Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group S N and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.
Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group S N and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.
Quantum Markovian master equation in N dimensions are systematically transformed into vector form by using the Lie algebra of the special unitary group SU(N). Density operators are represented through coherence vectors and the infinitesimal Kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable, in general. A complete classification of the spectrum of the latter and its Jordan canonical form, together with all types of associated stationary states, is given in detail. The results of this analysis are particularly suited for computations in practical applications.
In the thesis we present an analytic approach towards exact description for steady state density operators of nonequilibrium quantum dynamics in the framework of open systems. We employ the so-called quantum Markovian semi-group evolution, i.e. a general form of time-autonomous positivity and trace-preserving dynamical equation for reduced density operators, by only allowing unitarity-breaking dissipative terms acting at the boundaries of a system. Such setup enables to simulate macroscopic reservoirs for different values of effective thermodynamic potentials, causing incoherent transitions between quantum states which are modeled with aid of the Lindblad operators. This serves as a simple minimalistic model for studying quantum transport properties, either in the linear response domain or in more general regimes far from canonical equilibrium. We are mainly exploring possibilities of identifying nonequilibrium situations which are amenable to exact description within matrix product state representation, by exclusively focusing on steady states, i.e. fixed points of the Lindblad equation, of certain prototypic interacting integrable spin chains driven by incoherent polarizing processes. Finally, we define a concept of pseudo-local extensive almost-conserved quantities by allowing a violation of time-invariance up to boundary-localized terms. We elucidate the role of such quantities on non-ergodic behaviour of temporal correlation functions rendering anomalous transport properties. It turns out that such conservation laws can be generated by means of boundary universal quantum transfer operators of the fundamental integrable models.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.
The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).