Postselected Non-Hermitian Hamiltonian

• Postselected non-Hermitian Hamiltonians are effective generators of conditional quantum dynamics, obtained by restricting evolution to no-quantum-jump trajectories.
• They bridge quantum trajectory methods with equilibrium many-body formulations by equating effective Hamiltonians to retarded self-energies in correlated systems.
• Their unique spectral properties, biorthogonality, and emergence of exceptional points critically inform observable response functions and dynamical phase transitions.

Searching arXiv for the cited topic and related papers. A postselected non-Hermitian Hamiltonian is an effective generator of conditional quantum dynamics obtained by restricting an open-system evolution to a selected measurement record, most commonly a no-quantum-jump trajectory. In that setting the state evolves nonunitarily, its norm encodes the probability of the retained record, and physical observables are evaluated after conditional renormalization. The notion has become central across several domains because the same non-Hermitian structure appears in quantum trajectories, in retarded single-particle descriptions of correlated electrons, in exceptional-point physics, and in monitored many-body dynamics. A major development was the demonstration that the postselected effective Hamiltonian of an open quantum system can coincide exactly with the non-Hermitian single-particle Hamiltonian extracted from a correlated material’s retarded self-energy, thereby linking postselection-based and equilibrium many-body formulations of non-Hermiticity (Michishita et al., 2020).

1. Formal definition in monitored open quantum systems

For a Markovian open quantum system with Hamiltonian $H$ and jump operators $L_\mu$, the density matrix obeys the Lindblad equation

$$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$

In the quantum-trajectory picture this mixed-state evolution is unraveled into stochastic pure-state histories consisting of continuous no-jump segments interrupted by quantum jumps. Conditioning on a no-jump record yields the effective non-Hermitian Hamiltonian

$$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$

and the unnormalized conditional state evolves as

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$

The norm $\langle \psi(t)|\psi(t)\rangle$ is the no-jump survival probability, so conditional expectation values require renormalization: $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$ These relations define the postselected non-Hermitian Hamiltonian in its standard operational sense (Hayata et al., 2021).

The construction is physically meaningful only because it refers to a subensemble of measurement outcomes. Without postselection the correct description is the full Lindbladian mixed-state dynamics, not pure-state non-Hermitian evolution. This distinction becomes acute in many-body systems, where the no-jump probability can decay exponentially with particle number and time; for local jump rates of order $\Gamma$ per particle, the survival probability scales as $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$. Postselection is therefore not merely a technical convenience but an exponentially costly resource (Chaduteau et al., 9 Jul 2025).

A related complexity-theoretic formulation treats non-Hermitian evolution as conditional dynamics of the form $U=e^{-iHt}$, followed by state renormalization. In that framework a postselected state is represented as

$L_\mu$0

and the central control parameter is the singular-value structure of $L_\mu$1, which quantifies how far the conditional map is from unitary evolution (Barch et al., 3 Jun 2025).

2. Microscopic derivation and equivalence to correlated-electron self-energies

A microscopic derivation begins from a system-plus-bath von Neumann equation, traces out the bath, and obtains a quantum master equation for a chosen subsystem. In the Hubbard-model construction of a single electronic mode $L_\mu$2, the reduced dynamics is expressed directly through the bath self-energies. After postselection onto a fixed subsystem particle number, the gain and loss terms are eliminated and the reduced density matrix obeys a non-Hermitian equation generated by

$L_\mu$3

with the anti-Hermitian part arising microscopically from the retarded bath self-energy. Under the Markov approximation and long-time limit, the memory kernel becomes time local and the conditional dynamics reduces to a von Neumann equation with this effective non-Hermitian Hamiltonian (Michishita et al., 2020).

The same structure appears in equilibrium single-particle physics of strongly correlated electrons. There the retarded Green’s function satisfies

$L_\mu$4

which suggests the effective single-particle Hamiltonian

$L_\mu$5

The crucial result is that this $L_\mu$6 is identical to the postselected non-Hermitian Hamiltonian obtained from the open-system quantum master equation. For the density matrix relevant to the retarded Green’s function, the off-diagonal component such as $L_\mu$7 is annihilated by the gain/loss superoperators, so only the non-Hermitian generator survives. Consequently,

$L_\mu$8

which matches the familiar Dyson form (Michishita et al., 2020).

This equivalence clarifies a longstanding asymmetry between fields. In open systems, postselection is required to realize pure-state non-Hermitian evolution. In strongly correlated materials, by contrast, the same non-Hermitian structure emerges directly in equilibrium response functions because the relevant off-diagonal density-matrix elements eliminate the gain/loss superoperators structurally. The result is not a heuristic analogy but an identity between effective generators.

The equivalence also constrains when a static non-Hermitian band picture is valid. If $L_\mu$9 varies slowly with frequency, one may approximate $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$0. In strongly correlated regimes such as Mott or Kondo physics, however, retaining the full $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$1-dependence is essential; Markovian evaluation of $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$2 at a single energy can severely distort the spectral function (Michishita et al., 2020).

3. Spectral structure, biorthogonality, and exceptional points

Because $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$3 is non-Hermitian, right and left eigenvectors must be distinguished: $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$4 with biorthogonal normalization

$$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$5

Expectation values are then evaluated biorthogonally as $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$6, and Green’s-function residues involve projectors $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$7. Under postselected evolution, state norms decay and must be renormalized, which is consistent with this biorthogonal framework (Michishita et al., 2020).

Exceptional points arise when eigenvalues and eigenvectors coalesce. A widely studied route is $$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$8-symmetric non-Hermitian dynamics. In a two-level model with

$$\frac{d\rho}{dt}=-i[H,\rho]+\sum_\mu\left(L_\mu\rho L_\mu^\dagger-\frac12\{L_\mu^\dagger L_\mu,\rho\}\right).$$9

the eigenvalues are

$$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$0

and coalesce at $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$1, which marks spontaneous $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$2-symmetry breaking (Gopalakrishnan et al., 2020). The exceptional point is not merely spectral: near it the biorthogonal basis becomes ill-conditioned, and the late-time state-selection mechanism changes qualitatively.

A microscopic open-system realization of exceptional points was derived for the open Jaynes–Cummings model. Using a Moore–Penrose normalized $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$3 representation that removes the vacuum-sector singularity, the postselected no-jump Hamiltonian acquires sector-resolved eigenvalues

$$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$4

The exceptional-point condition is the vanishing of the discriminant. For independent losses on resonance this recovers the single-excitation condition $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$5, while equal correlated losses with orthogonal channel phase produce a second-order exceptional point at $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$6 in every excitation sector, a regime absent from the standard phenomenological model (Rodríguez-Lara, 12 Jun 2026).

A conceptually distinct but related perspective is provided by “automatic Hermiticity.” There a diagonalizable non-Hermitian Hamiltonian is rendered normal with respect to a positive metric $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$7, and long-time evolution selects the subspace with maximal imaginary part. On that surviving subspace the anti-$$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$8-Hermitian contribution acts only as a global rescaling and the effective generator becomes $$H_{\mathrm{eff}}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu,$$9-Hermitian. This is not the same as quantum-trajectory postselection, but it provides a parallel mechanism in which non-Hermitian evolution dynamically filters state space and yields an emergent Hermitian description (Nagao et al., 2010).

4. Response functions, non-Hermitian topology, and the limits of postselection

Once the identification $$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$0 is made, non-Hermitian physics can be diagnosed without postselection through standard response functions. The primary objects are

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$1

Linewidths and asymmetries reflect $$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$2 and its momentum or orbital dependence; peak splittings and anomalous dispersions can signal complex off-diagonal self-energies; and bulk Fermi arcs or anomalous gap filling can indicate exceptional degeneracies in $$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$3. Suggested probes include ARPES for $$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$4, STM for local spectra, and momentum-resolved EELS or RIXS for orbital-selective features (Michishita et al., 2020).

Topological classification requires additional care because the topology of the postselected non-Hermitian Hamiltonian need not match the topology of the full Lindbladian relaxation dynamics. In a translation-invariant quadratic fermion system with loss and gain, the postselected Bloch Hamiltonian is

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$5

whereas the single-particle matrix controlling the Lindbladian relaxation spectrum is

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$6

The sign difference in the gain term is decisive. With loss only, the Lindbladian and postselected winding numbers coincide; with gain only, they are opposite. When gain and loss compete, the Lindbladian winding can change sign at a point-gap closing that reverses the Lindbladian skin effect, while the postselected winding remains fixed and therefore misses the transition (Chaduteau et al., 9 Jul 2025).

This directly addresses a common misconception: postselected non-Hermitian topology does not exhaust the topology of an open quantum system. In the Hatano–Nelson-type example analyzed in the literature, gain–loss balance can close the point gap of the Lindbladian relaxation matrix and eliminate or reverse skin localization even though the postselected Hamiltonian retains a nonzero winding. Conversely, removing postselection can reveal a Lindbladian skin effect when the postselected non-Hermitian dynamics is topologically trivial (Chaduteau et al., 9 Jul 2025).

5. Thermodynamic relations, entanglement transitions, and many-body dynamical phases

Postselected non-Hermitian Hamiltonians also support nonequilibrium thermodynamic relations. In a two-level no-jump framework, the unnormalized transition probability is

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$7

and the normalized conditional probability is

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$8

For work defined through the Hermitian part of the Hamiltonian in the two-point measurement scheme, a conditional non-Hermitian Jarzynski equality holds if and only if

$$|\psi(t)\rangle=e^{-iH_{\mathrm{eff}}t}|\psi(0)\rangle.$$9

In the $\langle \psi(t)|\psi(t)\rangle$0-rotated hybrid $\langle \psi(t)|\psi(t)\rangle$1–$\langle \psi(t)|\psi(t)\rangle$2 family,

$\langle \psi(t)|\psi(t)\rangle$3

this condition is enforced by a parity-exchange symmetry. A trapped-$\langle \psi(t)|\psi(t)\rangle$4 experiment implemented $\langle \psi(t)|\psi(t)\rangle$5, used $\langle \psi(t)|\psi(t)\rangle$6, $\langle \psi(t)|\psi(t)\rangle$7, and $\langle \psi(t)|\psi(t)\rangle$8 repetitions per transition, and observed $\langle \psi(t)|\psi(t)\rangle$9 for cyclic protocols within the protected subspace. Under sinusoidal detuning the symmetry generally broke down, but it revived at $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$0 and $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$1 when the effective Floquet Hamiltonian re-entered the hybrid form (Yang et al., 11 May 2026).

In monitored many-body systems, postselection can itself drive phase transitions. For the non-Hermitian Hubbard model with onsite pair loss,

$$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$2

time strengthens postselection filtering through factors $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$3, so time itself becomes an intrinsic control parameter. Starting from a half-filled Néel state in three dimensions, the system undergoes a dissipation-induced dynamical phase transition into ferromagnetic order at a finite critical time. For $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$4, finite-size scaling yielded

$$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$5

and the critical behavior was consistent with the 3D classical XY universality class (Hayata et al., 2021).

A distinct but related transition occurs in continuously measured spin systems. There the postselection rate drives an exceptional-point transition between a mixed phase, where all modes share a common decay rate and volume-law entanglement persists, and a pure phase, where distinct decay rates select a unique longest-lived eigenvector with low entanglement. Mean-field theory gives the transition line

$$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$6

In the two-level limit, the exceptional point also controls purification dynamics, with critical algebraic behavior $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$7 exactly at the exceptional point (Gopalakrishnan et al., 2020).

6. Experimental realizations, quantum simulation, and practical constraints

Several platforms now realize postselected non-Hermitian Hamiltonians directly. In a three-level ladder $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$8, postselecting against the $$\langle \hat O\rangle_{\mathrm{cond}}(t)=\frac{\langle \psi(t)|\hat O|\psi(t)\rangle}{\langle \psi(t)|\psi(t)\rangle}.$$9 fluorescence jump leaves an effective two-level dynamics in the $\Gamma$0 subspace with

$\Gamma$1

For resonant $\Gamma$2- and $\Gamma$3-axis drives, this produces the matrices

$\Gamma$4

Continuous homodyne measurement of the $\Gamma$5 fluorescence then probes the competition between measurement backaction and postselection-induced loss. With $\Gamma$6 and $\Gamma$7, the exceptional-point scale is $\Gamma$8. Far from the exceptional point the ensemble-averaged stochastic dynamics agrees with the Liouvillian average, whereas near the exceptional point the discrepancy depends sensitively on the drive axis and measured quadrature. The same framework supports an optimal-path description obtained by extremizing a trajectory action (Nongthombam et al., 15 Oct 2025).

Gate-model simulation faces the central obstacle that iterative ancilla postselection makes the total success probability exponentially small in the number of Trotter steps. For the qubit model

$\Gamma$9

the traditional Trotter protocol implements $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$0 as a product of many postselected nonunitary steps, with total success bounded by $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$1. A fixed-depth variational circuit demonstrated on IBM superconducting qubits replaces this many-step protocol by a single trained dilation with one ancilla postselection, allowing simulation deep in the $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$2-broken regime and near the exceptional point while avoiding the exponential postselection overhead of the iterative approach (Jebraeilli et al., 19 Feb 2025).

From a computational perspective, non-Hermitian dynamics and postselection are approximately equivalent resources. If a non-unitary gate $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$3 has normalized singular-value radius

$P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$4

then, together with universal unitary gates, it can efficiently emulate postselection and hence decide every language in PostBQP. Conversely, any non-Hermitian evolution admits a compact unitary purification on a system–meter pair with $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$5 Trotter error per time step $P_{\mathrm{no\,jump}}(t)\sim e^{-N\Gamma t}$6. This implies that non-Hermitian dynamics neither automatically yields computational advantage nor evades efficient classical simulation; the decisive quantities are the singular-value structure of the conditional propagator and the structure of its unitary dilation (Barch et al., 3 Jun 2025).

The principal limitations are now clear. Postselection remains exponentially costly in large systems; mixed-state Lindbladian physics can differ qualitatively from the topology or dynamics inferred from the postselected Hamiltonian; Markovian reductions can fail badly in strongly correlated regimes with pronounced memory effects; and beyond single-particle observables, higher-order self-energies or vertex corrections enter. Postselected non-Hermitian Hamiltonians are therefore best understood as precise conditional generators with well-defined domains of validity rather than universal substitutes for full open-system dynamics.