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Register a new userTheory of Ergodic Quantum Processes
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The generic behavior of quantum systems has long been of theoretical and practical interest. Any quantum process is represented by a sequence of quantum channels. Random channels appear in a wide variety of applications, from quantum chaos to holographic dualities in theories of quantum gravity to operator dynamics, to random local circuits for their potential to demonstrate quantum supremacy. We consider general ergodic sequences of stochastic channels with arbitrary correlations and non-negligible decoherence. Ergodicity includes and vastly generalizes random independence. We obtain a theorem which shows that the composition of such a sequence of channels converges exponentially fast to a rank-one (entanglement breaking) channel. Using this, we derive the limiting behavior of translation invariant channels, and stochastically independent random channels. We then use our formalism to describe the thermodynamic limit of ergodic Matrix Product States. We derive formulas for the expectation value of a local observable and prove that the 2-point correlations of local observables decay exponentially. We then analytically compute the entanglement spectrum across any cut, by which the bipartite entanglement entropy (i.e., R{e}nyi or von Neumann) across an arbitrary cut can be computed exactly. Other physical implications of our results are that most Floquet phases of matter are meta-stable, and that noisy random circuits in the large depth limit will be trivial as far as their quantum entanglement is concerned. To obtain these results we bridge quantum information theory to dynamical systems and random matrix theory.
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Any quantum process is represented by a sequence of quantum channels. We consider ergodic processes, obtained by sampling channel valued random variables along the trajectories of an ergodic dynamical system. Examples of such processes include the effect of repeated application of a fixed quantum channel perturbed by arbitrary correlated noise, or a sequence of channels drawn independently and identically from an ensemble. Under natural irreducibility conditions, we obtain a theorem showing that the state of a system evolving by such a process converges exponentially fast to an ergodic sequence of states depending on the process, but independent of the initial state of the system. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk. Further applications and physical implications of our results are discussed in the companion paper [11].
In [arxiv:2106.02560] we proposed a reduced density matrix functional theory (RDMFT) for calculating energies of selected eigenstates of interacting many-fermion systems. Here, we develop a solid foundation for this so-called $boldsymbol{w}$-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights $boldsymbol{w}$ in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valones pioneering work on ground state RDMFT as the special case $boldsymbol{w}=(1,0,ldots)$. Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functionals domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.
We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valones pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functionals domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size.
Unitary dynamics with a strict causal cone (or light cone) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb-Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb-Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb-Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest.
Despite free-fermion systems being dubbed exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or measures of entanglement. We derive closed expressions for such nonlocal quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains known as generalised cluster models. While there is a bijection between general models in these fermionic classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results. In particular, (1) we derive closed expressions for the string correlation functions -- the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into the entanglement of the ground state; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit -- an independent and explicit construction for the BDI class is given in a concurrent work (Jones, Bibo, Jobst, Pollmann, Smith, Verresen, arXiv:2105.12143); (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. We show how general models in these classes can be obtained by taking limits of the models studied in this work, which can be used to apply limits of our results to the general case. To the best of our knowledge, these results constitute the first application of Days formula and Gorodetskys formula from the theory of Toeplitz determinants to the context of many-body quantum physics.
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