An algebraic characterisation of $n$-qubit pure-state invariants

Abstract

We study the set of $n$-qubit pure quantum states. Specifically, we consider their incarnation as density matrices in Bloch representation. Building on prior work, we define a family of quadratic functions in Pauli expectation values, and fully characterise all instances that result in state-independent purity constraints. The resulting family of pure-state invariants is surprisingly rich in structure, with strong ties to the geometry of symplectic polar spaces. After charting the general landscape, we define a sub-family of pure-state invariants that satisfies the dynamical constraint of being stable under state update. We prove that it corresponds with (different embeddings of) symplectic Lie algebras $\mathfrak{sp}(2^{n-1})\subset\mathfrak{su}(2^n)$, which admit a generating set of Pauli operators. We carefully distinguish the symplectic case from its orthogonal counterpart, using the Arf invariant of quadratic refinements of the symplectic form on the projective Pauli group, which we further identify with quaternionic structures on $\mathbb{C}^{2^n}$ that are both generated by and restricted to Pauli operators. As a consequence of this analysis, we reveal previously largely overlooked structure in the $n$-qubit Pauli group. We highlight this in a number of examples. We characterise all state-independent uncertainty relations in variance for Pauli observables, as well as all Pauli channels for which the output fidelity and output purity are state-independent. We further show that the orbits of the projective stabiliser groups of pure-state invariants, which for $n\geq2$ are proper subgroups of the Clifford group, form complex projective state $2$-designs, while their unitary lifts form symplectic $2$-designs.