Search

With the increasing demand for secure communication in optical space networks, it is essential to develop physical-layer scalable security solutions. In this context, we present the asymptotic security analysis of a keyless quantum private communication protocol that transmits classical information over quantum states. Different from the previous literature, our protocol sends dummy (decoy) states optimally obtained from the true information to deceive the eavesdropper. We analyze optical on-off keying (OOK) and binary phase shift keying (BPSK) for several detection scenarios. Our protocol significantly improves the protocol without decoy states whenever Bob is at a technological disadvantage with respect to Eve. Our protocol guarantees positive secrecy capacity when the eavesdropper gathers up to $90-99.9\%$ (depending on the detection scenario) of the photon energy that Bob detects, even when Eve is only limited by the laws of quantum mechanics. We apply our results to the design of an optical inter-satellite link (ISL) study case with pointing losses, and introduce a new design methodology whereby the link margin is guaranteed to be secure by our protocol. Hence, our design does not require knowing thr location of the eavesdropper and or channel state: the protocol aborts whenever the channel drops below the secured margin. Our protocol can be implemented with the state of the art space proof technology. Finally, we also show the potential secrecy advantage when using (not yet available) squeezed quantum states technology.

We present a quantum algorithm for computational fluid dynamics based on the Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a modified collision operator, assuming full relaxation to the local equilibrium within a single time step. Our quantum algorithm enables the computation of multiple time steps in the linearized case, specifically for solving the advection-diffusion equation, before necessitating a full state measurement. Moreover, our formulation can be extended to compute the non-linear equilibrium distribution function for a single time step prior to measurement, utilizing the measurement as an essential algorithmic step. However, in the non-linear case, a classical postprocessing step is necessary for computing the moments of the distribution function. We validate our algorithm by solving the one dimensional advection-diffusion of a Gaussian hill. Our results demonstrate that our quantum algorithm captures non-linearity.

We critically examine the often-made observation that "quantum winter [or some other winter] is coming", and the related admonition to prepare for this or that winter, inevitably bound to arrive. What we find based on even the most superficial look at the available evidence is that such statements not only are overblown hype, but are also factually wrong: Winter is here, and the real question is rather for how long it/they will stay., Comment: 2 pages

We consider the private classical capacity of a quantum wiretap channel, where the users (sender Alice, receiver Bob, and eavesdropper Eve) have access to the resource of a shared quantum state, additionally to their channel inputs and outputs. An extreme case is maximal entanglement or a secret key between Alice and Bob, both of which would allow for onetime padding the message. But here both the wiretap channel and the shared state are general. In the other extreme case that the state is trivial, we recover the wiretap channel and its private capacity [N. Cai, A. Winter and R. W. Yeung, Probl. Inform. Transm. 40(4):318-336, 2004]. We show how to use the given resource state to build a code for secret classical communication. Our main result is a lower bound on the assisted private capacity, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.

Following initial work by JaJa, Ahlswede and Cai, and inspired by a recent renewed surge in interest in deterministic identification (DI) via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by its output distributions as a subset in the probability simplex. Our main findings are that the maximum length of messages thus identifiable scales superlinearly as $R\,n\log n$ with the block length $n$, and that the optimal rate $R$ is bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension $d$ of a certain algebraic transformation of the output set: $\frac14 d \leq R \leq \frac12 d$. Remarkably, both the lower and upper Minkowski dimensions play a role in this result. Along the way, we present a "Hypothesis Testing Lemma" showing that it is sufficient to ensure pairwise reliable distinguishability of the output distributions to construct a DI code. Although we do not know the exact capacity formula, we can conclude that the DI capacity exhibits superactivation: there exist channels whose capacities individually are zero, but whose product has positive capacity. We also generalise these results to classical-quantum channels with finite-dimensional output quantum system, in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs., Comment: 39 pages, 5 figures

Motivated by deterministic identification via classical channels, where the encoder is not allowed to use randomization, we revisit the problem of identification via quantum channels but now with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new results as our main findings: first, we show that all four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher-Westmoreland Theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples, we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity, which in turn can be larger than pure-state-encoded simultaneous identification capacity., Comment: 25 pages, 1 figure

We propose an interferometric scheme for generating the totally antisymmetric state of $N$ identical bosons with $N$ internal levels (generalized singlet). This state is a resource for various problems with dramatic quantum advantage. The procedure uses a sequence of Fourier multi-ports, combined with coincidence measurements filtering the results. Successful preparation of the generalized singlet is confirmed when the $N$ particles of the input state stay separate (anti-bunch) on each multiport. The scheme is robust to local lossless noise and works even with a totally mixed input state., Comment: 5+2 pages, 1 figure

We introduce methods of characterizing entanglement, in which entanglement measures are enriched by the matrix representations of operators for observables. These observable operator matrix representations can enrich the partial trace over subsets of a system's degrees of freedom, yielding reduced density matrices useful in computing various measures of entanglement, which also preserve the observable expectation value. We focus here on applying these methods to compute observable-enriched entanglement spectra, unveiling new bulk-boundary correspondences of canonical four-band models for topological skyrmion phases and their connection to simpler forms of bulk-boundary correspondence. Given the fundamental roles entanglement signatures and observables play in study of quantum many body systems, observable-enriched entanglement is broadly applicable to myriad problems of quantum mechanics., Comment: 6+4 pages, 8 figures

We show that marginals of blocks of $t$ systems of any finitely correlated translation invariant state on a chain can be learned, in trace distance, with $O(t^2)$ copies -- with an explicit dependence on local dimension, memory dimension and spectral properties of a certain map constructed from the state -- and computational complexity polynomial in $t$. The algorithm requires only the estimation of a marginal of a controlled size, in the worst case bounded by the minimum bond dimension, from which it reconstructs a translation invariant matrix product operator. In the analysis, a central role is played by the theory of operator systems. A refined error bound can be proven for $C^*$-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The learning algorithm also works for states that are only close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states., Comment: 27+7 pages, 6 figures. Typos corrected, improved presentation

Quantum error correcting (QEC) stabilizer codes enable protection of quantum information against errors during storage and processing. Simulation of noisy QEC codes is used to identify the noise parameters necessary for advantageous operation of logical qubits in realistic quantum computing architectures. Typical quantum error correction techniques contain intermediate measurements and classical feedback that determine the actual noisy circuit sequence in an instance of performing the protocol. Dynamical subset sampling enables efficient simulation of such non-deterministic quantum error correcting protocols for any type of quantum circuit and incoherent noise of low strength. As an importance sampling technique, dynamical subset sampling allows one to effectively make use of computational resources to only sample the most relevant sequences of quantum circuits in order to estimate a protocol's logical failure rate with well-defined error bars. We demonstrate the capabilities of dynamical subset sampling with examples from fault-tolerant (FT) QEC. We show that, in a typical stabilizer simulation with incoherent Pauli noise of strength $p = 10^{-3}$, our method can reach a required sampling accuracy on the logical failure rate with two orders of magnitude fewer samples than direct Monte Carlo simulation. Furthermore, dynamical subset sampling naturally allows for efficient simulation of realistic multi-parameter noise models describing faulty quantum processors. It can be applied not only for QEC in the circuit model but any noisy quantum computing framework with incoherent fault operators including measurement-based quantum computation and quantum networks., Comment: 33 pages, 26 figures

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer, Liss and Mor [Phys. Rev. A 95:032308, 2017], and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces., Comment: 22 pages; 5 figures

Quantum cellular automata are alternative quantum-computing paradigms to quantum Turing machines and quantum circuits. Their working mechanisms are inherently automated, therefore measurement free, and they act in a translation invariant manner on all cells/qudits of a register, generating a global rule that updates cell states locally, i.e., based solely on the states of their neighbors. Although desirable features in many applications, it is generally not clear to which extent these fully automated discrete-time local updates can generate and sustain long-range order in the (noisy) systems they act upon. In special, whether and how quantum cellular automata can perform quantum error correction remain open questions. We close this conceptual gap by proposing quantum cellular automata with quantum-error-correction capabilities. We design and investigate two (quasi-)one dimensional quantum cellular automata based on known classical cellular-automata rules with density-classification capabilities, namely the local majority voting and the two-line voting. We investigate the performances of those quantum cellular automata as quantum-memory components by simulating the number of update steps required for the logical information they act upon to be afflicted by a logical bit flip. The proposed designs pave a way to further explore the potential of new types of quantum cellular automata with built-in quantum error correction capabilities., Comment: 5 pages, 3 figures + appendix

We approach two interconnected problems of quantum information processing in networks: Conference key agreement and entanglement distillation, both in the so-called source model where the given resource is a multipartite quantum state and the players interact over public classical channels to generate the desired correlation. The first problem is the distillation of a conference key when the source state is shared between a number of legal players and an eavesdropper; the eavesdropper, apart from starting off with this quantum side information, also observes the public communication between the players. The second is the distillation of Greenberger-Horne-Zeilinger (GHZ) states by means of local operations and classical communication (LOCC) from the given mixed state. These problem settings extend our previous paper [IEEE Trans. Inf. Theory 68(2):976-988, 2022], and we generalise its results: using a quantum version of the task of communication for omniscience, we derive novel lower bounds on the distillable conference key from any multipartite quantum state by means of non-interacting communication protocols. Secondly, we establish novel lower bounds on the yield of GHZ states from multipartite mixed states. Namely, we present two methods to produce bipartite entanglement between sufficiently many nodes so as to produce GHZ states. Next, we show that the conference key agreement protocol can be made coherent under certain conditions, enabling the direct generation of multipartite GHZ states.

The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical range. In this work, we explore geometric properties of this construct for measurements represented by tensor products of Pauli observables, also known as Pauli strings. The structure of pairwise commutation and anticommutation relations among a set of Pauli strings determines a graph $G$, sometimes also called the frustration graph. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range. Such an outer approximation can be very practical since ellipsoids can be handled analytically even in high dimensions. We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O. G\"uhne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph parameter that we call $\beta(G)$. Besides, we develop this approach in different directions, such as comparison with graph-theoretic approaches in other fields, applications in quantum information theory, numerical methods, properties of the new graph parameter, etc. Our approach suggests many open questions that we discuss briefly at the end., Comment: 14+3 pages, 5 figures

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author., Comment: 30 pages, 3 figures; v2 fixes an error in Definition 36 and various typos and minor issues throughout; v3 fixes some further minor points and provides a proof of Lemma 11 due to F. Dupuis, it is the version accepted by IJQI (special issue in honour of Alexander S. Holevo)

We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old [Dupuis et al. Commun. Math. Phys. 328:251-284 (2014)] and new methods [Dupuis, arXiv:2105.05342], we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of R\'enyi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, that is for only partially known i.i.d. source or channel, which are furthermore optimal for entanglement of assistance and state merging., Comment: This second version includes new results on the achievable rates of multi-party assisted entanglement distillation and entanglement of assistance in the compound setting, as well as further comments and corrections. The idea of the telescoping trick has independently and concurrently been discovered for multipartite decoupling by Hao-Chung Cheng, Li Gao and Mario Berta. 39 pages, 6 figures

We derive a measurement-independent asymptotic continuity bound on the observational entropy for general POVM measurements, making essential use of its property of bounded concavity. The same insight is used to obtain continuity bounds for other entropic quantities, including the measured relative entropy distance to a convex a set of states under a general set of measurements. As a special case, we define and study conditional observational entropy, which is an observational entropy in one (measured) subsystem conditioned on the quantum state in another (unmeasured) subsystem. We also study continuity of relative entropy with respect to a jointly applied channel, finding that observational entropy is uniformly continuous as a function of the measurement. But we show by means of an example that this continuity under measurements cannot have the form of a concrete asymptotic bound., Comment: 16 pages; v2 new Corollary 15 on measurement continuity

Topological qubits composed of unpaired Majorana zero-modes are under intense experimental and theoretical scrutiny in efforts to realize practical quantum computation schemes. In this work, we show the minimum four \textit{unpaired} Majorana zero-modes required for a topological qubit according to braiding schemes and control of entanglement for gate operations are inherent to multiplicative topological phases, which realize symmetry-protected tensor products -- and maximally-entangled Bell states -- of unpaired Majorana zero-modes known as multiplicative Majorana zero-modes. We introduce multiplicative Majorana zero-modes as topologically-protected boundary states of both one and two-dimensional multiplicative topological phases, using methods reliant on multiplicative topology to construct relevant Hamiltonians from the Kitaev chain model. We furthermore characterize topology in the bulk and on the boundary with established methods while also introducing techniques to overcome challenges in characterizing multiplicative topology. In the process, we explore the potential of these multiplicative topological phases for an alternative to braiding-based topological quantum computation schemes, in which gate operations are performed through topological phase transitions., Comment: 24 pages, 23 figures, and 2 tables in main text, 11 pages in supplementary materials

Non-classical features of quantum systems have the potential to strengthen the way we currently exchange information. In this paper, we explore this enhancement on the most basic level of single particles. To be more precise, we compare how well multi-party information can be transmitted to a single receiver using just one classical or quantum particle. Our approach is based on a multiple-access communication model in which messages can be encoded into a single particle that is coherently distributed across multiple spatial modes. Theoretically, we derive lower bounds on the accessible information in the quantum setting that strictly separate it from the classical scenario. This separation is found whenever there is more than one sender, and also when there is just a single sender who has a shared phase reference with the receiver. Experimentally, we demonstrate such quantum advantage in single-particle communication by implementing a multi-port interferometer with messages being encoded along the different trajectories. Specifically, we consider a two-sender communication protocol built by a three-port optical interferometer. In this scenario, the rate sum achievable with a classical particle is upper bounded by one bit, while we experimentally observe a rate sum of $1.0152\pm0.0034$ bits in the quantum setup., Comment: 18+10 pages. Comments welcome

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory state is a finite dimensional quantum state, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes., Comment: 19 pages, added 2 figures

When describing the effective dynamics of an observable in a many-body system, the repeated randomness assumption, which states that the system returns in a short time to a maximum entropy state, is a crucial hypothesis to guarantee that the effective dynamics is classical, Markovian and obeys local detailed balance. While the latter behaviour is frequently observed in naturally occurring processes, the repeated randomness assumption is in blatant contradiction to the microscopic reversibility of the system. Here, we show that the use of the repeated randomness assumption can be justified in the description of the effective dynamics of an observable that is both slow and coarse, two properties we will define rigorously. Then, our derivation will invoke essentially only the eigenstate thermalization hypothesis and typicality arguments. While the assumption of a slow observable is subtle, as it provides only a necessary but not sufficient condition, it also offers a unifying perspective applicable to, e.g., open systems as well as collective observables of many-body systems. All our ideas are numerically verified by studying density waves in spin chains., Comment: Final version

We show that entirely quantum Shannon theoretic methods, based on von Neumann entropies and their properties, can be used to derive Singleton bounds on the performance of entanglement-assisted hybrid classical-quantum (EACQ) error correcting codes. Concretely, we show that the triple-rate region of qubits, cbits and ebits of possible EACQ codes over arbitrary alphabet sizes is contained in the quantum Shannon theoretic rate region of an associated memoryless erasure channel, which turns out to be a polytope. We show that a large part of this region is attainable by certain EACQ codes, whenever the local alphabet size (i.e. Hilbert space dimension) is large enough, in keeping with known facts about classical and quantum minimum distance separable (MDS) codes: in particular, all of its extreme points and all but one of its extremal lines. The attainability of the remaining one extremal line segment is left as an open question., Comment: 18 pages, 1 figure; v2 has more details, more references, and self-contained proofs; in v3 some typos are fixed and ORCIDs added; v4 greatly expanded to 22 pages with more references, now 4 figures, improvements to the presentation

We consider a rate-distortion version of the quantum state redistribution task, where the error of the decoded state is judged via an additive distortion measure; it thus constitutes a quantum generalisation of the classical Wyner-Ziv problem. The quantum source is described by a tripartite pure state shared between Alice ($A$, encoder), Bob ($B$, decoder) and a reference ($R$). Both Alice and Bob are required to output a system ($\widetilde{A}$ and $\widetilde{B}$, respectively), and the distortion measure is encoded in an observable on $\widetilde{A}\widetilde{B}R$. It includes as special cases most quantum rate-distortion problems considered in the past, and in particular quantum data compression with the fidelity measured per copy; furthermore, it generalises the well-known state merging and quantum state redistribution tasks for a pure state source, with per-copy fidelity, and a variant recently considered by us, where the source is an ensemble of pure states [1], [2]. We derive a single-letter formula for the rate-distortion function of compression schemes assisted by free entanglement. A peculiarity of the formula is that in general it requires optimisation over an unbounded auxiliary register, so the rate-distortion function is not readily computable from our result, and there is a continuity issue at zero distortion. However, we show how to overcome these difficulties in certain situations.

We introduce the notion of $p$-adic quantum bit ($p$-qubit) in the context of the $p$-adic quantum mechanics initiated and developed by Volovich and his followers. In this approach, physics takes place in three-dimensional $p$-adic space rather than Euclidean space. Based on our prior work describing the $p$-adic special orthogonal group, we outline a programme to classify its continuous unitary projective representations, which can be interpreted as a theory of $p$-adic angular momentum. The $p$-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes $p$., Comment: 42 pages

Optical qubits uniquely combine information transfer in optical fibers with a good processing capability and are therefore attractive tools for quantum technologies. A large challenge, however, is to overcome the low efficiency of two-qubit logic gates. The experimentally achieved efficiency in an optical controlled NOT (CNOT) gate reached approximately 11% in 2003 and has seen no increase since. Here we report on a new platform that was designed to surpass this long-standing record. The new scheme avoids inherently probabilistic protocols and, instead, combines aspects of two established quantum nonlinear systems: atom-cavity systems and Rydberg electromagnetically induced transparency. We demonstrate a CNOT gate between two optical photons with an average efficiency of 41.7(5)% at a postselected process fidelity of 81(2)%. Moreover, we extend the scheme to a CNOT gate with multiple target qubits and produce entangled states of presently up to five photons. All these achievements are promising and have the potential to advance optical quantum information processing in which almost all advanced protocols would profit from high-efficiency logic gates., Comment: In this revision, we extend the discussion of the efficiency in Sec. IV.C and we add models in appendices B, E, and L-O

Under the influence of external environments, quantum systems can undergo various different processes, including decoherence and equilibration. We observe that macroscopic objects are both objective and thermal, thus leading to the expectation that both objectivity and thermalisation can peacefully coexist on the quantum regime too. Crucially, however, objectivity relies on distributed classical information that could conflict with thermalisation. Here, we examine the overlap between thermal and objective states. We find that in general, one cannot exist when the other is present. However, there are certain regimes where thermality and objectivity are more likely to coexist: in the high temperature limit, at the non-degenerate low temperature limit, and when the environment is large. This is consistent with our experiences that everyday-sized objects can be both thermal and objective., Comment: Published version: 17 + 3 pages, 4 figures, 1 table

Resource theories in quantum information science are helpful for the study and quantification of the performance of information-processing tasks that involve quantum systems. These resource theories also find applications in other areas of study; e.g., the resource theories of entanglement and coherence have found use and implications in the study of quantum thermodynamics and memory effects in quantum dynamics. In this paper, we introduce the resource theory of unextendibility, which is associated to the inability of extending quantum entanglement in a given quantum state to multiple parties. The free states in this resource theory are the k-extendible states, and the free channels are k-extendible channels, which preserve the class of k-extendible states. We make use of this resource theory to derive non-asymptotic, upper bounds on the rate at which quantum communication or entanglement preservation is possible by utilizing an arbitrary quantum channel a finite number of times, along with the assistance of k-extendible channels at no cost. We then show that the bounds obtained are significantly tighter than previously known bounds for quantum communication over both the depolarizing and erasure channels., Comment: 7 pages, 2 figures, see companion paper at arXiv:1803.10710

The theoretical measuring of information was famously initiated by Shannon in his mathematical theory of communication, in which he proposed a now widely used quantity, the entropy, measured in bits. Yet, in the same paper, Shannon also chose to measure the information in continuous systems in nats, which differ from bits by the use of the natural rather than the binary logarithm. We point out that there is nothing natural about the choice of logarithm basis, rather it is arbitrary. We remedy this problematic state of affairs by proposing a genuinely natural measure of information, which we dub gnats. We show that gnats have many advantages in information theory, and propose to adopt the underlying methodology throughout science, arts and everyday life., Comment: 2.5 pages, 3 figures

Quantum coherence, a basic feature of quantum mechanics residing in superpositions of quantum states, is a resource for quantum information processing. Coherence emerges in a fundamentally different way for nonidentical and identical particles, in that for the latter a unique contribution exists linked to indistinguishability which cannot occur for nonidentical particles. We experimentally demonstrate by an optical setup this additional contribution to quantum coherence, showing that its amount directly depends on the degree of indistinguishability and exploiting it to run a quantum phase discrimination protocol. Furthermore, the designed setup allows for simulating Fermionic particles with photons, thus assessing the role of particle statistics (Bosons or Fermions) in coherence generation and utilization. Our experiment proves that independent indistinguishable particles can supply a controllable resource of coherence for quantum metrology., Comment: 6 pages, 4 figures

We establish a second anti-blocker theorem for non-commutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimisation parameters for a given convex corner that generalise well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterise its maximum value in terms of a generalised fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behaviour of the introduced parameters with respect to tensoring. Specialising to non-commutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of non-commutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a non-commutative graph and compute the values of our parameters in some specific cases.

In recent years, the notion of Quantum Materials has emerged as a powerful unifying concept across diverse fields of science and engineering, from condensed-matter and cold atom physics to materials science and quantum computing. Beyond traditional quantum materials such as unconventional superconductors, heavy fermions, and multiferroics, the field has significantly expanded to encompass topological quantum matter, two-dimensional materials and their van der Waals heterostructures, Moire materials, Floquet time crystals, as well as materials and devices for quantum computation with Majorana fermions. In this Roadmap collection we aim to capture a snapshot of the most recent developments in the field, and to identify outstanding challenges and emerging opportunities. The format of the Roadmap, whereby experts in each discipline share their viewpoint and articulate their vision for quantum materials, reflects the dynamic and multifaceted nature of this research area, and is meant to encourage exchanges and discussions across traditional disciplinary boundaries. It is our hope that this collective vision will contribute to sparking new fascinating questions and activities at the intersection of materials science, condensed matter physics, device engineering, and quantum information, and to shaping a clearer landscape of quantum materials science as a new frontier of interdisciplinary scientific inquiry., Comment: 93 pages, 27 figures

We show that the positivity of the Wigner function of Gaussian states and measurements provides an elegant way to bound the discriminating power of "linear optics", which we formalise as Gaussian measurement operations augmented by classical (feed-forward) communication (GOCC). This allows us to reproduce and generalise the result of Takeoka and Sasaki [PRA 78:022320, 2008], which tightly characterises the GOCC norm distance of coherent states, separating it from the optimal distinguishability according to Helstrom's theorem. Furthermore, invoking ideas from classical and quantum Shannon theory we show that there are states, each a probabilistic mixture of multi-mode coherent states, which are exponentially reliably discriminated in principle, but appear exponentially close judging from the output of GOCC measurements. In analogy to LOCC data hiding, which shows an irreversibility in the preparation and discrimination of states by the restricted class of local operations and classical communication (LOCC), we call the present effect GOCC data hiding. We also present general bounds in the opposite direction, guaranteeing a minimum of distinguishability under measurements with positive Wigner function, for any bounded-energy states that are Helstrom distinguishable. We conjecture that a similar bound holds for GOCC measurements., Comment: 11 pages, 1 figure

Passive environment assisted communication takes place via a quantum channel modeled as a unitary interaction between the information carrying system and an environment, where the latter is controlled by a passive helper, who can set its initial state such as to assist sender and receiver, but not help actively by adjusting her behaviour depending on the message. Here we investigate the information transmission capabilities in this framework by considering Gaussian unitaries acting on Bosonic systems. We consider both quantum communication and classical communication with helper, as well as classical communication with free classical coordination between sender and helper (conferencing encoders). Concerning quantum communication, we prove general coding theorems with and without energy constraints, yielding multi-letter (regularized) expressions. In the search for cases where the capacity formula is computable, we look for Gaussian unitaries that are universally degradable or anti-degradable. However, we show that no Gaussian unitary yields either a degradable or anti-degradable channel for all environment states. On the other hand, restricting to Gaussian environment states, results in universally degradable unitaries, for which we thus can give single-letter quantum capacity formulas. Concerning classical communication, we prove a general coding theorem for the classical capacity under and energy constraint, given by a multi-letter expression. Furthermore, we derive an uncertainty-type relation between the classical capacities of the sender and the helper, helped respectively by the other party, showing a lower bound on the sum of the two capacities. Then, this is used to lower bound the classical information transmission rate in the scenario of classical communication between sender and helper., Comment: 36 pages, 2 figures

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately., Comment: 24 pages, 2 figures, published version, added Remark 13, expanded Section 3.1 and Example 23 significantly

A universal programmable quantum processor uses program quantum states to apply an arbitrary quantum channel to an input state. We generalize the concept of a finite-dimensional programmable quantum processor to infinite dimension assuming an energy constraint on the input and output of the target quantum channels. By proving reductions to and from finite-dimensional processors, we obtain upper and lower bounds on the program dimension required to approximately implement energy-limited quantum channels. In particular, we consider the implementation of Gaussian channels. Due to their practical relevance, we investigate the resource requirements for gauge-covariant Gaussian channels. Additionally, we give upper and lower bounds on the program dimension of a processor implementing all Gaussian unitary channels. These lower bounds rely on a direct information-theoretic argument, based on the generalization from finite to infinite dimension of a certain replication lemma for unitaries., Comment: 38 pages, 2 figures, published version

We study quantum hypothesis testing between orthogonal states under restricted local measurements in the many-copy scenario. For testing arbitrary multipartite entangled pure state against its orthogonal complement state via the local operation and classical communication (LOCC) operation, we prove that the optimal average error probability always decays exponentially in the number of copies. Second, we provide a sufficient condition for the LOCC operations to achieve the same performance as the positive-partial-transpose (PPT) operations. We further show that testing a maximally entangled state against its orthogonal complement and testing extremal Werner states both fulfill the above-mentioned condition. Hence, we determine the explicit expressions for the optimal average error probability, the optimal trade-off between the type-I and type-II errors, and the associated Chernoff, Stein, Hoeffding, and strong converse exponents. Then, we show an infinite asymptotic separation between the separable (SEP) and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB). The quantum states can be distinguished perfectly by PPT operations, while the optimal error probability, with SEP operations, admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is a UPB., Comment: Paper re-organized and revised

We consider a theory of quantum thermodynamics with multiple conserved quantities (or charges). To this end, we generalize the seminal results of Sparaciari et al. [PRA 96:052112, 2017] to the case of multiple, in general non-commuting charges, for which we formulate a resource theory of thermodynamics of asymptotically many non-interacting systems. To every state we associate the vector of its expected charge values and its entropy, forming the phase diagram of the system. Our fundamental result is the Asymptotic Equivalence Theorem (AET), which allows us to identify the equivalence classes of states under asymptotic approximately charge-conserving unitaries with the points of the phase diagram. Using the phase diagram of a system and its bath, we analyze the first and the second laws of thermodynamics. In particular, we show that to attain the second law, an asymptotically large bath is necessary. In the case that the bath is composed of several identical copies of the same elementary bath, we quantify exactly how large the bath has to be to permit a specified work transformation of a given system, in terms of the number of copies of the elementary bath systems per work system (bath rate). If the bath is relatively small, we show that the analysis requires an extended phase diagram exhibiting negative entropies. This corresponds to the purely quantum effect that at the end of the process, system and bath are entangled, thus permitting classically impossible transformations. For a large bath, system and bath may be left uncorrelated and we show that the optimal bath rate, as a function of how tightly the second law is attained, can be expressed in terms of the heat capacity of the bath. Our approach solves a problem from earlier investigations about how to store the different charges under optimal work extraction protocols in physically separate batteries., Comment: 38 pages, 5 figures; v3 is the accepted journal version, title is shortened and more explanations and references added in many places

Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness of adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and non-adaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3)We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to non-adaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond non-adaptive tensor product input strategies., Comment: v3 is the accepted version

Magic (non-stabilizerness) is a necessary but "expensive" kind of "fuel" to drive universal fault-tolerant quantum computation. To properly study and characterize the origin of quantum "complexity" in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic. Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions). Here we present an initiatory study of the magic of genuinely many-body quantum states that may be strongly entangled, with focus on the important case of many qubits, at a quantitative level. We first address the basic question of how "magical" a many-body state can be, and show that the maximum magic of an $n$-qubit state is essentially $n$, simultaneously for a range of "good" magic measures. We then show that, in fact, almost all $n$-qubit pure states have magic of nearly $n$. In the quest for explicit, scalable cases of highly entangled states whose magic can be understood, we connect the magic of hypergraph states with the second-order nonlinearity of their underlying Boolean functions. Next, we go on and investigate many-body magic in practical and physical contexts. We first consider a variant of MBQC where the client is restricted to Pauli measurements, in which magic is a necessary feature of the initial "resource" state. We show that $n$-qubit states with nearly $n$ magic, or indeed almost all states, cannot supply nontrivial speedups over classical computers. We then present an example of analyzing the magic of "natural" condensed matter systems of physical interest. We apply the Boolean function techniques to derive explicit bounds on the magic of certain representative 2D SPT states, and comment on possible further connections between magic and the quantum complexity of phases of matter., Comment: 18 pages, 3 figures. Close to published version

We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of generalized quantum Singleton bound [Brun et al., IEEE Trans. Inf. Theory 60(6):3073--3089 (2014)] was believed to hold for many years until recently one of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we rectify this state of affairs by proving the correct generalized quantum Singleton bound, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block length $n$ and code length $k$ for given minimum distance $d$ and we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less than $d$ letters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule: any pure QECC yields an EAQECC with the same distance and dimension, but of shorter block length., Comment: 10 pages, 5 figures. Theorem 7 and Corollaries 8 and 9 added. Accepted version

We construct new protocols for the tasks of converting noisy multipartite quantum correlations into noiseless classical and quantum ones using local operations and classical communications (LOCC). For the former, known as common randomness (CR) distillation, two new lower bounds on the "distillable common randomness", an operational measure of the total genuine (classical) correlations in a quantum state, are obtained. Our proof relies on a generalization of communication for omniscience (CO) [Csiszar and Narayan, IEEE Trans. Inf. Theory 50:3047-3061, 2004]. Our contribution here is a novel simultaneous decoder for the compression of correlated classical sources by random binning with quantum side information at the decoder. For the latter, we derive two new lower bounds on the rate at which Greenberger-Horne-Zeilinger (GHZ) states can be asymptotically distilled from any given pure state under LOCC. Our approach consists in "making coherent" the proposed CR distillation protocols and recycling of resources [Devetak et al. IEEE Trans. Inf. Theory 54(10):4587-4618, 2008]. The first lower bound is identical to a recent result by Vrana and Christandl [IEEE Trans. Inf. Theory 65(9):5945-5958, 2019], which is based on a combinatorial method to achieve the same rate. Our second lower bound generalises and improves upon this result, and unifies a number of other known lower bounds on GHZ distillation.

Godwin's law, i.e. the empirical observation that as an online discussion grows in time, the probability of a comparison with Nazis or Hitler quickly approaches unity, is one of the best-documented facts of the internet. Anticipating the quantum internet, here we show under reasonable model assumptions a polynomial quantum speedup of Godwin's law. Concretely, in quantum discussions, Hitler will be mentioned on average quadratically earlier, and we conjecture that under specific network topologies, even cubic speedups are possible. We also show that the speedup cannot be more than exponential, unless the polynomial hierarchy collapses to a certain finite level. We report on numerical experiments to simulate the appearance of the quantum Godwin law in future quantum internets; the most amazing finding of our studies is that -- unlike quantum computational speedups -- the quantum Godwin effect is not only robust against noise, but actually enhanced by decoherence. We have as yet no theoretical explanation, nor a good application, for this astonishing behaviour, which we dub quantum hyperpiesia., Comment: 6 pages, 10 figures. Comments are welcome but will likely be ignored

Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics. This tutorial focuses on the derivation of the first and second law for closed and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies. The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat and temperature. These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes. The consistency with previous results is demonstrated. The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically. An integral fluctuation theorem for entropy production is satisfied. In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond., Comment: Final version of the 24 pages tutorial

We consider the most general (finite-dimensional) quantum mechanical information source, which is given by a quantum system $A$ that is correlated with a reference system $R$. The task is to compress $A$ in such a way as to reproduce the joint source state $\rho^{AR}$ at the decoder with asymptotically high fidelity. This includes Schumacher's original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubit-ebit rate pairs.

It is by now well-established that there exist non-local games for which the best entanglement-assisted performance is not better than the best classical performance. Here we show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e. a face of maximum dimension, of the local Bell polytope. Indeed, using semidefinite programming duality, we prove upper bounds on the dimension of these faces, bounding it far away from the maximum. In the special case of non-local computation games, it had been shown before that they are not facet-defining; our result generalises and improves this. As a by-product of our analysis, we find a similar upper bound on the dimension of the faces of the convex body of quantum correlation matrices, showing that (except for the trivial ones expressing the non-negativity of probability) it does not have facets.

Even in the presence of conservation laws, one can perform arbitrary transformations on a system if given access to a suitable reference frame, since conserved quantities may be exchanged between the system and the frame. Here we explore whether these quantities can be separated into different parts of the reference frame, with each part acting as a `battery' for a distinct quantity. For systems composed of spin-$\frac12$ particles, we show that the components of angular momentum $S_x$, $S_y$ and $S_z$ (non-commuting conserved quantities) may be separated in this way, and also provide several extensions of this result. These results also play a key role in the quantum thermodynamics of non-commuting conserved quantities., Comment: Comments welcome ! V2 includes a new result on extracting the angular momentum components of an unknown spin

We define non-commutative versions of the vertex packing polytope, the theta convex body and the fractional vertex packing polytope of a graph, and establish a quantum version of the Sandwich Theorem of Gr\"{o}tschel, Lov\'{a}sz and Schrijver. We define new non-commutative versions of the Lov\'{a}sz number of a graph which lead to an upper bound of the zero-error capacity of the corresponding quantum channel that can be genuinely better than the one established previously by Duan, Severini and Winter. We define non-commutative counterparts of widely used classical graph parameters and establish their interrelation.

We show that the generalization of the relative entropy of a resource from states to channels is not unique, and there are at least six such generalizations. We then show that two of these generalizations are asymptotically continuous, satisfy a version of the asymptotic equipartition property, and their regularizations appear in the power exponent of channel versions of the quantum Stein's Lemma. To obtain our results, we use a new type of "smoothing" that can be applied to functions of channels (with no state analog). We call it "liberal smoothing" as it allows for more spread in the optimization. Along the way, we show that the diamond norm can be expressed as a D_max distance to the set of quantum channels, and prove a variety of properties of all six generalizations of the relative entropy of a resource., Comment: 5 pages (main text) + 20 pages (appendix), references added, published version

We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two different methods to show that finitely many measurement outcomes in every step are sufficient for approaching the optimal probability of discrimination. In the first method, each measurement of an optimal LOCC protocol, applied to a $d_{\rm loc}$-dim local system, is replaced by one with at most $2d_{\rm loc}^2$ outcomes, without changing the probability of success. In the second method, we decompose any LOCC protocol into a convex combination of a number of "slim protocols" in which each measurement applied to a $d_{\rm loc}$-dim local system has at most $d_{\rm loc}^2$ outcomes. To maximize any convex functions in LOCC (including the probability of state discrimination or fidelity of state transformation), an optimal protocol can be replaced by the best slim protocol in the convex decomposition without using shared randomness. For either method, the bound on the number of outcomes per measurement is independent of the global dimension, the number of parties, the depth of the protocol, how deep the measurement is located, and applies to LOCC protocols with infinite rounds, and the "measurement compression" can be done "top-down" -- independent of later operations in the LOCC protocol. The second method can be generalized to implement LOCC instruments with finitely many outcomes: if the instrument has $n$ coarse-grained final measurement outcomes, global input dimension $D_0$ and global output dimension $D_i$ for $i=1,...,n$ conditioned on the $i$-th outcome, then one can obtain the instrument as a convex combination of no more than $R=\sum_{i=1}^n D_0^2 D_i^2 - D_0^2 + 1$ slim protocols; in other words, $\log_2 R$ bits of shared randomess suffice., Comment: see related results by Scott Cohen

We initiate the systematic study of resource theories of quantum channels, i.e. of the dynamics that quantum systems undergo by completely positive maps, in abstracto: Resources are in principle all maps from one quantum system to another, but some maps are deemed free. The free maps are supposed to satisfy certain axioms, among them closure under tensor products, under composition and freeness of the identity map (the latter two say that the free maps form a monoid). The free maps act on the resources simply by tensor product and composition. This generalizes the much-studied resource theories of quantum states, and abolishes the distinction between resources (states) and the free maps, which act on the former, leaving only maps, divided into resource-full and resource-free ones. We discuss the axiomatic framework of quantifying channel resources, and show two general methods of constructing resource monotones of channels. Furthermore, we show that under mild regularity conditions, each resource theory of quantum channels has a distinguished monotone, the robustness (and its smoothed version), generalizing the analogous concept in resource theories of states. We give an operational interpretation of the log-robustness as the amount of heat dissipation (randomness) required for resource erasure by random reversible free maps, valid in broad classes of resource theories of quantum channels. Technically, this is based on an abstract version of the recent convex-split lemma, extended from its original domain of quantum states to ordered vector spaces with sufficiently well-behaved base norms (which includes the case of quantum channels with diamond norm or variants thereof). Finally, we remark on several key issues concerning the asymptotic theory., Comment: 24 pages, 2 figures