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arXiv:2204.04532

The entropy cones of $W_N$ and $W_N^d$ states

Howard J. Schnitzer

Published Apr 9, 2022• 0 citations
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Abstract

The quantum entropy cones (QEC) for $W_N$ states of qubits and $W_N^d$ states of qudits are computed. These cones emerge as symmetrized quantum entropy cones (SQEC) for arbitrary $N$ and $d$. Directed graph models are presented which describe the SQEC for $W_N$ states and $W_N^d$ states. Monogamous mutual information (MMI) is violated for all $N>3$.

DOI
10.48550/arxiv.2204.04532

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The NNN-qubit Dicke states $|{D}_{k}^{N}\ensuremath{\rangle}$, of Hamming weight kkk, are a class of entangled states which play an important role in quantum algorithm optimization. We present a general calculation of entanglement entropy in Dicke states, which we use to describe the $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy cone. We demonstrate that all $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy vectors emerge symmetrized and use this to define a min-cut protocol on star graphs which realizes $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy vectors. We identify the stabilizer group for all $|{D}_{k}^{N}\ensuremath{\rangle}$, under the action of the NNN-qubit Pauli group and two-qubit Clifford group, which we use to construct $|{D}_{k}^{N}\ensuremath{\rangle}$ reachability graphs. We use these reachability graphs to analyze and bound evolution of $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy vectors in Clifford circuits.
In this work, we generalize the graph-theoretic techniques used for the holographic entropy cone to study hypergraphs and their analogously-defined entropy cone. This allows us to develop a framework to efficiently compute entropies and prove inequalities satisfied by hypergraphs. In doing so, we discover a class of quantum entropy vectors which reach beyond those of holographic states and obey constraints intimately related to the ones obeyed by stabilizer states and linear ranks. We show that, at least up to 4 parties, the hypergraph cone is identical to the stabilizer entropy cone, thus demonstrating that the hypergraph framework is broadly applicable to the study of entanglement entropy. We conjecture that this equality continues to hold for higher party numbers and report on partial progress on this direction. To physically motivate this conjectured equivalence, we also propose a plausible method inspired by tensor networks to construct a quantum state from a given hypergraph such that their entropy vectors match.
The holographic entropy cone (HEC) characterizes the entanglement structure of quantum states which admit geometric bulk duals in holography. Due to its intrinsic complexity, to date it has only been possible to completely characterize the HEC for at most n=5n=5n=5 parties. For larger nnn, our knowledge of the HEC falls short of incomplete: almost nothing is known about its extremal elements. Here, we introduce a symmetrization procedure that projects the HEC onto a natural lower dimensional subspace. Upon symmetrization, we are able to conjecture precise properties that its extremal structure exhibits for general nnn. Further, by applying this symmetrization to the quantum entropy cone, we are able to quantify the typicality of symmetrized holographic entropies, which we find to be exponentially rare quantum entropies in the number of parties.
The monogamy of mutual information (MMI) is a quantum entropy inequality that enforces the non-positivity of tripartite information. We investigate the failure of MMI in graph states as a forbidden-subgraph phenomenon, conjecturing that every MMI-violating graph state is local-Clifford equivalent to one whose graph contains a four-star subgraph. We construct a family of star-like graphs whose states fail a specific class of MMI instances, and extend this analysis to general star topologies. Deriving adjacency matrix constraints that fix the MMI evaluation for these instances and interpreting them physically, we prove the forbidden-subgraph conjecture for this family of graphs. Finally, through an exhaustive search over graph representatives for all 888-qubit stabilizer entropy vectors, we establish that MMI failure is not reducible to the cases within our scope.
We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.