Completeness of Graphical Languages for Mixed State Quantum Mechanics

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-calculus, ZW-calculus, and so on. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-SMC of finite-dimensional Hilbert spaces. In recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of how to extend these languages beyond pure quantum mechanics to reason about mixed states and general quantum operations, i.e., completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map that allows one to get rid of a quantum system, an operation that is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction , which transforms any †-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction, we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However, this construction fails for some fringe cases like Clifford+T quantum mechanics, as the category does not have enough isometries.

Nonlocality of a type of multi-star-shaped quantum networks

The correlations in quantum networks have attracted strong interest due to the fact that linear Bell inequalities derived from one source are useless for characterizing multipartite correlations of general quantum networks. In this paper, { a type of multi-star-shaped quantum networks are introduced and discussed. Such a network consists of three-grade nodes: the first grade is named party (node) $A$, the second one consists of $m$ nodes marked $B^1,B^2,\ldots,B^m$, which are stars of $A$ and the third one consists of $m^2$ nodes $C^j_k (j,k=1,2,\ldots,m)$, where $C^j_k (k=1,2,\ldots,m)$ are stars of $B^j$. We call such a network a $3$-grade $m$-star quantum network and denoted by $SQN(3,m)$, being as a natural extension of bilocal networks and star-shaped networks.} We introduce and discussed the locality and strong locality of a $SQN(3,m)$ and derive the related nonlinear Bell inequalities, called $(3,m)$-locality inequalities and $(3,m)$-strong locality inequalities. To compare with the bipartite locality of quantum states, we define the separability of $SQN(3,m)$ that imply the locality and then locality of $SQN(3,m)$. When all of the shared states of the network are pure ones, we prove that $SQN(3,m)$ is nonlocal if and only if it is entangled.

Quantum Extremal Surfaces and the Holographic Entropy Cone

Quantum states with geometric duals are known to satisfy a stricter set of entropy inequalities than those obeyed by general quantum systems. The set of allowed entropies derived using the Ryu-Takayanagi (RT) formula defines the Holographic Entropy Cone (HEC). These inequalities are no longer satisfied once general quantum corrections are included by employing the Quantum Extremal Surface (QES) prescription. Nevertheless, the structure of the QES formula allows for a controlled study of how quantum contributions from bulk entropies interplay with HEC inequalities. In this paper, we initiate an exploration of this problem by relating bulk entropy constraints to boundary entropy inequalities. In particular, we show that requiring the bulk entropies to satisfy the HEC implies that the boundary entropies also satisfy the HEC. Further, we also show that requiring the bulk entropies to obey monogamy of mutual information (MMI) implies the boundary entropies also obey MMI.