The Quantum Formalization Drive

Two resources can be extracted from many copies of a shared quantum state $\rho$: pure entanglement (Bell pairs, at the rate $E_D$, the distillable entanglement) and a secret key that an eavesdropper holding the rest of the world cannot learn (at the rate $K_D$, the distillable key). One always has $K_D \ge E_D$, because Bell pairs can be measured to give key.…

Two long standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (1) a map between the initial and final system states, and (2) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information…

For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such as the full unitary group to closed subgroups like partitionings by factorisation into tensor products, where the…

In quantum mechanics, a fundamental law prevents quantum communications to simultaneously achieve high rates and long distances. This limitation is well known for point-to-point protocols, where two parties are directly connected by a quantum channel, but not yet fully understood in protocols with quantum repeaters. Here we solve this problem bounding the…

A 2-group is a categorified group: a monoidal category in which every object and every morphism is invertible. When the associativity and unit laws hold as equalities rather than up to isomorphism, the 2-group is called strict. Strict 2-groups appear as the fundamental 2-groups of homotopy 2-types, as symmetry objects of gauge theories with higher form…

A density operator $\rho$ on $\mathbb C^d$ describes a quantum system completely, but it cannot be read off a single copy: it has to be estimated from many identically prepared copies. Its spectrum $r_1 \ge \dots \ge r_d \ge 0$, $\sum_\alpha r_\alpha = 1$, is the part of $\rho$ that does not depend on a basis. It determines the entropy and purity of the state and the…

The no-cloning theorem says that no physical process turns an unknown quantum state $|\psi\rangle$ into two copies $|\psi\rangle|\psi\rangle$. Once perfect copies are excluded, the quantitative question is how good two imperfect copies can be, and how the quality of one copy trades off against the quality of the other. The answer bounds what an eavesdropper on…

A quantum channel has many capacities, one for each kind of information sent and each auxiliary resource allowed: classical bits or qubits, with or without shared entanglement, with or without a free classical side channel. Comparing these capacities is how quantum Shannon theory locates the value of each resource. Some comparisons look as if they should be…

Suppose a quantum system has been prepared $n$ times, either always in the state $\rho_0$ (with prior probability $\pi_0$) or always in the state $\rho_1$ (with prior $\pi_1$), and the task is to guess which. Helstrom's formula gives the smallest error probability over all measurements,

Two parties who do not share a reference frame can align one by sending quantum systems through an unknown rotation. Mathematically, a device applies an unknown matrix $g\in\mathrm{SU}(2)$ to each of $n$ qubits, and the task is to estimate $g$ from one measurement on the output. How fast the smallest achievable error falls with $n$ is a basic question of quantum…

Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the…

A quantum computer executes a fixed, finite instruction set of gates, while quantum algorithms are written with arbitrary gates. Every algorithm must therefore be compiled: each gate $U$ is replaced by a sequence of available instructions whose product approximates $U$. If an accuracy $\delta$ required sequences of length $O(1/\delta)$, compiling an $m$-gate…

The data processing inequality (DPI) for the conditional von Neumann entropy says that discarding part of the side information cannot make a quantum system less uncertain: for every tripartite state $\rho_{ABC}$,

A qubit coupled to an environment loses the phase relation between its basis states $|0\rangle$ and $|1\rangle$: the off-diagonal entries of its density matrix decay. This decoherence is the main obstacle to storing quantum information, and a large body of work tries to counter it with external control fields. Lidar and Schneider (arXiv:quant-ph/0410048,…