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Journal of Physics A: Mathematical and Theoretical - latest papers
Latest articles for Journal of Physics A: Mathematical and Theoretical
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Quantum gravity observables: observation, algebras, and mathematical structure *
The questions of describing observables and observation in quantum gravity appear to be centrally important to its physics. A relational approach holds significant promise, and a classification of different types of relational observables (gravitationally dressed, field relational, and more general) is outlined. Plausibly gravitationally dressed observables are particularly closely tied to the fundamental structure of the theory. These may be constructed in the quantum theory to leading order in Newton’s constant, and raise important questions about localization of information. Approximate localization is given by a ‘standard dressing’ construction of a ‘gravitational splitting’. It is also argued that such gravitational dressings give a generalization of the crossed product construction, reducing to this and yielding type II von Neumann algebras in special cases. Gravity therefore introduces a significantly more general alteration of the algebraic structure of local quantum field theory, also with apparent connections to holography, but whose implications have not been fully understood. In particular, properties of the algebra of gravitationally dressed observables suggest a possible role for other non-algebraic structure on the Hilbert space for quantum gravity.
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On the distinguishability of geometrically uniform quantum states
A geometrically uniform (GU) ensemble is a uniformly weighted quantum state ensemble generated from a fixed state by a unitary representation of a finite group G. In this work we analyze the problem of discriminating GU ensembles from various angles. Assuming that the representation of G is irreducible, we first show that a particular optimal measurement can be understood as the limit of weighted ‘pretty good measurements’ (PGMs). This naturally provides examples of state discrimination for which the unweighted PGM is provably sub-optimal. We extend this analysis to certain reducible representations, and use Schur–Weyl duality to discuss two particular examples of GU ensembles in terms of Werner-type and permutation-invariant generator states. For the case of pure-state GU ensembles we give a streamlined proof of optimality of the PGM first proved in Eldar et al (2004 IEEE Trans. Inf. Theory50 1198–207). We use this result to give simplified proofs of the optimality of the PGM, along with expressions for the corresponding success probabilities, for two tasks: the hidden subgroup problem (HSP) over semidirect product groups (first proved in Bacon et al (2005 46th Annual IEEE Symp. Foundations of Computer Science (FOCS’05) pp 469–78), and port-based teleportation (first proved in Mozrzymas et al (2019 New J. Phys.20 053006) and Leditzky (2022 Lett. Math. Phys.112 98). Finally, we consider the n-copy setting and adapt a result of Montanaro (2007 Commun. Math. Phys.273 619–36) to derive a compact and easily evaluated lower bound on the success probability of the PGM for this task. This result can be applied to the HSP to obtain a new proof for an upper bound on the sample complexity by Hayashi et al (2006 Quantum Inf. Comput.8 345–58).
- Preface to fields, gravity, strings and beyond: in memory of Stanley Deser
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Shannon information entropies of the Hulthén potential and their critical behavior near the system bound limit
The Shannon entropies in both coordinate and momentum spaces for some low-lying eigenstates of the Hulthén potential (HP) are investigated over a wide range of screening parameters where bound states exist. The system wave functions are expanded in terms of multiple groups of Slater-type orbitals and the Hamiltonian is solved by employing the Rayleigh–Ritz variational method. The accuracy of the obtained bound-state eigenenergies, Shannon entropies, and radial mean values are validated by comparing with the analytical solutions of the HP in s-wave states. The Bialynicki–Birula and Mycielski inequalities for Shannon entropies in both the coordinate and momentum spaces as well as their sum are examined in both weak and strong screening situations. It is shown that the variation of Shannon entropy sum is governed by its upper bound for the system in a well-defined bound state. In the critical bound region where the system undergoes a bound-continuum transition, the Shannon entropies and entropy sum differ in states by their orbital angular momenta and their asymptotic behavior can be properly shaped by the rigorous upper and lower bounds in terms of the radial and momentum expectation values and . We finally extended the calculation to a modified HP (MHP) that has been widely discussed in the literature and compared our results with previous predictions. Benchmark Shannon entropies are provided for further investigation of the information measures of MHPs.
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Entanglement in directed graph states
We investigate a family of quantum states defined by directed graphs, where the oriented edges represent interactions between ordered qubits. As a measure of entanglement, we adopt the entanglement distance—a quantity derived from the Fubini–Study metric on the system’s projective Hilbert space. We demonstrate that this measure is entirely determined by the vertex degree distribution and remains invariant under vertex relabeling, underscoring its topological nature. Consequently, the entanglement depends solely on the total degree of each vertex, making it insensitive to the distinction between incoming and outgoing edges. These findings offer a geometric interpretation of quantum correlations and entanglement in complex systems, with promising implications for the design and analysis of quantum networks.