Biorthogonal universal quantum control

Abstract

As a paradigm for describing open-quantum-system dynamics, the non-Hermitian Schrödinger equation generally modifies state norms, complicating coherent control over the system's Hilbert space. With a general instantaneous frame spanned by biorthogonal ancillary bases, we develop a biorthogonal quantum-control theory for discrete-variable systems governed by time-dependent non-Hermitian Hamiltonian. In particular, we derive a necessary and sufficient condition for extracting the transitionless pathways that transport parallel in ket and bra states. When this condition is satisfied by a set of projectors constructed with biorthogonal basis states, the non-Hermitian Hamiltonian becomes diagonal in the relevant biorthogonal subspace, enabling arbitrary initial state to evolve as a superposition of controlled transitionless passages. We further identify the parametric constraints under which the transformation linking the initial and final states becomes unitary. As a result, the final system state is automatically normalized even though the transitional dynamics is nonunitary. Our theory is justified by the bidirectional population transfer and a holomonic quantum gate implemented by a non-Hermitian three-level system in $Λ$ type. For the population transfer, controlled gain and loss reduce the required coherent pulse area, enabling shorter transfer times than the Hermitian benchmark with the same average Rabi frequency. Our work integrates the universal quantum control (UQC) into the biorthogonal representation, displaying full-rank passage engineering with coherent operations in non-Hermitian systems.

The paper addresses how to control multiple transitionless pathways simultaneously in a finite-dimensional system governed by a time-dependent non-Hermitian Hamiltonian, while obtaining a unitary transformation at the final time despite intermediate norm changes. It extends an ancillary-frame control construction to complete pairs of biorthogonal right and left bases. A nonunitary moving-frame transformation introduces a non-Hermitian gauge potential. The central analytical result is that an ancillary projector $\Pi_k=|\mu_k^R\rangle\langle\mu_k^L|$ obeys $\dot\Pi_k=-i[H,\Pi_k]$ if and only if its pathway decouples in the rotating frame. Imposing this condition on a complete basis diagonalizes the rotating-frame generator and gives an explicit propagator for arbitrary initial states. Orthonormal endpoint bases and cancellation of the imaginary accumulated pathway phases provide sufficient conditions for a unitary endpoint map. The construction is illustrated analytically and numerically by bidirectional population transfer in a two-level system and a single-qubit gate in a three-level $\Lambda$ system. For transfer, the required coherent drive is reduced by a factor $1/\cosh\alpha_i(t)$ relative to the corresponding Hermitian pathway, allowing shorter transfer times at fixed average Rabi frequency by using controlled gain and loss as additional resources. The gate protocol cancels integrated dynamical and imaginary geometric contributions over a cycle. Both examples require time-dependent gain/loss rates that change sign. The discussion identifies classical optical and magnonic analog simulations as implementation routes and notes that restricted controls may permit endpoint-unitary control only within selected subspaces rather than the full Hilbert space.