A self-contained treatment of the Bogoliubov-Valatin transformation for homogeneous fermionic Hamiltonians is presented. The aim is to provide a quick reference that may also serve as supplementary material for a graduate-level course, and that can be understood with quantum mechanics knowledge up to the level of the second quantization's rules. The objective of the transformation is to cast a quadratic Hamiltonian into a diagonal form that resembles the Hamiltonian of a system of non-interacting particles. To obtain this, the first step consists in putting its coefficient matrix into its canonical form; the transformation can always be performed on fermionic Hamiltonians, only some care must be taken when this form is singular. Having explained how to cast a general matrix into its standard form, a complete description of the transformation is provided; a novel procedure is proposed here for the singular matrix case.
Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) provide an efficient framework for implementing polynomials of block-encoded matrices, and thus offer a systematic approach to quantum algorithm design. However, despite a number of recent advances, important limitations remain. In particular, QSP can only transform unitary matrices, by applying a polynomial to their eigenvalues, while QSVT is a singular-value transformation and thus one can only obtain the polynomial of Hermitian matrices. As a consequence, these techniques do not directly apply to an arbitrary non-Hermitian matrix that is not diagonalizable. In this work, we propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues. To this end, we introduce the notion of an
n-regular block encoding, namely, a block encoding whose
k-th power reproduces the
k-th power of the encoded matrix for every
0<k<n. We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most
n to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an
n-regular one using only
O(logn) ancillary qubits and operations. Finally, we show that this construction induces the desired transformation on the eigenvalues associated with the Jordan normal form of the matrix; for non-diagonalizable matrices this yields the full matrix function
P(A), whose action on each Jordan block involves the derivatives of
P, and not merely the map
λ↦P(λ).
We study hypergeometric functions of nilpotent operators in finite-dimensional settings, motivated by the algebraic structure of exceptional points in non-Hermitian quantum mechanics. Our starting point is the following exact result: if N is a nilpotent operator of index m+1 in an associative algebra over C, then every generalized hypergeometric function pFq evaluated at N reduces to a finite polynomial in N of degree at most m, without any analytic convergence requirement. This "functional collapse" is distinct from the classical parameter-termination mechanism and arises purely from the nilpotent structure of the argument. The main result is a "nilpotent depth criterion" (Theorem 2): if the first non-constant coefficient of a formal series F appears in degree r >= 1, then the nilpotent part F(N) - F(0)I has nilpotency index bounded above by ceil((m+1)/r). We apply this criterion to Hamiltonians at exceptional points, where H = lambda I + N with N^{m+1} = 0. Theorem 3 establishes that a function F analytic at lambda reduces the Jordan depth of the exceptional point from m+1 to at most ceil((m+1)/r), where r is the contact order of F at lambda. As consequences: the time evolution operator e^{tH} preserves the full Jordan depth for all t != 0; a function with a zero of order m+1 at lambda annihilates the entire Jordan structure; and the order of the pole of the modified resolvent is reduced from m+1 to at most m+1-r. Results are illustrated with explicit 3x3 Jordan block computations for 1F1, 2F1, and the time evolution operator, confirming sharpness of the bounds.
We consider quantum mechanics with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells, in particular, biorthogonal bases. The 'self-orthogonality' phenomenon is clarified in terms of a correct spectral decomposition and it is shown that 'self-orthogonal' states never jeopardize a resolution of identity and thereby quantum averages of observables. The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated. Some puzzles with zero-binorm bound states in a continuous spectrum are unravelled with the help of a correct resolution of identity.
Abstract The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k -point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.