The Quantum Formalization Drive

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…

Quantum teleportation moves an unknown qubit from Alice to Bob using one shared, perfectly entangled pair and two classical bits (Bennett et al. 1993). In practice the pairs are distributed through a noisy channel and arrive as a mixed state $M$, and teleporting through an impure pair degrades the transmitted state. Bennett, Brassard, Popescu, Schumacher,…

Quantum algorithms are usually written as unitary circuits on pure states, yet many natural computational steps are not unitary: measuring, discarding a register, or calling a probabilistic subroutine whose output is random even on a classical input. Aharonov, Kitaev and Nisan (arXiv:quant-ph/9806029v1, 1998) set up a circuit model in which every gate is a…

A classical-quantum channel sends a classical letter $i$ by preparing a quantum state $S_i$; the receiver measures the state to guess the letter. This is the basic model of transmitting classical information with quantum carriers (photons, spins), and its capacity is the quantity that quantum information theory compares with Shannon's classical capacity. A. S.…

A quantum channel is a completely positive, trace-preserving linear map between finite-dimensional matrix algebras. One central question in quantum Shannon theory was whether the minimal output entropy $S_{\min}(\Phi) = \inf_\rho S(\Phi(\rho))$ is additive under tensor products, $S_{\min}(\Phi\otimes\Psi) = S_{\min}(\Phi) + S_{\min}(\Psi)$. Shor showed in 2004 (…

We consider quantum key distribution (QKD) and entanglement distribution using a single-sender multiple-receiver pure-loss bosonic broadcast channel. We determine the unconstrained capacity region for the distillation of bipartite entanglement and secret key between the sender and each receiver, whenever they are allowed arbitrary public classical…

We provide a versatile upper bound on the number of maximally entangled qubits, or private bits, shared by two parties via a generic adaptive communication protocol over a quantum network when the use of classical communication is not restricted. Although our result follows the idea of Azuma et al. [Nat. Comm. 7, 13523 (2016)] of splitting the network into…

The role of quantum coherence and the environment in the dynamics of excitation energy transfer is not fully understood. In this work, we introduce the concept of dynamical contributions of various physical processes to the energy transfer efficiency. We develop two complementary approaches, based on a Green's function method and energy transfer…

Hereunder we continue the study of the representation theory of the algebra of permutation operators acting on the $n$-fold tensor product space, partially transposed on the last subsystem. We develop the concept of partially reduced irreducible representations, which allows to simplify significantly previously proved theorems and what is the most important…

Following [OW16], we continue our analysis of: (1) "Quantum tomography", i.e., learning a quantum state, i.e., the quantum generalization of learning a discrete probability distribution; (2) The distribution of Young diagrams output by the RSK algorithm on random words. Regarding (2), we introduce two powerful new tools: (i) A precise upper bound on the…

A fault-tolerant quantum computer executes only a discrete set of gates. The standard choice is the Clifford and T library: the Hadamard gate $H$, the phase gate $S$, CNOT, and the gate $T = \mathrm{diag}(1, e^{i\pi/4})$, which together are universal and are well suited to error-corrected implementation. Every other gate, in particular every single-qubit rotation…

Light carries information in electromagnetic field modes. At optical frequencies, photon statistics affect the capacities of thermal-noise and broadcast communication channels. Bounds on these capacities depend on identifying the smallest entropy that a beam splitter can produce under specified input conditions. Guha, Erkmen, and Shapiro explain why two…

Gelfand duality identifies compact Hausdorff spaces with commutative unital C*-algebras: a space $X$ corresponds to the algebra $C(X)$ of continuous complex functions, and continuous maps correspond contravariantly to unital *-homomorphisms (Gelfand 1941; in categorical form, Negrepontis 1971). Measure theory has an analogous expectation. A measurable space…