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Entanglement distillation is a fundamental information processing task whose implementation is key to quantum communication and modular quantum computing. Noise experienced by such communication and computing platforms occurs not only in the form of Pauli noise such as dephasing (sometimes called $T_2$) but also non-Pauli noise such as amplitude damping (sometimes called $T_1$). We initiate a study of practical and asymptotic distillation over what we call the joint damping-dephasing noise channel. In the practical setting, we propose a distillation scheme that completely isolates away the damping noise. In the asymptotic setting we derive lower bounds on the entanglement sharing capacities including the coherent and reverse coherent information. Like the protocol achieving the reverse coherent information, our scheme uses only backward classical communication. However, for realistic damping noise ($T_1 \neq 2T_2$) our strategy can exceed the reverse coherent strategy, which is the best known for pure damping. In the forward communication setting we numerically exceed the single-letter coherent information strategy by observing the channel displays non-additivity at the two-letter level. The work shows non-additivity can also be found in realistic noise models with magnitudes of non-additivity similar to those found in more idealized noise channels., Comment: 17 page, 8 figures, comments are welcome

Answering a question posed by Vladimir Tkachuk, we prove that every $T_1$ connected first-countable space is a continuous open image of a connected metrizable space.

In [5] we studied spaces with a Lusin $\pi$-base and $\pi$-spaces and posed the following question: Does the class of continuous open images of spaces with a Lusin $\pi$-base equal the class of continuous open images of $\pi$-spaces? We give a negative answer to this question., Comment: arXiv admin note: text overlap with arXiv:2208.06288

We prove that: 1. If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable. 2. A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally Hausdorff and has a countable pseudocharacter. 3. Let $Y$ be a dense-in-itself space with the following property: $\forall y\in Y\ \exists Q(y) \subseteq Y\ [y \text{ is a non-isolated q-point in } Q(y)]$. If $Y$ is an open-closed image of a submetrizable space, then $Y$ is submetrizable. 4. There exist a submetrizable space $X$, a regular hereditarily paracompact non submetrizable first-countable space $Y$, and an open-closed map $f\colon X \to Y$.

Assessment of practical quantum information processing (QIP) remains partial without understanding limits imposed by noise. Unfortunately, mere description of noise grows exponentially with system size, becoming cumbersome even for modest sized systems of imminent practical interest. We fulfill the need for estimates on performing noisy quantum state preparation, verification, and observation. To do the estimation we propose fast numerical algorithms to maximize the expectation value of any $d$-dimensional observable over states of bounded purity. This bound on purity factors in noise in a measurable way. Our fastest algorithm takes $O(d)$ steps if the eigendecomposition of the observable is known, otherwise takes $O(d^3)$ steps at worst. The algorithms also solve maximum likelihood estimation for quantum state tomography with convex and even non-convex purity constraints. Numerics show performance of our key sub-routine (it finds in linear time a probability vector with bounded norm that most overlaps with a fixed vector) can be several orders of magnitude faster than a common state-of-the-art convex optimization solver. Our work fosters a practical way forward to asses limitations on QIP imposed by quantum noise. Along the way, we also give a simple but fundamental insight, noisy systems (equivalently noisy Hamiltonians) always give higher ground-state energy than their noiseless counterparts., Comment: 28 pages, 7 figures, comments are welcome

We analyze a bilateral trade model in which the buyer's value for the product and the seller's costs are uncertain, the seller chooses the product price, and the product is recommended by an algorithm based on its value and price. We characterize an algorithm that maximizes the buyer's expected payoff and show that the optimal algorithm underrecommends the product at high prices and overrecommends at low prices. Higher algorithm precision increases the maximal equilibrium price and may increase prices across all of the seller's costs, whereas informing the seller about the buyer's value results in a mean-preserving spread of equilibrium prices and a mean-preserving contraction of the buyer's payoff.

Entanglement and coherence are fundamental properties of quantum systems, promising to power near future quantum technologies, such as quantum computation, quantum communication and quantum metrology. Yet, their quantification, rather than mere detection, generally requires reconstructing the spectrum of quantum states, i.e., experimentally challenging measurement sets that increase exponentially with the system size. Here, we demonstrate quantitative bounds to operationally useful entanglement and coherence that are universally valid, analytically computable, and experimentally friendly. Specifically, our main theoretical results are lower and upper bounds to the coherent information and the relative entropy of coherence in terms of local and global purities of quantum states. To validate our proposal, we experimentally implement two purity detection methods in an optical system: shadow estimation with random measurements and collective measurements on pairs of state copies. The experiment shows that both the coherent information and the relative entropy of coherence of pure and mixed unknown quantum states can be bounded by purity functions. Our research offers an efficient means of verifying large-scale quantum information processing., Comment: 17 pages, 8 figures. arXiv admin note: text overlap with arXiv:1707.09928

A topological space is called a submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.

I study several aspects of the path(st) integral we formulated in previous papers on energetic causal sets with Cortes and others. The focus here is on quantum field theories, including the standard model of particle physics. I show that the the theory can be extended to a quantum field theory, cut off in momentum space. Fields of spin 0, 1/2 and 1 may be naturally included, which allows us to formulate the standard model in this framework. The theory is at first formulated in momentum space. Under certain conditions, spacetime can emerge in a semiclassical limit. The theory comes with a $uv$ cutoff in momentum space, $\mu$, hence that is also a scale for lorentz invariance to break down. Traditionally, m is taken to be. a Planck energy, but we explore as a possibility making m smaller., Comment: 13 pages, 4 Figures

Sharing entanglement across quantum interconnects is fundamental for quantum information processing. We discuss a practical setting where this interconnect, modeled by a quantum channel, is used once with the aim of sharing high fidelity entanglement. For any channel, we provide methods to easily find both this maximum fidelity and optimal inputs that achieve it. Unlike most metrics for sharing entanglement, this maximum fidelity can be shown to be multiplicative. This ensures a complete understanding in the sense that the maximum fidelity and optimal inputs found in our one-shot setting extend even when the channel is used multiple times, possibly with other channels. Optimal inputs need not be fully entangled. We find the minimum entanglement in these optimal inputs can even vary discontinuously with channel noise. Generally, noise parameters are hard to identify and remain unknown for most channels. However, for all qubit channels with qubit environments, we provide a rigorous noise parametrization which we explain in-terms of no-cloning. This noise parametrization and a channel representation we call the standard Kraus decomposition have pleasing properties that make them both useful more generally., Comment: Essentially matches published version

We study spaces that can be mapped onto the Baire space (i.e. the countable power of the countable discrete space) by a continuous quasi-open bijection. We give a characterization of such spaces in terms of Souslin schemes and call these spaces $\pi$-spaces. We show that every space that has a Lusin $\pi$-base is a $\pi$-space and that every second-countable $\pi$-space has a Lusin $\pi$-base. The main result of this paper is a characterization of continuous open images of $\pi$-space.

We show that a class of matrix theories can be understood as an extension of quantum field theory which has non-local interactions. This reformulation is based on the Wigner-Weyl transformation, and the interactions take the form of Moyal product on a doubled geometry. We recover local dynamics on the spacetime as a low-energy limit. This framework opens up the possibility for studying novel high-energy phenomena, including the unification of gauge and geometric symmetries in a gauge theory., Comment: 24 pages

We investigate solutions to the TAP equation, a phenomenological implementation of the Theory of the Adjacent Possible. Several implementations of TAP are studied, with potential applications in a range of topics including economics, social sciences, environmental change, evolutionary biological systems, and the nature of physical laws. The generic behaviour is an extended plateau followed by a sharp explosive divergence. We find accurate analytic approximations for the blow-up time that we validate against numerical simulations, and explore the properties of the equation in the vicinity of equilibrium between innovation and extinction. A particular variant, the two-scale TAP model, replaces the initial plateau with a phase of exponential growth, a widening of the TAP equation phenomenology that may enable it to be applied in a wider range of contexts., Comment: 8 pages, 13 figures, companion to arXiv:2204.09378 and arXiv:2204.09379 (simultaneous release) for development of field of Biocosmology. Develops mathematical formalism used in arXiv:2204.09378; v2: updating title for context

Cosmologists wish to explain how our Universe, in all its complexity, could ever have come about. For that, we assess the number of states in our Universe now. This plays the role of entropy in thermodynamics of the Universe, and reveals the magnitude of the problem of initial conditions to be solved. The usual budgeting accounts for gravity, thermal motions, and finally the vacuum energy whose entropy, given by the Bekenstein bound, dominates the entropy budget today. There is however one number which we have not accounted for: the number of states in our complex biosphere. What is the entropy of life and is it sizeable enough to need to be accounted for at the Big Bang? Building on emerging ideas within theoretical biology, we show that the configuration space of living systems, unlike that of their fundamental physics counterparts, can grow rapidly in response to emerging biological complexity. A model for this expansion is provided through combinatorial innovation by the Theory of the Adjacent Possible (TAP) and its corresponding TAP equation, whose solutions we investigate, confirming the possibility of rapid state-pace growth. While the results of this work remain far from being firmly established, the evidence we provide is many-fold and strong. The implications are far-reaching, and open a variety of lines for future investigation, a new scientific field we term biocosmology. In particular the relationship between the information content in life and the information content in the Universe may need to be rebuilt from scratch., Comment: 35 pages, 1 figure, minor updates to conclusions, reference addition. See also companion paper arXiv:2204.09379

The Universe contains everything that exists, including life. And all that exists, including life, obeys universal physical laws. Do those laws then give adequate foundations for a complete explanation of biological phenomena? We discuss whether and how cosmology and physics must be modified to be able to address certain questions which arise at their intersection with biology. We show that a universe that contains life, in the form it has on Earth, is in a certain sense radically non-ergodic, in that the vast majority of possible organisms will never be realized. We argue from this that complete explanations in cosmology require a mixture of reductionist and functional explanations., Comment: 28 pages. See also companion paper arXiv:2204.09378

A monopoly seller is privately and imperfectly informed about the buyer's value of the product. The seller uses information to price discriminate the buyer. A designer offers a mechanism that provides the seller with additional information based on the seller's report about her type. We establish the impossibility of screening for welfare purposes, i.e., the designer can attain any implementable combination of buyer surplus and seller profit by providing the same signal to all seller types. We use this result to characterize the set of implementable welfare outcomes, study the seller's incentive to acquire third-party data, and demonstrate the trade-off between buyer surplus and efficiency.

We study information design in games in which each player has a continuum of actions. We show that an information structure is designer-optimal whenever the equilibrium play it induces can also be implemented in a principal-agent contracting problem. We use this observation to solve three novel applications. In an investment game, the optimal structure fully informs a single investor while providing no information to others. In an expectation polarization game, the optimal structure fully informs half of the players while providing no information to the other half. In a price competition game, the optimal structure is noise-free Gaussian and recommends prices linearly in the states. Our analysis further informs on the robustness and uniqueness of the optimal information structures.

Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel's capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture," which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In the companion paper [Phys. Rev. Lett. 130, 200801 (2023); arXiv:2202.08377], we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before., Comment: Comments: 42 pages, 4 figures. v2: matches published version. See also the companion paper arXiv:2202.08377

Determining capacities of quantum channels is a fundamental question in quantum information theory. Despite having rigorous coding theorems quantifying the flow of information across quantum channels, their capacities are poorly understood due to super-additivity effects. Studying these phenomena is important for deepening our understanding of quantum information, yet simple and clean examples of super-additive channels are scarce. Here we study a family of channels called platypus channels. Its simplest member, a qutrit channel, is shown to display super-additivity of coherent information when used jointly with a variety of qubit channels. Higher-dimensional family members display super-additivity of quantum capacity together with an erasure channel. Subject to the "spin-alignment conjecture" introduced in the companion paper [IEEE Trans. Inf. Theory 69(6), pp. 3825-3849, 2023; arXiv:2202.08380], our results on super-additivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, super-additivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves super-additivity in all examples. Our results show that super-additivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity., Comment: Comments: 25 pages, 9 figures. v2: matches published version. See also the companion paper arXiv:2202.08380

This paper describes a continuation of the program of causal views, in which the world consists of nothing but a vast number of partial views of its past. Each view is associated to an event, and is a representation of the immediate causal past of that event. These consists mainly of processes that transfer energy, momentum and other charges to it from its past events. There is fundamentally no space or spacetime, just a large number of events, which are the causes of events to come. This is a development of energetic causal set theories, developed with Marina Cortes. Momentum and energy are fundamental, and are conserved under their transformation from present events to future events. As a result Minkowski spacetime emerges, in a way that preserves causal relations. The locality of events as constructed in the emergent spacetime is a consequence of the conservation of energy-momentum fundamentally. In this paper we propose that the views of events can be represented in terms of degrees of freedom on punctured two surfaces-each puncture corresponding to an immediate past event. This opens up to the use of mathematical tools from twistor theory and the nascent area of celestial spheres. We show that This makes possible versions of the theory that are relativistically invariant., Comment: 27 pages, no figures

We propose a new framework for matrix theories that are equivalent to field theories on a toroidal spacetime. The correspondence is accomplished via infinite Toeplitz matrices whose entries match the field degrees of freedom on an energy-momentum lattice, thereby replacing the background geometry with matrix indices. These matrix theories can then be embedded into the purely cubic action of a single matrix and combined into a common universality class. We reconstruct the Standard Model action in this framework and discuss its extensions within the same class., Comment: 34 pages

Hamiltonian learning is an important procedure in quantum system identification, calibration, and successful operation of quantum computers. Through queries to the quantum system, this procedure seeks to obtain the parameters of a given Hamiltonian model and description of noise sources. Standard techniques for Hamiltonian learning require careful design of queries and $O(\epsilon^{-2})$ queries in achieving learning error $\epsilon$ due to the standard quantum limit. With the goal of efficiently and accurately estimating the Hamiltonian parameters within learning error $\epsilon$ through minimal queries, we introduce an active learner that is given an initial set of training examples and the ability to interactively query the quantum system to generate new training data. We formally specify and experimentally assess the performance of this Hamiltonian active learning (HAL) algorithm for learning the six parameters of a two-qubit cross-resonance Hamiltonian on four different superconducting IBM Quantum devices. Compared with standard techniques for the same problem and a specified learning error, HAL achieves up to a $99.8\%$ reduction in queries required, and a $99.1\%$ reduction over the comparable non-adaptive learning algorithm. Moreover, with access to prior information on a subset of Hamiltonian parameters and given the ability to select queries with linearly (or exponentially) longer system interaction times during learning, HAL can exceed the standard quantum limit and achieve Heisenberg (or super-Heisenberg) limited convergence rates during learning., Comment: 53 pages, 21 figures