Depicting qudit quantum mechanics and mutually unbiased qudit theories

Abstract

We generalize the ZX calculus to quantum systems of dimension higher than two. The resulting calculus is sound and universal for quantum mechanics. We define the notion of a mutually unbiased qudit theory and study two particular instances of these theories in detail: qudit stabilizer quantum mechanics and Spekkens-Schreiber toy theory for dits. The calculus allows us to analyze the structure of qudit stabilizer quantum mechanics and provides a geometrical picture of qudit stabilizer theory using D-toruses, which generalizes the Bloch sphere picture for qubit stabilizer quantum mechanics. We also use our framework to describe generalizations of Spekkens toy theory to higher dimensional systems. This gives a novel proof that qudit stabilizer quantum mechanics and Spekkens-Schreiber toy theory for dits are operationally equivalent in three dimensions. The qudit pictorial calculus is a useful tool to study quantum foundations, understand the relationship between qubit and qudit quantum mechanics, and provide a novel, high level description of quantum information protocols.

Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

“…Despite being already well established in [5,26] and independently introduced as a typical special case of qudit ZX-calculus in [24], to the best of our knowledge, there are no completeness results available for qutrit ZX-calculus. Without this kind of results, how can we even know that the rules of a so-called ZX-calculus are useful enough for quantum computing?…”

Qutrit Dichromatic Calculus and Its Universality

“…In [3], Coecke and Duncan developed dichromatic ZX-calculus for qubit systems. To extend the graphical calculus to higher dimensions, Ranchin considered qudit (d-dimensional quantum system) ZXcalculus [8]. At almost the same time, the authors of this paper investigated the theory and application of qutrit ZX-calculus [1].…”

“…At almost the same time, the authors of this paper investigated the theory and application of qutrit ZX-calculus [1]. Unlike in [8] and [1], we introduce two new rules P1 and P2 in this paper. The necessity of these two rules is demonstrated by depicting in dichromatic calculus the simplest quantum speed-up algorithm with a single qutrit [6].…”

“…Due to Brylinski [2], to prove the universality of qudit ZX calculus, it suffices to prove that the qudit ZX calculus contains all single qudit unitary transformations. Such a proof given in [8] is based on the fact [7] that the d-dimensional phase gates Z d and X d are sufficient to simulate all single qudit unitary transforms. For our understanding, only part of the whole family of Z d phase gates can be represented by Λ X phase gates (i.e., X phase gates) in [8].…”

“…Such a proof given in [8] is based on the fact [7] that the d-dimensional phase gates Z d and X d are sufficient to simulate all single qudit unitary transforms. For our understanding, only part of the whole family of Z d phase gates can be represented by Λ X phase gates (i.e., X phase gates) in [8]. Actually, we have a counterexample that some Z d phase gates cannot be realized by Λ X only.…”