The Quantum Formalization Drive

A quantum computation on $n$ qubits is a unitary operator $U$ on $\mathbb C^{2^n}$, but hardware applies only a small library of gates. The standard library is every one-qubit gate together with the two-qubit controlled-NOT (CNOT). One-qubit gates are cheap, while two-qubit gates are slow and error-prone, so the cost of a circuit is usually measured by its number…

In quantum statistical inference one measures the expected values $\alpha=(\langle u_1,\rho\rangle,\dots,\langle u_r,\rho\rangle)$ of finitely many observables $u_1,\dots,u_r$ on an unknown state $\rho$ and, among all states compatible with $\alpha$, selects the one of maximal von Neumann entropy, $\rho^*(\alpha)$. The map $\alpha\mapsto\rho^*(\alpha)$ is the maximum-en…

A register of qubits that all couple to their environment in the same way, through the collective spin operators $S_x, S_y, S_z$, has subspaces on which this coupling acts trivially. Information stored there is immune to the noise, and such decoherence-free subspaces and subsystems (DFSs) were proposed as a passive alternative to active error correction (Zanar…

An open quantum system is a system $A$ that interacts with an environment $B$. When the joint state starts as a product $\rho\otimes\omega$ with $\omega$ fixed, and the joint evolution is a unitary $U$, the reduced dynamics $\rho\mapsto\operatorname{tr}_B[U(\rho\otimes\omega)U^\dagger]$ is a completely positive (CP), trace-preserving map, and it has a Kraus form …

An entanglement measure assigns to each bipartite or multipartite quantum state a number that does not increase under local operations and classical communication (LOCC). One property that physicists ask of such a measure, and that most known measures lack or do not have proved, is additivity: the measure of two independent copies placed side by side, …

A general $n$-qubit quantum computation is described by $2^n$ complex amplitudes, and following them on a classical computer costs exponential time and space. Daniel Gottesman's 1998 paper The Heisenberg Representation of Quantum Computers shows that a large and practically important class of quantum networks escapes this cost. Instead of the state, one follows…

Entanglement shared between two distant parties is the resource behind teleportation and entanglement-based cryptography, but these protocols need it in the pure singlet form $\Psi_-=\tfrac1{\sqrt2}(|{\uparrow\downarrow}\rangle-|{\downarrow\uparrow}\rangle)$. In practice decoherence leaves the parties with a mixed state $\varrho$, and distillation (Bennett et…

A quantum computer stores information in a system that cannot be perfectly isolated: it is coupled to an environment whose degrees of freedom are neither controlled nor observed. The coupling turns pure states of the system into mixed ones (decoherence), and this is the main obstacle to building reliable quantum memories. One response, developed in the late…

A decoherence-free subsystem (DFS) is a part of a quantum system that the coupling to its environment cannot touch: when the system operators of the system-environment interaction all respect a common block structure, information stored in the degeneracy of one block evolves trivially, however strong the coupling is (Zanardi–Rasetti 1997; Lidar–Chuang–Whaley…

Every experiment that manipulates a quantum system does so through some apparatus: a laser mode driving an atom, a gate electrode biasing a quantum dot, a coil producing a magnetic field. The apparatus is itself a quantum system, and the system and apparatus interact through a fixed Hamiltonian. Whether this interaction lets the apparatus act as a classical…

State redistribution asks how cheaply one party can hand part of a shared quantum state to another. Alice and Bob share many copies of a state; Alice holds two shares, $A$ and $C$, and wants to transfer $A$ to Bob, who already holds $B$, using quantum communication and pre-shared entanglement, without disturbing the correlations with anything else. Devetak and Yard…

A private state is the quantum analogue of a shared secret key: two parties, Alice and Bob, hold key systems $A$ and $B$ of dimension $K$ whose standard-basis measurement outcomes are perfectly correlated, uniformly random and independent of any eavesdropper holding a purification. Distilling such states from many copies of a noisy state $\rho_{AB}$ is the task of q…

In classical statistics the Fisher information of a parametric family of probability distributions bounds how well its parameter can be estimated (the Cramér–Rao inequality), and it can only decrease when the data are processed by a Markov kernel. Quantum estimation theory, quantum metrology and information geometry need the analogue for families of density…

Quantum discord was introduced by Ollivier and Zurek (quant-ph/0105072) and by Henderson and Vedral (quant-ph/0105028) as the part of the correlations of a bipartite quantum state that no local measurement can reveal. It is nonzero for some unentangled states, and it has been proposed as a resource behind mixed-state quantum computation with little…

In quantum tomography of a large system one rarely measures an informationally complete set of observables. For an $n$-qubit system with $n$ large, one typically knows only the expectation values of a few local observables, and many states are compatible with that data. The principle of maximum entropy (Jaynes, 1957; Wichmann, 1963) selects among them the state…

We extend the concept of the relative error to mixed-state cloning and related physical operations, in which the ancilla contains some a priori information about the input state. The lower bound on the relative error is obtained. It is shown that this result contributes to the stronger no-cloning theorem.

Previous work on symmetric group equivariant neural networks generally only considered the case where the group acts by permuting the elements of a single vector. In this paper we derive formulae for general permutation equivariant layers, including the case where the layer acts on matrices by permuting their rows and columns simultaneously. This case…

The Peres-Horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states. The partial transpose operation admits, in the continuous case, a geometric interpretation as mirror reflection in phase space. This recognition leads to uncertainty principles, stronger than the traditional…

An optical transmitter irradiates a target region containing a bright thermal-noise bath in which a low-reflectivity object might be embedded. The light received from this region is used to decide whether the object is present or absent. The performance achieved using a coherent-state transmitter is compared with that of a quantum illumination transmitter,…