The Quantum Formalization Drive

We initiate a systematic study to provide upper bounds on device-independent key, secure against a non-signaling adversary (NSDI). We employ the idea of "squashing" on the secrecy monotones and show that squashed secrecy monotones are the upper bounds on the NSDI key. Our technique for obtaining upper bounds is based on the non-signaling analog of quantum…

We present a new protocol for practical quantum cryptography, tailored for an implementation with weak coherent pulses. The key is obtained by a very simple time-of-arrival measurement on the data line; an interferometer is built on an additional monitoring line, allowing to monitor the presence of a spy (who would break coherence by her intervention).…

We analyze the security and feasibility of a protocol for Quantum Key Distribution (QKD), in a context where only one of the two parties trusts his measurement apparatus. This scenario lies naturally between standard QKD, where both parties trust their measurement apparatuses, and Device-Independent QKD (DI-QKD), where neither does, and can be a natural…

Following the various statements of [DW16] to their logical conclusion, this note explicitly argues the following statement, implicit in [DW16]: for positive semi-definite operators $C_{1},\ldots,\,C_{L}$, a unitary $V_{C_{i}}$ commuting with $C_{i}$, and $p\geq1$, the quantity

We introduce a class of variational states to describe quantum many-body systems. This class generalizes matrix product states which underly the density-matrix renormalization group approach by combining them with weighted graph states. States within this class may (i) possess arbitrarily long-ranged two-point correlations, (ii) exhibit an arbitrary degree…

We provide an analysis of a new family of device independent quantum key distribution (QKD) protocols with several novel features: (a) The bits used for the secret key do not come from the results of the measurements on an entangled state but from the choices of settings; (b) Instead of a single security parameter (a violation of some Bell inequality) a set…

We consider a quantum system strongly driven by forces that are periodic in time. The theorem concerns the probability $P(e)$ of observing a given energy change $e$ after a number of cycles. If the system is thermostated by a (quantum) thermal bath, $e$ is the total amount of energy transferred to the bath, while for an isolated system $e$ is the increase in energy…

Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels becomes near perfect for data transmission and the other fraction becomes near useless for this task. Mahdavifar and Vardy have recently exploited…

We prove that a unitary matrix has an exact representation over the Clifford+T gate set with local ancillas if and only if its entries are in the ring Z[1/sqrt(2),i]. Moreover, we show that one ancilla always suffices. These facts were conjectured by Kliuchnikov, Maslov, and Mosca. We obtain an algorithm for synthesizing a exact Clifford+T circuit from any…

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences such as the strong sub-additivity of the von Neumann entropy, the Golden-Thompson trace inequality and the monotonicity of the Holevo quantity.The…

In this article I discuss Charlie's Bennett's influence in quantum information theory and our answers to the question whether quantum entanglement is `monogamous'.

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an…

A quantum information source emits pure states $|\psi_x\rangle$ of a finite-dimensional system $A$ at random, with probabilities $p_X(x)$. To a receiver who does not know the labels $x$, each emission is described by the density operator $\rho = \sum_x p_X(x)|\psi_x\rangle\langle\psi_x|$. Storing or transmitting such states is expensive: noiseless qubit channels are…

Converse theorems in quantum Shannon theory follow one pattern. A good protocol produces a state that is close, in trace distance, to the ideal state it should produce. To bound the rate of the protocol, that statement about distance has to become a statement about entropies: if two states are close, their entropies, conditional entropies and mutual…

The Holevo limit $H(\rho)$ is the largest rate, in bits per channel use, at which classical information can be sent reliably through a channel whose outputs are quantum states. Holevo (doi:10.1109/18.651037) and Schumacher–Westmoreland (doi:10.1103/PhysRevA.56.131) proved in 1996–97 that it is achievable, but their receiver is a collective "square-root"…

The fidelity $F(\rho_1,\rho_2)$ of two quantum states measures how well one can be mistaken for the other. Its standard form is Uhlmann's transition probability Uhlmann 1976,

Ordinary gauge theory assigns group elements to curves: a connection $A$ with values in the Lie algebra $\mathfrak g$ of a Lie group $G$ is integrated along paths to give holonomies. Higher gauge theory goes one dimension up and also assigns labels to surfaces, so that extended objects (strings, membranes) can be transported in the same way as point particles.…

Quantum Shannon theory proves coding theorems by allowing a small error and showing that it vanishes for long blocks. The error has to be measured, and the standard measure for states is the trace distance $\|\rho-\sigma\|_1$: by the Helstrom–Holevo analysis it is linearly related to the best probability of telling $\rho$ from $\sigma$ with one measurement. For…

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…