The Quantum Formalization Drive

The quantum capacity of a channel is the largest rate, in qubits per channel use, at which quantum information can be sent through it with vanishing error. Unlike the classical capacity of a classical channel, it is not given by a single-letter formula in general: the quantum capacity theorem (Lloyd, Shor, Devetak) identifies it with the regularized coherent…

A dynamic information measure assigns a number to a quantum channel: the largest amount of a given kind of correlation that one use of the channel can create. Such a measure is a usable capacity formula only when it is additive, that is, when its value for two channels used together equals the sum of the individual values. Without additivity, the capacity…

Measuring a quantum system generally changes it. A qubit prepared in $|0\rangle$ and measured along the $X$ axis is left in $|+\rangle$ or $|-\rangle$, each with probability one half, and the original state is lost. The same qubit measured along the $Z$ axis returns its outcome with certainty and is left untouched. The gentle measurement lemma turns this intuition…

Every coding theorem for sending classical data over a quantum channel has to answer the same question: given a family of quantum states that a sender can prepare, how many of them can a receiver tell apart reliably, and with which measurement? The packing lemma is a one-shot answer. It asks for an ensemble of states and a set of projectors satisfying four…

A Pauli channel is the simplest model of noise on a qubit: with probability $p_I$ nothing happens, and with probabilities $p_X$, $p_Y$, $p_Z$ the bit-flip $X$, the bit-and-phase flip $Y$ or the phase flip $Z$ is applied. Depolarizing noise, pure dephasing and independent bit-flip noise are all special cases, and twirling reduces any qubit channel to one of this form, so…

The monotonicity of quantum relative entropy, $D(\rho\|\sigma)\ge D(\mathcal N(\rho)\|\mathcal N(\sigma))$ for every quantum channel $\mathcal N$, is the inequality from which the converse parts of the coding theorems of quantum Shannon theory are derived: strong subadditivity, concavity of conditional entropy, joint convexity of relative entropy and the Holevo…

Estimating an unknown phase shift is the basic task of quantum metrology: interferometers, atomic clocks and magnetometers all imprint a phase on a probe and read it out by a measurement. When the probe is a collection of qubits, the phase acts as a rotation about the $z$ axis, and the question is how well $N$ copies of the probe state can reveal it. For pure…

How much entanglement does a quantum state carry? For a bipartite pure state the accepted answer is the entanglement of formation $E_F$: the von Neumann entropy of either reduced density matrix, introduced by Bennett, Bernstein, Popescu, Schumacher, DiVincenzo, Smolin and Wootters in 1996 (quant-ph/9511030, quant-ph/9604024). For mixed states and for more…

Two parts of a quantum system can share correlations with no classical explanation, and entanglement measures quantify them. Several measures have an operational meaning: the entanglement cost, for instance, is the pure entanglement needed to create a state. The squashed entanglement of Christandl and Winter (quant-ph/0308088) has unusually good formal…

Quantum error-correcting codes protect quantum information against small perturbations, and the stabilizer formalism is the standard way to build them for qubits: an abelian group $S$ of Pauli operators not containing $-I$ fixes a code space, and the operators commuting with $S$ act on the encoded information (Gottesman 1997). For qubits the formalism runs…

We derive the optimal exponent of the error probability of the quantum fixed-length pure state source coding in both cases of blind coding and visible coding. The optimal exponent is universally attained by Jozsa et al. (PRL, 81, 1714 (1998))'s universal code. In the direct part, a group representation theoretical type method is essential. In the converse…

During what Vietnamese call the American War, Alexander Grothendieck spent three weeks teaching mathematics in and near Hanoi. Hoàng Xuân Sính took notes on his lectures and later did her thesis work with him by correspondence. In her thesis she developed the theory of "Gr-categories", which are monoidal categories in which all objects and morphisms have…

The no-cloning theorem (Wootters–Zurek and Dieks, 1982) says that no physical process copies an unknown pure quantum state. For mixed states the question is subtler, because copying is not the only way to hand a state to two parties. A process broadcasts a state $\rho$ onto two systems if each system, looked at alone, is in the state $\rho$; the joint state may…

Two probability distributions $p_0$ and $p_1$ on the same finite set can be told apart only through samples. How well they can be told apart is the starting point of C. A. Fuchs's dissertation Distinguishability and Accessible Information in Quantum Theory (arXiv:quant-ph/9601020), which takes classical measures of distinguishability and later optimizes them…

A quantum channel is the most general physical transformation of a quantum system: a linear, completely positive, trace-preserving map $S$ sending density matrices (positive semidefinite matrices of trace one) to density matrices. A central question of quantum Shannon theory is whether using two channels jointly, with entangled inputs, can ever do better than…

A quantum computer or a quantum channel stores information in the state of a physical system that interacts with its environment. The interaction corrupts the state, and unlike classical bits a quantum state can neither be copied nor measured without disturbing it. Quantum error-correcting codes circumvent this by encoding the information into a subspace of…

Chapter 2 of Fuchs's dissertation (arXiv:quant-ph/9601020) collects the classical measures of distinguishability that the later chapters optimize over quantum measurements. One of them, the Shannon mutual information, rests on the Shannon information $H(p)$. Section 2.4.1 derives $H(p)$ axiomatically (Daróczy's theorem); Section 2.4.2 gives it an operational…