenvalue has a conjugate partner under this symmetry.
If an individual eigenvector is also an eigenstate of PT up to a phase, then the same one-dimensional eigenspace has eigenvalues E and E*, forcing E to be real. If PT maps the eigenvector into a different eigenspace, a complex-conjugate pair is possible.
This derivation explains why the statement PT symmetry guarantees a real spectrum is too strong. The symmetry of the operator and the symmetry realised by its eigenvectors are separate facts. In the dimer, their change is visible at γ=g.
10. Right eigenvectors of the dimer
For an eigenvalue E, a convenient unnormalised right vector is rE=(g,E−iγ)ᵀ. Substitute it into H to check the equation. The first component becomes iγg+g(E−iγ)=gE. The second uses E²=g²−γ² to become E(E−iγ).
In the unbroken regime, set E=±Ω with Ω=√(g²−γ²). Both vectors have ordinary squared norm 2g² because |±Ω−iγ|²=Ω²+γ²=g². Their ordinary inner product is generally nonzero, so equal normalisations do not make them orthogonal.
As Ω approaches zero, the two directions become the same. A biorthogonal left basis must then acquire large coefficients to keep its overlaps equal to Kronecker deltas. That blow-up signals an ill-conditioned basis, not a failure of the finite matrix H to exist.
11. At the exceptional point, construct the missing direction
Set g=γ=1. Then H=[[i,1],[1,−i]], with H²=0 but H≠0. Solving Hv=0 gives the eigenvector v=(1,−i)ᵀ. There is only one independent eigenvector.
A generalised eigenvector w satisfies Hw=v. Choose w=(0,1)ᵀ. Direct multiplication gives Hw=(1,−i)ᵀ, as required. The pair v,w forms a Jordan chain.
In the basis with columns v and w, H takes the Jordan form [[0,1],[0,0]]. The matrix sends w to v and v to zero. This description replaces the missing second eigenvector with the exact way the matrix moves a generalised direction into the surviving eigenspace.
The method is more reliable than asking an eigensolver for two normalised eigenvectors exactly at the exceptional point. Numerical software may return nearly parallel vectors that differ only because of floating-point perturbations. A rank test and the equation Hw=v identify the structure directly.
12. Exact evolution at the exceptional point
Since H²=0, the matrix exponential terminates: exp(−iHt)=I−itH. The eigenvector v stays fixed because Hv=0. The generalised vector evolves as w→w−itv.
More generally, a Jordan block with eigenvalue E₀ produces polynomial factors multiplying exp(−iE₀t). A second-order block contributes a term proportional to t; a larger block can produce higher powers. This is why the spectrum alone misses important time dependence at a defective point.
For the unit dimer, starting from e₁=(1,0)ᵀ gives ψ(t)=(1+t,−it)ᵀ, whose ordinary squared norm is 1+2t+2t². The growth is fully finite for finite t and follows from the matrix exponential. In a gain-loss amplitude model it is an amplitude effect. It is not automatically a normalised quantum-state probability.
A common passive realisation adds uniform loss. If Hpass=H−iI, evolution acquires the factor e⁻ᵗ and the squared norm becomes e⁻²ᵗ(1+2t+2t²). It decreases from one, consistent with an appropriate conditional loss interpretation. The Jordan polynomial survives, but its physical probability is weighted by the decay.
13. Exact propagator away from the exceptional point
When Ω²=g²−γ² is nonzero, the even powers of H are powers of Ω² times I and the odd powers are powers of Ω² times H. Summing the series gives U(t)=cos(Ωt)I−i[sin(Ωt)/Ω]H.
For γ>g, write Ω=iκ with κ=√(γ²−g²). The trigonometric functions become hyperbolic: U(t)=cosh(κt)I−i[sinh(κt)/κ]H. At the threshold, sin(Ωt)/Ω→t, recovering I−itH smoothly.
This exact form is a good numerical reference near the exceptional point because it avoids subtracting two enormous biorthogonal contributions that nearly cancel. Near Ω=0, evaluate the ratio with a stable sinc-type expression or use a direct matrix exponential. A formula can be algebraically correct but numerically fragile.
14. A positive metric in the unbroken regime
For γ<g, define η=I+(γ/g)Y. It is Hermitian, with eigenvalues 1±γ/g, both positive. Direct Pauli algebra gives H†η=ηH.
Therefore the η-norm ⟨ψ|η|ψ⟩ is conserved under iψ̇=Hψ. Differentiating it gives i⟨ψ|(H†η−ηH)|ψ⟩=0. In this metric, H is self-adjoint.
This construction does not prove that an arbitrary laboratory intensity detector measures the η-norm. It proves that the same matrix admits a consistent alternative inner-product description in the unbroken regime. To use that as a quantum model, observables must satisfy the corresponding self-adjointness condition and probabilities must be defined consistently.
At γ=g, one eigenvalue of η reaches zero. The metric is no longer positive definite and its inverse ceases to exist. The ordinary similarity to a Hermitian problem becomes singular precisely where the dimer becomes defective. The limiting failure is therefore visible in both the eigenvector basis and the metric.
15. Similarity is not the same as unitary equivalence
If η is positive definite, write η=S†S with S invertible. Then h=SHS⁻¹ is Hermitian when H†η=ηH. The vector φ=Sψ has ordinary norm equal to the η-norm of ψ.
A unitary change of basis preserves ordinary lengths and angles. A general similarity transformation need not. It preserves eigenvalues but can change the appearance of amplitudes, localisation and conditioning. Physical equivalence therefore requires transforming the metric and observables, not only the Hamiltonian.
This distinction will reappear in the asymmetric chain. A non-Hermitian chain can be similar to a Hermitian chain while its ordinary right eigenvectors accumulate strongly at one boundary. The similarity is nonunitary, so that boundary concentration is not contradicted by equality of the open-chain eigenvalues.
16. Square-root splitting near the exceptional point
Fix γ=1 and change the coupling to g=1+δ. The eigenvalues are E±=±√(2δ+δ²). For small positive δ, the leading splitting scales as √δ rather than δ.
For δ=0.01, the eigenvalues are approximately ±0.141774, giving a separation about 0.283549. The change looks large relative to the 0.01 coupling perturbation. That is genuine spectral sensitivity of the eigenvalue problem.
But the derivative dE/dδ becoming large does not yet establish better parameter estimation. A measurement needs a signal distribution, noise model, observation duration and resource budget. Near a defective point, eigenvectors are also ill-conditioned and the measurement response can be accompanied by amplified fluctuations. We will return to that distinction in the sensing section.
17. Encircling a branch point is not the same as adiabatically moving a state
The elementary model H(z)=[[0,1],[z,0]] has eigenvalues ±√z. Let z=re^(iφ). Following one analytic square-root branch as φ increases by 2π exchanges the two eigenvalue labels. Two turns restore the label.
This is a statement about analytic continuation of eigenvalues in parameter space. It is not automatically the trajectory of a physically evolving state driven around the same loop. Non-Hermitian evolution changes relative amplitudes through growth and decay; slow driving can amplify tiny transitions between instantaneous modes.
To predict state transfer during encircling, solve the time-dependent evolution or use a justified non-Hermitian adiabatic analysis. A sketch of two sheets of a square root cannot replace the dynamical calculation. Geometric permutation, dynamical following and observed switching are related but distinct claims.
18. Non-Hermitian evolution as one branch of a quantum experiment
A Markovian open system can obey ρ̇=−i[H_S,ρ]+Σk(LkρLk†−½{Lk†Lk,ρ}). Define H_eff=H_S−iΣkLk†Lk/2. The same equation becomes ρ̇=−i(H_effρ−ρH_eff†)+ΣkLkρLk†.
The first part alone is the unnormalised no-jump evolution. The second part returns probability to branches in which a jump occurs. Omitting the second term changes the ensemble being described. It does not produce a complete trace-preserving replacement for the original master equation.
This connection provides a direct bridge to Guide 52 and Guide 54. The effective matrix is useful because it captures the conditional branch between detections, while the full channel describes the unconditioned experiment. [1]
19. Work through spontaneous decay
Let L=√κ|g⟩⟨e| and take HS=0. Then H_eff=−iκ|e⟩⟨e|/2. An initial vector a|g⟩+b|e⟩ evolves without a jump into a|g⟩+b e^(−κt/2)|e⟩.
Its squared norm is P_nojump=|a|²+|b|²e^(−κt). This is the probability of the no-jump record. Conditioned on that record, divide the vector by √Pnojump. A jump occurs with complementary probability and leaves the system in |g⟩.
The unconditioned excited population is |b|²e^(−κt). The conditional excited population is |b|²e^(−κt)/P_nojump. They answer different questions and generally differ. Reporting the conditional value without the acceptance probability conceals how many attempted runs contributed to it.
This calculation is also a diagnostic for simulation software. An unnormalised branch solver should lose trace. A full Lindblad solver should preserve it. Renormalising a branch at every step and calling the result the unconditioned state changes the physical model.
20. Normalised conditional evolution is nonlinear
Let Γ=ΣLk†Lk and let ρc be the normalised no-jump state. Differentiating the normalisation gives ρ̇c=−i[H_S,ρc]−½{Γ,ρc}+Tr(Γρc)ρc.
The last term is nonlinear in ρc. It reflects conditioning on a record whose probability depends on the state. The underlying physical instrument, including every record, remains a linear completely positive map. Nonlinearity in a normalised conditional update does not establish a fundamental nonlinear quantum dynamics.
For the decay example, seeing no emission gradually favours the part of the state that was less likely to emit. This has an information-update interpretation as well as a dynamical one. The observer learned something from an absence of a click, and the conditional state incorporates that information.
21. A passive version of the PT dimer
Subtract uniform imaginary loss from the balanced dimer: Hpass=H_PT−iγI=[[0,g],[g,−2iγ]]. The second component has loss and the first has no direct loss. In a single-excitation space with a sink state, a jump L=√(4γ)|sink⟩⟨2| produces the effective loss term −2iγ|2⟩⟨2|.
The right eigenvectors and exceptional-point position are unchanged by the scalar shift. The eigenvalues move downward by −iγ. The propagator is U_pass(t)=e^(−γt)U_PT(t).
After normalising the surviving branch, some relative-amplitude dynamics can resemble the balanced PT model. But the success probability has changed and must be retained. A rescaled plot that removes e⁻ᵞᵗ can show relative growth even though the actual survival probability never exceeds one.
This is a useful way to understand passive PT experiments without claiming that a loss-only quantum trajectory physically creates probability. What grows is a relative or rescaled component, not the total probability of an unconditioned closed system.
22. Non-normal matrices can amplify before they decay
A matrix is normal when it commutes with its adjoint. Hermitian matrices are normal; many non-Hermitian matrices are not. For a non-normal generator, nonorthogonal eigenmodes can combine to produce transient growth even when every eigenvalue predicts eventual decay.
Consider the ordinary amplitude equation ẋ=Ax with A=[[-1,K],[0,−2]]. Its eigenvalues are −1 and −2. Starting from x(0)=(0,1)ᵀ gives x₂=e⁻²ᵗ and x₁=K(e⁻ᵗ−e⁻²ᵗ).
At t=ln2, x₁=K/4 and x₂=1/4. For K=10, the vector norm has grown well above its initial value even though both asymptotic exponents are negative. This is a generic linear-amplitude example, not an assertion that the Euclidean norm of an arbitrary quantum no-jump process may grow. Physical interpretation still matters.
The example shows why fitting only the slowest eigenvalue can miss the practically important part of a response. Transient amplification, eigenvector condition numbers and singular values can matter as much as eigenvalue real parts over finite observation windows.
23. Resolvents and response near a defective point
The resolvent R(z)=(zI−H)⁻¹ describes how a linear system responds at complex spectral parameter z. For a diagonalizable matrix it has a sum of simple-pole terms |rn⟩⟨ln|/(z−En).
For the nilpotent Jordan block J with J²=0, (zI−J)⁻¹=I/z+J/z². A second-order pole appears. This is the frequency-domain partner of the t factor in the time-domain propagator.
A large resolvent norm can indicate strong response to forcing or sensitivity to perturbations even away from the exact eigenvalues. The pseudospectrum records regions where a small matrix perturbation could move an eigenvalue. For non-normal matrices it can be far larger than a set of small circles around the spectrum.
When studying a nearly exceptional device, inspect response functions and conditioning rather than only drawing two eigenvalue curves. The physically measured quantity may depend on an input-output transfer function with poles, zeros and noise couplings that are not determined by eigenvalue separation alone.
24. An asymmetric hopping chain that can be solved exactly
Take L sites with open boundaries and H=Σj[t_R|j+1⟩⟨j|+t_L|j⟩⟨j+1|], where tR and tL are positive but unequal. Define r=√(t_R/t_L) and a diagonal matrix S with entries Sjj=rʲ.
Then S⁻¹HS has equal hopping √(tRtL) in both directions. It is the ordinary Hermitian open chain. Its eigenvalues are Em=2√(t_Rt_L)cos[mπ/(L+1)] for m=1,…,L.
The original right eigenvectors are proportional to rʲ sin[mπj/(L+1)]. When r>1, their ordinary amplitudes accumulate toward the right boundary. The spectrum can be entirely real while an extensive number of eigenvectors are concentrated near one edge.
This simple algebra exhibits the non-Hermitian skin effect. It differs from an ordinary isolated topological edge state because it involves many bulk-associated modes. Yao and Wang showed how such boundary sensitivity requires revisiting conventional Bloch-based topological classifications in relevant non-Hermitian models. [4]
25. Open and periodic boundaries are not interchangeable
For the periodic version of the same asymmetric chain, a plane wave with amplitudes eⁱᵏʲ has energy E(k)=t_R e⁻ⁱᵏ+t_L eⁱᵏ. As k varies, this generally traces an ellipse in the complex plane.
For example, tR=2 and tL=0.5 give r=2 and √(tRtL)=1. The open-chain eigenvalues are real numbers 2cos[mπ/(L+1)], while the periodic dispersion is 2.5cosk−1.5i sink.
The nonunitary similarity that symmetrises the open chain does not preserve a uniform periodic closing link. Carrying it around the ring accumulates a large mismatch. Therefore the open-boundary result cannot be obtained merely by treating the periodic plane-wave eigenvectors as though they remained valid after cutting a bond.
When a numerical result seems sensitive to a tiny boundary link, check whether the perturbation is small in the representation that controls the eigenvector conditioning. Non-normal systems can magnify effects that would be innocuous in a Hermitian problem.
26. Do not confuse skin accumulation with Anderson localisation
Our asymmetric chain has no random onsite disorder. Its boundary accumulation follows from directional hopping and a nonunitary similarity transform. Anderson localisation instead concerns interference-induced localisation in disordered or related structured Hermitian settings.
The two effects can coexist, but their diagnostic questions differ. Ask whether the mode centre follows a boundary, whether changing boundary conditions reorganises many states, whether a reciprocal Hermitian limit removes the accumulation, and whether disorder is essential to the observed pattern.
Guide 75 develops the distinction between Anderson and interacting many-body localisation. A decaying amplitude profile alone does not identify the mechanism that produced it.
27. Spectral sensitivity is not automatically measurement advantage
The dimer gives a √δ eigenvalue response near the exceptional point. To convert that into a sensing claim, specify how the unknown δ changes the measured data. A detector might measure output power, a quadrature, a switching probability or a spectrum obtained over a finite time. Each has a different noise and resource model.
Gain can amplify the desired signal and also amplify or add fluctuations. A conditioning protocol can make surviving events highly sensitive while reducing the survival fraction. A long integration time can resolve closely spaced lines but consumes time that a comparator must also be allowed to use.
Lau and Clerk analysed two-mode non-Hermitian sensors while accounting for measurement and noise. Their results show that enhanced response, gain, nonreciprocity and proximity to an exceptional point must be distinguished rather than compressed into one claim of universal enhancement. [5]
The appropriate comparison fixes resources such as total experimental attempts, detected or injected energy where relevant, observation time, bandwidth and allowed controls. It then compares estimation error or a well-defined information measure. Eigenvalue slopes alone omit too much of the experiment.
28. Reading a contemporary sensing report without overstating it
A 2026 preprint by Assouly and colleagues reports a four-mode superconducting system that emulates a PT dimer without engineered dissipation. The authors report enhanced sensitivity near the exceptional point in a non-quantum-limited regime. That is a specific experimental finding under a stated regime, not evidence that all exceptional-point sensors beat all conventional sensors at a quantum limit. [6]
The distinction is constructive. The report explores a different physical implementation from a simple gain-loss pair and tests a measured signal-to-noise response. It should be read alongside, not as an automatic contradiction of, bounds established for other architectures and resource assumptions.
When comparing papers, write down their model classes first. Two results can both be correct because one concerns reciprocal two-mode linear sensors, another uses nonreciprocity, and another implements effective dynamics inside a larger closed system. A contradiction requires the same assumptions and the same measured objective.
29. A numerical checklist for nearly defective matrices
Compute right eigenvalue residuals ||Hr−Er|| and left residuals independently. Normalise the left-right overlaps only after pairing the eigenvalues correctly. Reconstruct H from the biorthogonal decomposition when the eigenbasis is well conditioned.
Then inspect the singular values of the eigenvector matrix V. A large condition number warns that small perturbations can produce large coefficient changes. Repeating a diagonalisation with higher precision or a different tolerance can reveal numerical instability, but agreement of printed eigenvalues alone is not sufficient.
At an exact exceptional point, use a rank test and Jordan-chain equations for the small model, or robust matrix-function algorithms for evolution. Compare the direct exponential with the analytic expression whenever one is available. For the dimer, H²=(g²−γ²)I provides an unusually strong internal check.
Finally verify the physical norm rule. A passive no-jump model must have nonincreasing branch norm. A full Lindblad calculation must preserve trace. A modified-metric calculation must preserve its stated η-norm. These checks test different physical problems and should not be substituted for one another.
30. A complete interpretation audit
Before presenting a non-Hermitian result, answer five questions in ordinary language. What do the components of the vector represent? Why is the effective generator non-Hermitian? Which outcomes or degrees of freedom have been omitted? What norm or metric has operational meaning? Which observable will be compared with data?
For a no-emission qubit, the answers are conditional state amplitudes, elimination of detected-jump branches, emission records, survival probability plus conditional normalisation, and a measurement conditioned on no click. For a classical waveguide pair, the answers may instead be optical mode amplitudes, gain/loss coupling, external pump and reservoirs, measured intensity, and a photodetector signal.
The matrices may be related, but the experimental stories are not identical. A careful explanation can transfer the useful mathematics without transferring an unjustified probability interpretation. This is the most important practical skill in the subject.
31. Practice laboratory with worked answers
Problem 1: decomposition. For H=[[iγ,g],[g,−iγ]], identify H₀ and K in H=H₀+iK.
Answer. H₀=gX and K=γZ. The ordinary norm derivative is 2γ⟨ψ|Z|ψ⟩. Its sign depends on the state, so the balanced dimer is not a generic loss-only no-jump Hamiltonian in the ordinary norm.
Problem 2: spectra. With g=2 and γ=1.2, find the eigenvalues.
Answer. g²−γ²=4−1.44=2.56, so E±=±1.6. The spectrum is real, but H remains non-Hermitian because γ is nonzero. Reality of the spectrum is not a test for H=H†.
Problem 3: degeneracy. Compare A=[[0,0],[0,0]] and B=[[0,1],[0,0]]. Both have two zero eigenvalues. Which is defective?
Answer. A has a two-dimensional zero eigenspace and is diagonalizable. B has only vectors proportional to (1,0)ᵀ in its kernel and is defective. Equal eigenvalues are not enough to identify an exceptional point.
Problem 4: Jordan evolution. If B²=0, calculate exp(−iBt).
Answer. The series terminates at first order: I−itB. Applied to (0,1)ᵀ, it gives (−it,1)ᵀ. A polynomial time dependence is present even though both eigenvalues are zero.
Problem 5: positive metric. For g=1 and γ=0.6, find the eigenvalues of η=I+0.6Y.
Answer. They are 1.6 and 0.4. The metric is positive definite and H†η=ηH. As γ approaches one, the smaller metric eigenvalue approaches zero and the metric becomes ill-conditioned.
Problem 6: left basis. For H=[[1,3],[0,2]], use right vectors (1,0)ᵀ and (3,1)ᵀ. Find the left rows.
Answer. The inverse of V=[[1,3],[0,1]] is [[1,−3],[0,1]]. The left rows are (1,−3) and (0,1). Their overlaps with the right columns give the identity matrix.
Problem 7: conditioning. Start a decaying two-level system in |+⟩ with κt=ln2. Find the no-jump probability and the conditional excited population.
Answer. Since e⁻ᵏᵗ=1/2, P_nojump=1/2+(1/2)(1/2)=3/4. The unnormalised excited weight is 1/4, so the conditional excited population is (1/4)/(3/4)=1/3. The unconditional excited population is 1/4. Mixing these answers would mix two different ensembles.
Problem 8: passive PT. Explain what changes when −iγI is added to the balanced dimer.
Answer. Every eigenvalue shifts by −iγ, while eigenvectors and the exceptional-point location remain unchanged. Every amplitude receives e⁻ᵞᵗ. Relative conditional dynamics can match after normalisation, but survival probabilities do not remain the same.
Problem 9: asymmetric chain. Let tR=2 and tL=0.5. Find r and the open-chain hopping after similarity transformation.
Answer. r=√4=2, and the symmetric hopping is √1=1. Right eigenvector amplitudes acquire the envelope 2ʲ before normalisation. The open spectrum is real, but the ordinary right eigenvectors accumulate near the right boundary.
Problem 10: sensor claim. A device shows eigenvalue splitting proportional to √δ. What information is still needed to establish improved parameter estimation?
Answer. A measurement likelihood or output statistic, its noise, observation time, conditioning probability if applicable, input resources and a comparator with the same allowed resources. The eigenvalue scaling establishes spectral response, not the final uncertainty of an estimator.
32. How to study the subject without learning misleading slogans
First master the triangular matrix and its left-right basis. Use it to reconstruct a general vector and evolve it. This establishes why biorthogonality is an algebraic tool rather than an automatic replacement for every probability rule.
Next solve the PT dimer in three regimes. Check H², the eigenvectors, the metric and the exact exponential. At the exceptional point, construct the Jordan chain instead of attempting to force a diagonal basis. These calculations expose the core structure more clearly than a collection of dramatic eigenvalue plots.
Then derive a conditional decay process from a Lindblad equation. Compute both the branch probability and the normalised branch state. Keep a separate calculation for the unconditional density matrix. This step prevents the most common physical misinterpretation of non-Hermitian evolution.
Finally compare a spectral claim, a dynamical claim and a sensing claim for the same model. The spectrum describes eigenvalues. Dynamics needs eigenvectors, initial conditions and time. Sensing needs a measurement and noise model. Each layer adds information; none can be inferred completely from the preceding layer.
Frequently asked questions
Does non-Hermitian mean unphysical? No. Effective, conditional and classical-wave descriptions often use non-Hermitian matrices legitimately. What is unphysical is assigning probabilities inconsistently or omitting necessary environmental and conditioning information.
Does PT symmetry guarantee real eigenvalues? Not by itself. In the dimer, γ<g gives a real pair and γ>g gives a conjugate imaginary pair. The operator symmetry remains, but the eigenvectors realise it differently.
Is every exceptional point a quantum phase transition? No. An exceptional point is a defective spectral degeneracy. Whether it also marks a many-body phase transition requires a separate model, order of limits and physical diagnostic.
Can a real spectrum coexist with edge accumulation? Yes. The asymmetric open chain provides an exact example. Its similarity to a Hermitian chain is nonunitary and therefore does not preserve ordinary eigenvector amplitudes.
What is the safest summary? A non-Hermitian model can encode useful physics, but its interpretation requires more than an eigenvalue calculation. State the model, metric or conditional ensemble, observables and resource accounting.
Sources and further study
[1] Yuto Ashida, Zongping Gong and Masahito Ueda, Non-Hermitian Physics, Advances in Physics 69, 249–435 (2020). Foundations, effective open-system descriptions and non-Hermitian topology.
[2] W. D. Heiss, The physics of exceptional points, Journal of Physics A 45, 444016 (2012). Defective spectral degeneracies and their consequences.
[3] Carl M. Bender and Stefan Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Physical Review Letters 80, 5243 (1998).
[4] Shunyu Yao and Zhong Wang, Edge states and topological invariants of non-Hermitian systems, Physical Review Letters 121, 086803 (2018). Skin effects and non-Bloch boundary analysis.
[5] Hoi-Kwan Lau and Aashish A. Clerk, Non-Hermitian Quantum Sensing: Fundamental Limits and Non-Reciprocal Approaches, published as Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing, Nature Communications 9, 4320 (2018).
[6] Réouven Assouly and colleagues, Enhanced Sensitivity near a Quantum Exceptional Point in the Absence of Engineered Dissipation, arXiv preprint first submitted 14 June 2026. The cited result concerns its stated non-quantum-limited regime; it is not presented here as a universal quantum-limit advantage.
Continue through Quantum Mathematics — Batch 19
Guide 73: Quantum Floquet Systems, Periodic Driving, Quasienergies, Magnus Expansion and Floquet Engineering develops repeated driven evolution. Guide 75: Many-Body Localization, Anderson Localization, l-Bits, Entanglement Growth and Thermalisation Breakdown examines localisation without confusing it with a skin effect. Guide 76: Non-Markovian Quantum Dynamics, Memory Kernels, Nakajima–Zwanzig Equations and Information Backflow develops complete reduced dynamics with memory.