We report analytical solutions to the unitary bound problem for coherence/polarization transfer in IS two-spin-12 systems by means of unitary operations. Theoretical upper bounds for the transfer efficiency along with the associated optimum transformation operators are obtained analytically by decomposing the unitary operator as a product of exponentials in the special unitary Lie group SU(4). Addressing NMR spectroscopy as a specific example, the method is demonstrated for the non-Hermitian transfers I−→S− and 2I−Sz→S− being relevant for heteronuclear single-quantum coherence (HSQC) experiments as well as the double- to single-quantum transfer I−S−→I−Sβ+IβS− being representative for coherence-order and spin-state-selective transfer in INADEQUATE CR experiments. Furthermore, using a Lagrangian function approach it is demonstrated how the method enables analytical description of two-dimensional bounds for Iz→Sz cross polarization.
The effect of spin permutation symmetry on the maximum achievable efficiency of coherence transfer between pairs of Hermitian or non-Hermitian spin operators is investigated. It is shown that spin permutation symmetry may tighten significantly the quantum mechanical bounds hitherto known to apply for coherence transfer processes. This feature is demonstrated for the heteronuclear coherence transfer processes S x → F x , S - → F - and 2F z S x → F x , 2F z S - → F - (F = Σ n i=1 I i ) in I2S and I3S (n = 2,3) spin-1/2 systems. The bounds arising from permutation symmetry apply for non-selective multidimensional NMR experiments.
The non-Hermitian Hamiltonian describes the effective dynamics of a system coupled to a continuously measured bath, and can exhibit anti-unitary symmetries that give rise to exceptional points and broken phases with complex eigenvalues, features unique to non-Hermitian systems. Going beyond conventional Hermitian physics, we analyze the impact of non-Hermiticity in the quantum state transmission by employing a non-Hermitian spin chain that functions as a quantum data bus. By deriving a general expression for the fidelity of quantum state transfer for a U(1)-symmetric non-Hermitian Hamiltonian, we analyze PT-symmetric XX and SSH models, complemented by a numerical study of the RT-symmetric iXY model. We demonstrate that, in several parameter regimes, the transfer fidelity in the non-Hermitian setting exceeds the classical threshold and can even exceed the performance of the corresponding Hermitian models. In particular, for the SSH model with dominant inter-cell coupling, the broken phase supports near-unit-fidelity quantum state transfer, a level of performance that the corresponding Hermitian model fails to attain. Moreover, we establish a correspondence between the non-Hermitian and Hermitian descriptions by identifying related parameter regions in which the fidelity fails to surpass the classical bound.
Non-unitary operations generated by an effective non-Hermitian Hamiltonian can be used to create quantum state manipulations which are impossible in Hermitian quantum mechanics. These operations include state preparation (or cooling) and non-orthogonal state discrimination. In this work we put a lower bound on the resources needed for the construction of some given non-unitary evolution. Passive systems are studied in detail and a general feature of such systems is derived. After interpreting our results using the singular value decomposition, several examples are studied analytically. In particular, we put a lower bound on the resources needed for non-Hermitian state preparation and non-orthogonal state discrimination.
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).