Biorthogonal Quantum Mechanics

• Biorthogonal quantum mechanics is defined using dual eigenbases and a metric operator, enabling a consistent treatment of non-Hermitian systems.
• It provides a rigorous framework for spectral analysis and quantum information, particularly in PT-symmetric and noncommutative models.
• The approach facilitates precise computation of observables and probabilities, supporting applications in quantum phase transitions and resource theories.

Biorthogonal quantum mechanics is a formal extension of quantum theory in which the orthogonality of eigenvectors, a defining feature of standard (Hermitian) quantum mechanics, is generalized to biorthogonality. This framework enables a consistent quantum description of systems governed by non-Hermitian (but diagonalizable) operators, as well as providing a precise algebraic foundation for noncommutative geometries, pseudo-Hermitian, and PT-symmetric models. Biorthogonal quantum mechanics also has deep implications in mathematical physics, quantum information, open quantum systems, and the spectral analysis of non-Hermitian Hamiltonians.

1. Foundations: Biorthogonal Bases and Metric Structures

Consider an operator $H$ (not necessarily Hermitian) acting on a complex vector space or Hilbert space $\mathcal{H}$, possibly infinite-dimensional. If $H$ is diagonalizable (i.e., possesses a complete set of eigenvectors), there exists a set of right eigenvectors $|\psi_n\rangle$ and left eigenvectors $\langle\phi_n|$, satisfying

$$H|\psi_n\rangle = E_n |\psi_n\rangle, \qquad H^\dagger|\phi_n\rangle = E_n^* |\phi_n\rangle,$$

with the biorthogonality relation

$\langle\phi_m|\psi_n\rangle = \delta_{mn},$

and the resolution of identity

$$\sum_{n} |\psi_n\rangle \langle\phi_n| = \mathbb{I}.$$

This construction replaces the symmetric orthonormal basis of standard quantum mechanics with a pair of dual (generally nonorthogonal) bases. Physical states can be expanded in the right basis, and observables or expectation values are consistently defined by combining right and left representatives (Brody, 2013, Edvardsson et al., 2022, Bagarello et al., 2017). Gauge degrees of freedom remain; the biorthogonal pairing is invariant under simultaneous rescalings $|\psi_n'\rangle = \lambda_n |\psi_n\rangle$, $\langle\phi_n'| = \lambda_n^{-1} \langle\phi_n|$.

An associated metric operator $\eta$ can be introduced: $$\eta = \sum_n |\phi_n\rangle \langle\phi_n| \qquad \Rightarrow \qquad (\chi,\psi)_\eta = \langle\chi|\eta|\psi\rangle,$$ acting as an inner product with respect to which $\{ |\psi_n\rangle \}$ is orthonormal (Brody, 2013, Edvardsson et al., 2022, Chen, 2021).

2. Observables, Probabilities, and the Born Rule

In biorthogonal quantum mechanics, a "physical" state is specified by a vector $|\Psi\rangle$ and its associated dual $\langle\widetilde{\Psi}| = \langle\Psi|\eta$, yielding a biorthogonal inner product: $$(\Psi, \Psi)_\text{bio} = \langle\widetilde{\Psi} | \Psi\rangle = \sum_n |c_n|^2,$$ if $|\Psi\rangle = \sum_n c_n |\psi_n\rangle$.

Expectation values of an observable $O$ are given by

$$\langle O \rangle = \frac{\langle\widetilde{\Psi}|O|\Psi\rangle}{\langle\widetilde{\Psi}|\Psi\rangle}.$$

For a "projective" measurement (onto eigenstate $|\psi_n\rangle$), the transition probability is

$P_n = |\langle\phi_n|\Psi\rangle|^2 = |c_n|^2,$

so the Born rule is preserved modulo replacement of the standard inner product by the biorthogonal pairing (Brody, 2013, Hawton et al., 2015, Edvardsson et al., 2022). Care must be taken in defining observables: they must be "self-adjoint" with respect to the metric, i.e., $O^\dagger = \eta O \eta^{-1}$.

3. Biorthogonal Systems Beyond Hermitian Quantum Mechanics

The biorthogonal framework is crucial for non-Hermitian, PT-symmetric, and pseudo-Hermitian quantum systems. Such Hamiltonians may exhibit real spectra (in the unbroken symmetry phase), parameterized by quasi-Hermiticity relations: $O^\dagger = \eta O \eta^{-1},$ enabling a mapping to a Hermitian theory under a similarity transformation (Chen, 2021, Edvardsson et al., 2022, Bagarello et al., 2016, Bagarello et al., 2017). This structure underlies the rigorous definition of observables, Gibbs states, and dynamical evolution (via para-unitary operators) in non-Hermitian systems.

In infinite dimensions, or under less restrictive assumptions, completeness is captured by the weaker concept of a G-quasi basis, where the dual families resolve identity and expectation values on a dense subdomain $\mathcal{G}$ of $\mathcal{H}$ (Bagarello et al., 2017).

4. Biorthogonal Quantum Mechanics in Noncommutative and Deformed Quantum Systems

Biorthogonal quantum mechanics provides explicit diagonalizations and completeness structures for quantum systems with noncommuting coordinates and momenta. For example, biorthogonal families of deformed complex Hermite polynomials $\widetilde H_{m,n}(z, \bar z)$ emerge via GL(2, $\mathbb{C}$) transformations applied to standard complex Hermite polynomials: $$\widetilde H_{m,n}(z,\bar z) = \sqrt{m!n!} \langle z,\bar z | \frac{(\widetilde a_1^\dagger)^m}{m!}\frac{(\widetilde a_2^\dagger)^n}{n!} |0,0\rangle,$$ where generators $\widetilde a_1^\dagger, \widetilde a_2^\dagger$ are linear combinations of the standard bosonic creation operators (Balogh et al., 2013).

A biorthogonal dual family $\{ \widetilde K_{m,n} \}$ is constructed as

$\widetilde K_{m,n} = [T_g^*]^{-1} H_{m,n},$

with biorthogonality relation

$$\int_\mathbb{C} \widetilde H_{m,n}(z,\bar z) \; \overline{\widetilde K_{p,q}(z,\bar z)} e^{-|z|^2} d^2 z = \pi m! n! \delta_{m,p} \delta_{n,q}.$$

These families diagonalize noncommutative quantum models in the Bargmann representation; orthogonality is supplanted by biorthogonality, but completeness and closure properties remain intact (Balogh et al., 2013).

In such systems, observables, transition amplitudes, and expectation values are systematically computed by integrating products of states from the biorthogonal pairs, and spectral decompositions use the biorthogonal dual families.

5. Applications: Quantum Geometry, Open Systems, and Quantum Information

a. Quantum Phase Transitions and Topological Invariants

Biorthogonal quantum mechanics underpins the definition of real-space topological invariants such as biorthogonal polarization: $$\mathcal{P} = \mathrm{Tr} \left( \langle \psi_L | \hat{P} | \psi_R \rangle \right),$$ which remains quantized even when traditional bulk-boundary correspondence fails in non-Hermitian topological phases. The biorthogonal density $\psi_L^*(n) \psi_R(n)$ penetrates the bulk at phase transitions, precisely signaling the relocation of edge modes (Edvardsson et al., 2020, Kunst et al., 2018).

b. Entanglement and Quantum Geometric Tensor

Entanglement entropies for subregions must be evaluated with biorthogonal reduced density matrices, leading to modifications of standard metrics (e.g., von Neumann, SVD, Tu–Tzeng–Chang entropies) and extracting quantum geometric tensors directly from biorthogonal pairings (Lu et al., 27 Jul 2025).

c. Photon Position and Covariant Localization

In relativistic quantum theory, the biorthogonal formalism supports a strictly localized, manifestly covariant position operator, with eigenstates constructed from positive- and negative-frequency field modes: $$|\psi_x^\pm\rangle = \hat{\Phi}^\pm(x)|0\rangle, \qquad |\phi_x^\pm\rangle = \hat{\Pi}^\pm(x)|0\rangle,$$ satisfying biorthogonality and completeness. The biorthogonal transition probability coincides with the first-order Glauber correlation function, unifying photon counting and field energy density concepts (Hawton et al., 2015, Hawton et al., 2016).

d. Non-unitary Operations and Quantum Computation

The biorthogonal representation enables explicit quantum dilation schemes for non-unitary and pseudo-Hermitian operators, extending the quantum circuit model to simulate non-contracting quantum dynamics efficiently. The protocol exploits the biorthogonal basis and associated metric to perform non-unitary maps with optimal resource scaling when eigenvalues satisfy $|\lambda_i| > 1$ (Koukoutsis et al., 2024).

e. Quantum Resource Theories

Resource theories quantifying nonclassicality and superposition in non-orthogonal bases leverage the biorthogonal density matrix. The off-diagonal components in the biorthogonal basis define "genuine quantum superposition" and quantify intra-basis indistinguishability effects, with measures generalizing $l_1$-coherence (Pusuluk, 2022).

6. Advanced Mathematical Structures: Algebras, Gibbs States, and Quasi-Bases

The biorthogonal framework supports the construction of generalized Gibbs states and KMS-like thermal equilibrium conditions in non-Hermitian quantum statistical mechanics: $$\rho_{\mathrm{bi}} = \frac{1}{Z_{\mathrm{bi}}} \sum_{n} e^{-\beta E_n} |\phi_n\rangle\langle\psi_n|, \qquad Z_{\mathrm{bi}} = \sum_{n} e^{-\beta E_n},$$ with observables averaging over the biorthogonal measure. Algebraic dynamics and Heisenberg equations extend straightforwardly, with the caveat that biorthogonality must be enforced at each step (Bagarello et al., 2016).

A central role is played by the classification of complete biorthogonal sets: from Riesz bases with bounded frame operators (leading to well-behaved spectral decompositions), to G-quasi bases allowing only partial or weak completeness on a dense domain. These notions generalize the spectral theory of Hamiltonians and enable the spectral resolution of non-self-adjoint operators possessing real spectra (Bagarello et al., 2017).

7. Outlook, Limitations, and Open Problems

The biorthogonal quantum formalism provides a consistent generalization of quantum mechanics to non-Hermitian, noncommutative, and pseudo-Hermitian realms, with robust applications across quantum information, topological phases, and many-body dynamics. However, foundational issues persist:

• Gauge dependence: Care must be exercised to ensure physical results are invariant under rescaling of the biorthogonal basis. Gauge-independent metric choices (e.g., canonical $\eta$) restore full invariance (Edvardsson et al., 2022).
• Physical meaning of basis vectors: The representation of spatial or site basis vectors is inherently metric-dependent and cannot be carried unchanged across distinct non-Hermitian Hamiltonians (Edvardsson et al., 2022).
• Defective and exceptional-point cases: The full extension of biorthogonal constructions to defective operators (with coalesced eigenvectors) remains an active area of study.
• Expectation values vs. probabilities: In certain generalized or finite-field versions, as in biorthogonal quantum mechanics over Galois fields, expectation values are well defined but micro-probabilities may be fundamentally indeterminate (Chang et al., 2012).

Biorthogonal quantum mechanics thus serves as a unifying mathematical structure for analyzing and generalizing quantum theory beyond Hermitian foundations, with profound implications for both theoretical and applied physics (Brody, 2013, Edvardsson et al., 2022, Bagarello et al., 2017, Balogh et al., 2013).