Communications in Mathematical Physics最新文献

Hadamard states were originally introduced for quantised Klein–Gordon fields and occupy a central position in the theory of quantum fields on curved spacetimes. Subsequently they have been developed for other linear theories, such as the Dirac, Proca and Maxwell fields, but the particular features of each require slightly different treatments. This paper gives a generalised definition of Hadamard states for linear bosonic and fermionic theories encompassing a range of theories that are described by Green-hyperbolic operators with ‘decomposable’ Pauli–Jordan propagators, including theories whose bicharacteristic curves are not necessarily determined by the spacetime metric. The new definition reduces to previous definitions for normally hyperbolic and Dirac-type operators. We develop the theory of Hadamard states in detail, showing that our definition propagates under the equation of motion, and is also stable under pullbacks and suitable pushforwards. There is an equivalent formulation in terms of Hilbert space valued distributions, and the generalised Hadamard condition on 2-point functions constrains the singular behaviour of all n-point functions. For locally covariant theories, the Hadamard states form a covariant state space. It is also shown how Hadamard states may be combined through tensor products or reduced by partial tracing while preserving the Hadamard property. As a particular application it is shown that state updates resulting from nonselective measurements preserve the Hadamard condition. The treatment we give was partly inspired by a recent work of Moretti, Murro and Volpe (MMV) (Ann H Poincaré 24: 3055–3111, 2023) on the neutral Proca field. Among the other applications, we revisit the neutral Proca field and prove a complete equivalence between the MMV definition of Hadamard states and an older work of Fewster and Pfenning (J Math Phys 44:4480–4513, 2003).

We consider the action of a finite group G by locality preserving automorphisms (quantum cellular automata) on quantum spin chains. We refer to such group actions as “symmetries”. The natural notion of equivalence for such symmetries is stable equivalence, which allows for stacking with factorized group actions. Stacking also endows the set of equivalence classes with a group structure. We prove that the anomaly of such symmetries provides an isomorphism between the group of stable equivalence classes of symmetries with the cohomology group (H^3(G,U(1))), consistent with previous conjectures. This amounts to a complete classification of locality preserving symmetries on spin chains. We further show that a locality preserving symmetry is stably equivalent to one that can be presented by finite depth quantum circuits with covariant gates if and only if the slant product of its anomaly is trivial in (H^2(G, U(1)[G])).

This paper presents a conceptual and efficient geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice (mathbb {Z}^n). It is shown that, under the typical assumption of Haag duality, the monoidal (C^*)-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of (mathbb {Z}^n). Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder (mathbb {R}^1times mathbb {S}^{n-1}). While the sphere (mathbb {S}^{n-1}) arises geometrically as the angular coordinates of cones, the origin of the line (mathbb {R}^1) is analytic and rooted in Haag duality. The usual braided (for (n=2)) or symmetric (for (nge 3)) monoidal (C^*)-categories of superselection sectors are recovered by removing a point of the sphere (mathbb {R}^1times (mathbb {S}^{n-1}setminus text {pt}) cong mathbb {R}^n) and using the equivalence between (mathbb {E}_n)-algebras and locally constant prefactorization algebras defined on open disks in (mathbb {R}^n). The non-trivial homotopy groups of spheres induce additional algebraic structures on these (mathbb {E}_n)-monoidal (C^*)-categories, which in the case of (mathbb {Z}^2) is given by a braided monoidal self-equivalence arising geometrically as a kind of ‘holonomy’ around the circle (mathbb {S}^1). The locally constant prefactorization algebra structures discovered in this work generalize, under some mild geometric conditions, to other discrete spaces and thereby provide a clear link between the geometry of the localization regions and the algebraic structures on the category of superselection sectors.

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

We extend the cohomological setting developed by Batalin, Fradkin and Vilkovisky (BFV), which produces a resolution of coisotropic reduction in terms of hamiltonian dg manifolds, to the case of nested coisotropic embeddings (Chookrightarrow C_circ hookrightarrow F) inside a symplectic manifold F. To this, we naturally assign (underline{C}) and (underline{C_circ }), as well as the respective BFV dg manifolds. We show that the data of a nested coisotropic embedding defines a natural graded coisotropic embedding inside the BFV dg manifold assigned to (underline{C}), whose reduction can further be resolved using the BFV prescription. We call this construction double BFV resolution, and we use it to prove that “resolution commutes with reduction” for a large class of nested coisotropic embeddings. We then deduce a quantisation of (underline{C}), from the (graded) geometric quantisation of the double BFV Hamiltonian dg manifold (when it exists), following the quantum BFV prescription. As an application, we provide a well defined candidate space of (physical) quantum states of three-dimensional Einstein–Hilbert theory, which is thought of as a partial reduction of the Palatini–Cartan model for gravity.

We investigate the extent to which two particles can be maximally entangled when they are also similarly entangled with other particles on a complete graph, focusing on Werner, isotropic, and Brauer states. To address this, we formulate and solve optimization problems that draw on concepts from many-body physics, computational complexity, and quantum cryptography. We approach the problem by formalizing it as a semi-definite program (SDP), which we solve analytically using tools from representation theory. Notably, we determine the exact maximum values for the projection onto the maximally entangled state and the antisymmetric Werner state, thereby resolving long-standing open problems in the field of quantum extendibility. Our results are achieved by leveraging SDP duality, the representation theory of symmetric, unitary and orthogonal groups, and the Brauer algebra.

This paper concerns the Couette flow for 2-D compressible Navier-Stokes equations (N-S) in an infinitely long flat torus (mathbb {T}times mathbb {R}). Compared to the incompressible flow, the compressible Couette flow has a stronger lift-up effect and weaker dissipation. To the best of our knowledge, there has been no work on the nonlinear stability in the cases of high Reynolds number until now and only linear stability was known in Antonelli et al. (Ann PDE 7(2):24, 2021) and Zeng et al. (SIAM J Math Anal 54(5):5698–5741, 2022). In this paper, we study the nonlinear stability of 2-D compressible Couette flow in Sobolev space at high Reynolds numbers. Moreover, we also show the enhanced dissipation phenomenon and stability threshold for the compressible Couette flow. First, We decompose the perturbation into zero and non-zero modes and obtain two systems for these components, respectively. Different from Antonelli et al. (Ann PDE 7(2):24, 2021) and Zeng et al. (SIAM J Math Anal 54(5):5698–5741, 2022), we use the anti-derivative technique to study the zero-mode system. We introduce a kind of diffusion wave to remove the excessive mass of the zero-modes and construct coupled diffusion waves along characteristics to improve the resulting time decay rates of error terms, and derive a new integrated system (2.25). Secondly, we observe a cancellation with the new system (2.25) so that the lift-up effect is weakened. Thirdly, the large time behavior of the zero-modes is obtained by the weighted energy method and a weighted inequality on the heat kernel (Huang et al. in Arch Ration Mech Anal 197:89–116, 2010). In addition, with the help of the Fourier multipliers method, we can show the enhanced dissipation phenomenon for the non-zero modes by commutator estimates to avoid loss of derivatives. Finally, we complete the higher-order derivative estimates to close the a priori assumptions by the energy method and show the stability threshold.