The Quantum Formalization Drive

In 1915, George D. Birkhoff proved the existence of a retrograde periodic orbit in each bounded component of the planar circular restricted three-body problem and asked whether its double cover bounds a disk-like global surface of section. Such a surface turns a three-dimensional flow into a two-dimensional return map and was intended as a route to a direct…

The quantum relative entropy is frequently used as a distance, or distinguishability measure between two quantum states. In this paper we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More precisely, we derive lower and upper bounds on the relative entropy in terms of various…

Optical communication sends classical information on modes of the electromagnetic field, and a fundamental limit on its rates is a statement about bosonic channels: quantum channels acting on harmonic-oscillator modes under a photon-number budget. For one sender and one receiver over a pure-loss channel the classical capacity is known and coherent states…

In this paper a general definition of quantum conditional entropy for infinite-dimensional systems is given based on recent work of Holevo and Shirokov arXiv:1004.2495 devoted to quantum mutual and coherent informations in the infinite-dimensional case. The properties of the conditional entropy such as monotonicity, concavity and subadditivity are also…

The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi…

The area law asks how the correlations between a region of a many-body system and the rest of the system scale with the size of the region. For ground states of quantum lattice systems the question is usually phrased through the entanglement entropy of the region. Interest in it comes from several directions: the entropy of a black hole scales with the area…

Fault-tolerant quantum computers implement Clifford gates cheaply. Universality needs one non-Clifford ingredient, usually a distilled copy of the magic state $|H\rangle=\cos(\pi/8)|0\rangle+\sin(\pi/8)|1\rangle$, and distillation is expensive (Bravyi–Kitaev 2005). Algorithms, however, ask for arbitrary single-qubit rotations. The standard route compiles such…

A quantum channel is a completely positive, trace-preserving linear map $\Lambda$ on matrices; it models how a quantum system is changed by noise. Its minimal entropy output $S[\Lambda]$ measures how close the channel can come to preserving purity: the smallest von Neumann entropy of any output state. The additivity question asks whether …

In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show…

Quantum error correction protects quantum information against noise by encoding it into a larger system and undoing the damage with a recovery channel. The classical criterion for exact correctability is the Knill–Laflamme condition (Knill, Laflamme 1997). Realistic codes, however, rarely correct errors perfectly, and many useful schemes (approximate codes,…

The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have derived many such inequalities in our previous work (K.M.R. Audenaert and J. Eisert, J. Math. Phys. 46, 102104 (2005)).…

Quantum Max Cut assigns a qubit to each vertex of a weighted graph and a two-qubit energy term to each edge. It asks for a quantum state with high total energy. The problem is a quantum analogue of graph Max Cut, but states can be entangled and edge terms act on a Hilbert space whose dimension grows exponentially with the number of vertices. Approximation…

Classical information can be sent by choosing a quantum state, transmitting it, and asking a receiver to measure it. An ideal noiseless channel seems to have no finite information limit if its carrier can occupy arbitrarily many quantum states. A physical carrier, however, has a limited average energy, and a channel carrying massive particles also has a…

Two orthonormal bases of C^d are mutually unbiased when every transition amplitude has squared modulus 1/d. At most d+1 such bases can coexist, and complete families are known in prime-power dimensions through finite-field constructions. Dimension six is the smallest famous composite case where existence of the complete seven-base family remains unknown.

Let $G$ be a compact connected Lie group. Its path group is the group of paths in $G$ starting at the identity, with pointwise multiplication. Unitary representations of path and current groups were studied by Gelfand, Graev and Vershik (Compositio Math. 35, 1977) and by Albeverio and Høegh-Krohn (Compositio Math. 36, 1978), in connection with quantum field…

A state of a composite quantum system is entangled when it cannot be prepared by two parties who share only classical randomness. Deciding entanglement is hard in general. The Peres–Horodecki criterion gives a computable necessary condition for separability: the partial transpose of a separable density operator is again a density operator (Peres 1996; Horode…

Thermal states of local Hamiltonians are the basic objects of quantum many-body physics, and estimating such a state from experimental data is the problem of Gibbs state learning (also called Hamiltonian learning from thermal states). It matters for the certification of quantum simulators and for characterizing quantum devices. A generic state of $n$ qubits…

In classical statistics a statistic is sufficient for a family of distributions when nothing is lost by passing to it: the original data can be regenerated from the statistic by a Markov kernel that does not depend on the unknown parameter. The quantum analogue concerns a coarse-graining $T$, a trace-preserving map sending density matrices on one system to…

Quantum key distribution produces a classical key $S$ shared by two parties, while an adversary may hold a quantum system $E$ correlated with it. For a long time, security of such a key was measured by the accessible information $I_{\mathrm{acc}}(S;E)$: the largest mutual information between $S$ and the outcome of any measurement the adversary can perform on $E$. A…

Entanglement is the resource behind teleportation, dense coding and much of quantum cryptography, and quantifying it is a basic task of quantum information theory. Many entanglement measures have been proposed, and for mixed states they genuinely disagree. For pure states of two parties they all reduce to one quantity, the entropy of the reduced state.…

We consider the simulation of a quantum channel by two parties who share a resource state and may apply local operations assisted by classical communication (LOCC). One specific type of such LOCC is standard teleportation, which is however limited to the simulation of Pauli channels. Here we show how we can easily enlarge this class by means of a minimal…

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…