The Quantum Formalization Drive

We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation ($U$) of a finite group ($G$), whenever $U \otimes U^c$ is simply reducible (with $U^c$ being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of…

Variational quantum machine learning is an extensively studied application of near-term quantum computers. The success of variational quantum learning models crucially depends on finding a suitable parametrization of the model that encodes an inductive bias relevant to the learning task. However, precious little is known about guiding principles for the…

Shannon's entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule:

We show that the quantum $α$-relative entropies with parameter $α\in (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretation to the quantum $α$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in…

Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly…

Two parties, Alice and Bob, who share many copies of a noisy entangled state can try to convert them into fewer, nearly perfect maximally entangled pairs. This is entanglement purification (or distillation). Its best rate, the distillable entanglement, determines how much quantum information can be sent through a noisy channel when classical communication…

A sender encodes one bit in the quantum state of a system: $\hat\rho_0$ with probability $1-t$ and $\hat\rho_1$ with probability $t$. The receiver measures the system. How much information about the bit can any measurement extract? The answer, the accessible information $I(t)$, is the basic figure of merit of a binary quantum communication channel and the starting…

Quantum optics generates entanglement between continuous variable systems: modes of the electromagnetic field described by quadratures $\hat x_j,\hat p_j$ with $[\hat x_j,\hat p_{j'}]=i\delta_{jj'}$, rather than qubits. Two-mode squeezed light underlies continuous variable teleportation (Furusawa et al., Science 282, 706, 1998), and the noise in such…

A quantum system is prepared in one of two known states, and an observer must guess which one by measuring it. This is the simplest quantum decision problem, and its optimal error probability is the most operational way to say how distinguishable two quantum states are. It underlies the analysis of quantum hypothesis testing, of state discrimination in…

An unknown quantum state cannot be copied perfectly (Wootters–Zurek and Dieks, 1982), but it can be copied approximately. A universal cloner takes $m$ identical qubits in an unknown pure state and outputs $m+l$ qubits whose individual states are as close as possible to the input, with a quality independent of the input. Bužek and Hillery (PRA 54, 1844 (1996))…

A classical-quantum channel sends a classical letter $i$ to a quantum state $\rho_i$. It models any scheme that stores classical messages in quantum systems: optical communication with coherent or thermal states, or a quantum memory read out by a measurement. Holevo (1998) and Schumacher–Westmoreland (1997) showed that its capacity, the largest rate at which…

Teleportation (Bennett et al. 1993) sends an unknown state of a $d$-dimensional system using one maximally entangled pair and $2\log d$ bits of classical communication, and both resources are necessary. Remote state preparation (RSP) is the same task when the sender, Alice, knows the state she wants the receiver, Bob, to hold. Lo (PRA 62, 012313, 2000) asked…

A forecaster, an economist or any expert who sells a probability distribution cannot be checked directly: a single outcome of an experiment never reveals whether the reported distribution was the expert's true state of knowledge. The only safeguard is to pay according to a scheme under which lying does not pay in expectation. This is the problem of keeping…

The conditional entropy $S(\rho^{12}\mid\rho^2)$ of a bipartite quantum state measures how much uncertainty about the first system remains for an observer holding the second. It enters the quantum Slepian–Wolf and state-merging protocols, the coherent information that governs quantum channel capacity, and entanglement measures such as squashed entanglement.…

A probability assignment can never be confirmed outright, but it can be strengthened or undermined by observation when the phenomenon is repeatable. The Law of Large Numbers says that relative frequencies approach the assigned probabilities. The next question is quantitative: how fast does the probability of an "incorrect" frequency go to zero?

A sender who encodes classical messages into quantum states chooses an ensemble: signal states $\rho_k$ prepared with probabilities $p_k$. Holevo proved in 1973 that the information a receiver can extract from such an ensemble, whatever measurement is used, is at most the Holevo quantity $\chi$ (Holevo 1973). Holevo (1998) and Schumacher and Westmoreland (1997)…

A quantum measurement turns a quantum state into classical data. How much of that data is information about the state, and how much is randomness the measurement device adds on its own? For a measurement repeated on many copies of a source, Andreas Winter (arXiv:quant-ph/0109050) gives a sharp answer: the output splits into intrinsic data, correlated with…

A pure entangled state of two parties can be converted reversibly, in the limit of many copies, into singlets: $N$ copies of $\psi_{AB}$ yield about $N\,S(\rho_A)$ singlets and can be recovered from them (Bennett, Bernstein, Popescu, Schumacher 1996). For bipartite pure states the singlet is therefore the only kind of entanglement. For three or more parties the…

A quantum channel is a completely positive, trace-preserving linear map between matrix algebras: the most general physical process taking quantum states on one system to quantum states on another. A basic question of quantum Shannon theory is how much classical information a channel can carry. If the sender is not allowed to entangle inputs across channel…

Simple quantum hypothesis testing asks how well one can decide, from $n$ independent copies of a quantum system, whether its state is $\rho$ (the null hypothesis) or $\sigma$ (the alternative). It is the simplest problem in which the non-commutativity of quantum states is unavoidable, and it underlies the analysis of classical-quantum channel coding, wire-tap…

Cryptographic protocols are usually proved secure by showing that what an adversary sees in the real protocol is indistinguishable from what it would see in an ideal one, typically with a distinguishing advantage that decays exponentially in a security parameter. Classically there are several ways to measure how distinguishable two probability distributions…

A quantum computer must apply gates to encoded data while errors keep occurring. On a stabilizer code the Pauli operators, and more generally the gates of the Clifford group, can be applied transversally or by simple fault-tolerant circuits, but the Clifford group alone is not universal. Before 1999 the non-Clifford gates needed for universality, such as…

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given $n$ copies of a mixed state $ρ\in\mathbb{C}^{d\times d}$ and the goal is to distinguish whether $ρ$'s spectrum satisfies some property $\mathcal{P}$ or is at least $ε$-far in $\ell_1$-distance from satisfying $\mathcal{P}$. This problem was promoted in the…

Schur-Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the tensor powers of all unitaries is spanned by the permutations of the tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object…

We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto's, given as a transform of the Renyi capacities with parameters alpha>1. It is important to note that, unlike in the classical case, there are many inequivalent ways to define…