Abstract
The Pauli groups are ubiquitous in quantum information theory because of their usefulness in describing quantum states and operations and their readily understood symmetry properties. In addition, the most well-understood quantum error correcting codes -- stabilizer codes -- are built using Pauli operators. The eigenstates of these operators -- stabilizer states -- display a structure (e.g., mutual orthogonality relationships) that has made them useful in examples of multi-qubit non-locality and contextuality. Here, we apply the graph-theoretical contextuality formalism of Cabello, Severini and Winter to sets of stabilizer states, with particular attention to the effect of generalizing two-level qubit systems to odd prime d-level qudit systems. While state-independent contextuality using two-qubit states does not generalize to qudits, we show explicitly how state-dependent contextuality associated with a Bell inequality does generalize. Along the way we note various structural properties of stabilizer states, with respect to their orthogonality relationships, which may be of independent interest.
Highlights
The Pauli operators are ubiquitous in quantum information theory, typically used as an operator basis to decompose multi-particle states or circuits
One significant difference between nonlocality and general contextuality is that arbitrary states can exhibit contextuality when the set of tests comprise an example of state-independent contextuality (SIC), e.g., the projectors associated with the Peres–Mermin magic square [12,13]
We begin by concentrating on the set of bipartite separable stabilizer states, {Π}sep, and show that they are insufficient for exhibiting SIC
Summary
The Pauli operators are ubiquitous in quantum information theory, typically used as an operator basis to decompose multi-particle states or circuits. The Gottesman–Knill theorem tells us we cannot see better-than-classical computational performance using quantum circuits restricted to (i) operating on Pauli eigenstates, (ii) using Pauli measurements; and (iii) using gates that inter-convert Pauli operators (Clifford gates). In this model of fault-tolerant computation, the use of qudits appears to offer an advantage over qubits in terms of the efficiency associated with a magic state distillation routine [6]. Nonlocality can be understood as a special type of contextuality, and so Bell inequalities can be recast in the graph-based contextuality formalism We derive such a decomposition for a family of two-qudit. The remaining subsection provides some additional considerations related to graph-based contextuality that should be borne in mind, and discusses how these apply to our current investigation
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