We study the action of quantum channels on states on a von Neumann algebra $\mathcal{M}$ with conserved quantities encoded by a subalgebra $\mathcal{N}$. We define a phase as an equivalence class of normal states on $\mathcal{M}$ under the action of quantum channels preserving $\mathcal{N}$. First, we establish a relative quasi-entropic characterization of phase equivalence using the Petz recovery map. We further introduce the operator algebra (OA) symmetry of a state and define an equivalence relation between OA symmetries via bishifts of biprojections. When $\mathcal{N}\subseteq\mathcal{M}$ is an irreducible finite-index subfactor of type II$_1$, we prove that phase equivalence implies OA symmetry equivalence. Consequently, OA symmetry breaking provides a sufficient mechanism for detecting phase transitions. Furthermore, combining finite-index subfactor theory with quantum Fourier analysis, we prove that the cardinality of each phase is finite and obtain an explicit upper bound depending only on the Jones index.
We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and th...
Corey Jones, Nikita Sopenko, Ryan Thorngren· 2 citations
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$, or equivalently with a commut...
We establish that the skew-symmetric bicharacter on the group of $\ell$-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental $\ell$-weights and show that this invariance continues to hold under...
Berlekamp's algorithm factors a squarefree polynomial $f\in\mathbb{F}_q[x]$ by deterministic linear algebra, reducing the problem to splitting an explicit commutative algebra $B\cong\mathbb{F}_q^r$ into its $r$ simple factors. For large odd $q$, the standard efficient splitting step is randomized, while known derandomi...
For every odd prime power $q$ and $D=q+1$, we construct an infinite family of Ramanujan quantum expanders of degree $D$. The construction transfers Morgenstern's Ramanujan Cayley graphs on $\operatorname{PSL}_2(\mathbb{F}_{p})$ for $p$ which is an even power of $q$, through the odd irreducible subrepresentation of the...
Standard finite quantum de Finetti theorems approximate the $k$-system marginals of permutation-invariant states of $n$-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of oper...
Liu Ye, Bjarne Bergh, Nilanjana Datta· 1 citation
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