The Quantum Formalization Drive

Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes that obey (are covariant with respect to) continuous symmetries in a certain sense cannot correct erasure errors perfectly (a well-known result in this regard being the Eastin-Knill theorem in the context of…

According to a fundamental result in quantum computing, any unitary transformation on a composite system can be generated using so-called 2-local unitaries that act only on two subsystems. Beyond its importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts…

We present a proof for the quantum channel coding theorem which relies on the fact that a randomly chosen code space typically is highly suitable for quantum error correction. In this sense, the proof is close to Shannon's original treatment of information transmission via a noisy classical channel.

By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite-dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann…

We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of smooth min and max Renyi…

Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable…

Two quantum states cannot in general be told apart perfectly, and how well they can be told apart depends on the measurement. Classically, the statistical overlap (Bhattacharyya coefficient) $\sum_b\sqrt{p_0(b)p_1(b)}$ of two probability distributions controls how quickly repeated samples separate them. The quantum question is which measurement makes the…

The Shannon information function $H(p) = -\sum_i p_i \ln p_i$ measures the average information gained when one samples a discrete probability distribution $p$. Its use as a measure of distinguishability (the mutual information of Chapter 2 of C. A. Fuchs's thesis, arXiv:quant-ph/9601020) is only as convincing as the reasons for the formula itself. Section 2.4.1…

We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with $n$ copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every $n\in\mathbb{N}$ there exist non-trivial maps with this property and that for two-dimensional Hilbert…

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into $n$ physical subsystems, we say that the code is…

We find a strong-converse bound on the private capacity of a quantum channel assisted by unlimited two-way classical communication. The bound is based on the max-relative entropy of entanglement and its proof uses a new inequality for the sandwiched Rényi divergences based on complex interpolation techniques. We provide explicit examples of quantum channels…

We analyse the problem of transmitting a number of unknown quantum states or one composite system in one go. We derive a lower bound on the performance of such process, measured in the entanglement fidelity. The obtained bound is effectively computable and outperforms the explicit values of the entanglement fidelity calculated for the pre-existing variants…

The Polyanskiy-Verdú paradigm provides an elegant way of using generalized-divergences to obtain strong converses and thus far has remained confined to protocols involving channels (classical or quantum). In this paper, drawing inspirations from it, we provide strong converses for protocols involving LOCC (local operations and classical communication). The…

We obtain a lower bound on the maximum number of qubits, $Q^{n, ε}(\mathcal{N})$, which can be transmitted over $n$ uses of a quantum channel $\mathcal{N}$, for a given non-zero error threshold $ε$. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the…

We develop a theoretical framework for $S_n$-equivariant convolutional quantum circuits with SU$(d)$-symmetry, building on and significantly generalizing Jordan's Permutational Quantum Computing (PQC) formalism based on Schur-Weyl duality connecting both SU$(d)$ and $S_n$ actions on qudits. In particular, we utilize the Okounkov-Vershik approach to prove Harrow's…

We prove that several known upper bounds on the classical capacity of thermal and additive noise bosonic channels are actually strong converse rates. Our results strengthen the interpretation of these upper bounds, in the sense that we now know that the probability of correctly decoding a classical message rapidly converges to zero in the limit of many…

This note quantifies the continuity properties of entanglement: how much does entanglement vary if we change the entangled quantum state just a little? This question is studied for the pure state entanglement of a bipartite system and for the entanglement of formation of a bipartite system in a mixed state.

The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary $t$-designs. Unitary $t$-designs are probability distributions that mimic Haar…

Quantum entanglement is a key physical resource in quantum information processing that allows for performing basic quantum tasks such as teleportation and quantum key distribution, which are impossible in the classical world. Ever since the rise of quantum information theory, it has been an open problem to quantify entanglement in an…

The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. However, both quantities are given by the infinite regularization of respectively the coherent and the private information. Here, we construct a family of channels for which the private and coherent information can remain strictly…

In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on…

Port-based teleportation (PBT) is a protocol in which Alice teleports an unknown quantum state to Bob using measurements on a shared entangled multipartite state called the port state and forward classical communication. In this paper, we give an explicit proof that the so-called pretty good measurement, or square-root measurement, is optimal for the PBT…

For a given quantum state $ρ$ and two quantum operations $Φ$ and $Ψ$, the information encoded in the quantum state $ρ$ is quantified by its von Neumann entropy $§(ρ)$. By the famous Choi-Jamiołkowski isomorphism, the quantum operation $Φ$ can be transformed into a bipartite state, the von Neumann entropy $§^{\mathrm{map}}(Φ)$ of the bipartite state describes the…

We find a tight upper bound for the classical capacity of quantum thermal noise channels that is within $1/\ln 2$ bits of Holevo's lower bound. This lower bound is achievable using unentangled, classical signal states, namely displaced coherent states. Thus, we find that while quantum tricks might offer benefits, when it comes to classical communication they…

Given a quantum gate implementing a $d$-dimensional unitary operation $U_d$, without any specific description but $d$, and permitted to use $k$ times, we present a universal probabilistic heralded quantum circuit that implements the exact inverse $U_d^{-1}$, whose failure probability decays, exponentially in $k$. The protocol employs an adaptive strategy, proven…