The Quantum Formalization Drive

Convexity and concavity of trace functions of matrices are a basic tool of mathematical physics and quantum information. Lieb's concavity theorem (Lieb 1973), that $(A,B)\mapsto\operatorname{Tr}K^*A^pKB^{1-p}$ is jointly concave for $0<p\le1$, settled the Wigner–Yanase–Dyson conjecture on the concavity of skew information and underlies the strong subadditivity…

Two parties who share a partially secret string $Z$ can turn it into a short, highly secret key by privacy amplification: one of them publicly picks a hash function $F$ and both compute $S=F(Z)$ (Bennett, Brassard, Robert 1988). The step ends almost every quantum key distribution protocol, so the security of those protocols rests on what an adversary who holds a…

Quantum key distribution lets two parties who share no secret agree on one over a public quantum channel, with security resting on physics and not on a computational assumption. The BB84 scheme of Bennett and Brassard is the standard protocol, and the question that drove the field through the 1990s was whether it stays secure when the eavesdropper applies…

Geometric quantum machine learning builds the symmetries of a learning task into a quantum model. A parametrized circuit is made equivariant when it commutes with a representation of the symmetry group, and a measurement is made invariant in the same way. The tutorial of Ragone et al., Representation Theory for Geometric Quantum Machine Learning, organizes…

Rényi relative entropies $D_\alpha(\rho\|\sigma)$ are one-parameter deformations of the Umegaki relative entropy. In quantum information theory they give the strong converse exponents of hypothesis testing and of channel coding, and the sandwiched Rényi relative entropy

General relativity has no fixed causal structure, and quantum theory has indefiniteness: a theory of quantum gravity may have to be probabilistic about causal structure itself. L. Hardy's causaloid framework addresses this by asking for predictions that make no reference to a background causal order. The 2008 paper Formalism Locality in Quantum Theory and…

Quantum metrology studies how well unknown parameters of a physical process can be estimated when the probe is a quantum system. With $N$ uncorrelated repetitions, or total interrogation time $T$ under noise, the estimation error of a parameter typically scales as $1/\sqrt{T}$ (the standard scaling). Noiseless coherent evolution allows Heisenberg scaling (HS), an…

Schur-Weyl duality says that on $V^{\otimes r}$, with $V=\mathbb C^n$, the general linear group and the symmetric group span each other's commutants. It is the standard route from the representation theory of $\mathfrak S_r$ to that of $\mathrm{GL}_n$. Brauer extended it to orthogonal and symplectic groups, and Benkart, Chakrabarti, Halverson, Leduc, Lee and…

A quantum computer is built from gates acting on one or two qubits, and the two-qubit gates are the expensive ones: they need a physical interaction between qubits, take longer and fail more often than single-qubit gates. Single-qubit gates are comparatively cheap. A question asked by everyone who designs a quantum architecture is therefore: what is left of…

Group schemes of finite type over a field split into two complementary kinds. Affine group schemes are the linear groups, and their theory is extensive. Anti-affine groups are the group schemes $G$ with $\mathcal O(G)=k$: every regular function is constant. Any connected group scheme $G$ has a smallest normal subgroup scheme $G_{\mathrm{ant}}$ with affine quotient,…

Quadratic fermion Hamiltonians $H=\sum C_{ij}b_i^\dagger b_j+\sum(A_{ij}b_i^\dagger b_j^\dagger+\text{h.c.})$ describe BCS superconductivity and the Hubbard model in mean-field approximation, and several exactly solvable spin chains after a Jordan–Wigner transformation. Their ground states, and all their eigenstates with a definite set of quasi-particle…

Qubits that sit close together, closer than the wavelength of the environment modes they couple to, are disturbed by that environment in the same way. The resulting noise is collective decoherence: its error operators are the collective spin operators $S_\alpha=\sum_i\tfrac12\sigma_\alpha^{(i)}$ rather than independent errors on each qubit. Chapter 8 of D. M.…

A weak measurement of a quantum system, or the amplification of a weak signal, always adds noise. The review of Clerk, Devoret, Girvin, Marquardt and Schoelkopf (arXiv:0810.4729, Rev. Mod. Phys. 82, 1155, 2010) derives the size of this unavoidable noise in a linear-response framework that assumes almost nothing about the detector: some stationary quantum…

A non-local unitary transformation of two qubits occurs when some Hamiltonian interaction couples them. Here we characterize the amount, as measured by time, of interaction required to perform two--qubit gates, when also arbitrarily fast, local unitary transformations can be applied on each qubit. The minimal required time of interaction, or interaction…