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Measurement-based uncomputation (MBU) is a technique used to perform probabilistic uncomputation of quantum circuits. We formalize this technique for the case of single-qubit registers, and we show applications to modular arithmetic. First, we present formal statements for several variations of quantum circuits performing non-modular addition: controlled addition, addition by a constant, and controlled addition by a constant. We do the same for subtraction and comparison circuits. This addresses gaps in the current literature, where some of these variants were previously unexplored. Then, we shift our attention to modular arithmetic, where again we present formal statements for modular addition, controlled modular addition, modular addition by a constant, and controlled modular addition by a constant, using different kinds of plain adders and combinations thereof. We introduce and prove a "MBU lemma" in the context of single-qubit registers, which we apply to all aforementioned modular arithmetic circuits. Using MBU, we reduce the Toffoli count and depth by $10\%$ to $15\%$ for modular adders based on the architecture of [VBE96], and by almost $25\%$ for modular adders based on the architecture of [Bea02]. Our results have the potential to improve other circuits for modular arithmetic, such as modular multiplication and modular exponentiation, and can find applications in quantum cryptanalysis., Comment: 45 pages, 25 figures

Amplification (mitigation) is the increase (decrease) in the change of thermodynamic quantities when an initial thermal state is thermalized to a different temperature in the presence of constraints, studied thus far only for permutationally invariant baths. In this manuscript, we generalize amplification and mitigation to accommodate generic strong symmetries of open quantum systems and connect the phenomenon to Landauer's erasure. We exemplify our general theory with a new bath-induced battery charging protocol that overcomes the passivity of KMS-preserving transitions., Comment: Harshank Shrotriya and Midhun Krishna contributed equally, 8 pages, comments welcome

Quantum coherence is an indispensable resource for quantum technological applications. It is known to be distillable from a noisy form using operations that cannot create coherence. However, distillation exacts a hidden coherent measurement cost, whose extent has not previously been estimated. Here we show that this cost (quantified by an equivalent number of Hadamard measurements) is related to what we call the irretrievable coherence: the difference between the coherence of formation and the distillable coherence. We conjecture (and make partial progress towards proving) that when distilling from many copies of a given noisy coherent state, the coherent measurement cost scales extensively in the number of copies, at an asymptotic rate exactly equalling the input's irretrievable coherence. This cost applies to any application whereof coherence distillation is an incidental outcome (e.g. incoherent randomness extraction), but the implications are more dramatic if pure coherence is the only desired outcome: the measurement cost may often be higher than the distilled yield, in which case coherence should rather be prepared afresh than distilled from a noisy input., Comment: 24+5 pages, 1 figure

From correlations in measurement outcomes alone, can two otherwise isolated parties establish whether such correlations are atemporal? That is, can they rule out that they have been given the same system at two different times? Classical statistics says no, yet quantum theory disagrees. Here, we introduce the necessary and sufficient conditions by which such quantum correlations can be identified as atemporal. We demonstrate the asymmetry of atemporality under time reversal, and reveal it to be a measure of spatial quantum correlation distinct from entanglement. Our results indicate that certain quantum correlations possess an intrinsic arrow of time, and enable classification of general quantum correlations across space-time based on their (in)compatibility with various underlying causal structures., Comment: 29 pages

The amount of information that a noisy channel can transmit has been one of the primary subjects of interest in information theory. In this work we consider a practically-motivated family of optical quantum channels that can be implemented without an external energy source. We optimize the Holevo information over procedures that encode information in attenuations and phase-shifts applied by these channels on a resource state of finite energy. It is shown that for any given input state and environment temperature, the maximum Holevo information can be achieved by an encoding procedure that uniformly distributes the channel's phase-shift parameter. Moreover for large families of input states, any maximizing encoding scheme has a finite number of channel attenuation values, simplifying the codewords to a finite number of rings around the origin in the output phase space. The above results and numerical evidence suggests that this property holds for all resource states. Our results are directly applicable to the quantum reading of an optical memory in the presence of environmental thermal noise., Comment: 32 pages, 11 figures; Accepted for publication in Quantum

Quantum information-processing techniques enable work extraction from a system's inherently quantum features, in addition to the classical free energy it contains. Meanwhile, the science of computational mechanics affords tools for the predictive modeling of non-Markovian classical and quantum stochastic processes. We combine tools from these two sciences to develop a technique for predictive work extraction from non-Markovian stochastic processes with quantum outputs. We demonstrate that this technique can extract more work than non-predictive quantum work extraction protocols, on one hand, and predictive work extraction without quantum information processing, on the other. We discover a phase transition in the efficacy of memory for work extraction from quantum processes, which is without classical precedent. Our work opens up the prospect of machines that harness environmental free energy in an essentially quantum, essentially time-varying form.

We show that both the classical as well as the quantum definitions of the Fisher information faithfully identify resourceful quantum states in general quantum resource theories, in the sense that they can always distinguish between states with and without a given resource. This shows that all quantum resources confer an advantage in metrology, and establishes the Fisher information as a universal tool to probe the resourcefulness of quantum states. We provide bounds on the extent of this advantage, as well as a simple criterion to test whether different resources are useful for the estimation of unitarily encoded parameters. Finally, we extend the results to show that the Fisher information is also able to identify the dynamical resourcefulness of quantum operations., Comment: 5+6 pages

Following Frauchiger and Renner's discovery of a conflict between quantum mechanics and certain commonsense reasoning axioms, much work has gone into finding alternative axiomatizations that can avoid the conflict. However, this body of work is largely based on specific interpretations of quantum mechanics, and at times employs specialized formalism that may be inaccessible to mainstream quantum information experts. Taking an interpretation-agnostic approach, we propose a simple operational principle called superpositional solipsism to aid rational agents in making situational inferences. We show that the principle leads to sound inferences in all operationally-relevant instances. Along the way, we discuss certain subtleties about the Frauchiger--Renner result that may have gone hithertofore unnoticed., Comment: 12+2 pages. Feedback appreciated. Expected to be revised with more extensive literature survey

Thermodynamic resources, beyond their well-known usefulness in work extraction and other thermodynamic tasks, are often important also in tasks that are not evidently thermodynamic. Here we develop a framework for identifying such resources in diverse applications of bosonic continuous-variable systems. Introducing the class of bosonic linear thermal operations to model operationally-feasible processes, we apply this model to identify uniquely quantum properties of bosonic states that refine classical notions of thermodynamic resourcefulness. Among these are (1) a suite of temperature-like quantities generalizing the equilibrium temperature to quantum, non-equilibrium scenarios; (2) signal-to-noise ratios quantifying a system's capacity to carry information in phase-space displacement; and (3) well-established non-classicality measures quantifying the resolution in sensing and parameter estimation tasks., Comment: 9 pages (6 figures), 4-page supplemental material

We characterize the operational capabilities of quantum channels which can neither create nor detect quantum coherence vis-\`a-vis efficiently manipulating coherence as a resource. We study the class of dephasing-covariant operations (DIO), unable to detect the coherence of any input state, as well as introduce an operationally-motivated class of channels $\rho$-DIO which is tailored to a specific input state. We first show that pure-state transformations under DIO are completely governed by majorization, establishing necessary and sufficient conditions for such transformations and adding to the list of operational paradigms where majorization plays a central role. We then show that $\rho$-DIO are strictly more powerful: although they cannot detect the coherence of the input state $\rho$, the operations $\rho$-DIO can distill more coherence than DIO. However, the advantage disappears in the task of coherence dilution as well as generally in the asymptotic limit, where both sets of operations achieve the same rates in all transformations., Comment: 6+4 pages; v2: minor additions and clarifications

Quantum physics, which describes the strange behavior of light and matter at the smallest scales, is one of the most successful descriptions of reality, yet it is notoriously inaccessible. Here we provide an approachable explanation of quantum physics using simple thought experiments. We derive all relevant quantum predictions using minimal mathematics, without introducing the advanced calculations that are typically used to describe quantum physics. We focus on the two key surprises of quantum physics, namely wave-particle duality, a term that was introduced to capture the fact that single quantum particles in some respects behave like waves and in other respects like particles, and entanglement, which applies to two or more quantum particles and brings out the inherent contradiction between quantum physics and seemingly obvious assumptions regarding the nature of reality. Following arguments originally made by John Bell and Lucien Hardy, we show that the so-called local hidden variables are inadequate at explaining the behavior of entangled quantum particles. This means that one either has to give up on hidden variables, i.e., the idea that the outcomes of measurements on quantum particles are determined before an experiment is actually carried out, or one has to relinquish the principle of locality, which requires that no causal influences should be faster than the speed of light and is a cornerstone of Einstein's theory of relativity. Finally, we describe how these remarkable predictions of quantum physics have been confirmed in experiments. We have successfully used the present approach in a course that is open to all undergraduate students at the University of Calgary, without any prerequisites in mathematics or physics., Comment: 37 two-column pages, 30000 words, 30 figures and sub-figures. Comments are welcome. Typos corrected

The information-carrying capacity of a memory is known to be a thermodynamic resource facilitating the conversion of heat to work. Szilard's engine explicates this connection through a toy example involving an energy-degenerate two-state memory. We devise a formalism to quantify the thermodynamic value of memory in general quantum systems with nontrivial energy landscapes. Calling this the thermal information capacity, we show that it converges to the non-equilibrium Helmholtz free energy in the thermodynamic limit. We compute the capacity exactly for a general two-state (qubit) memory away from the thermodynamic limit, and find it to be distinct from known free energies. We outline an explicit memory--bath coupling that can approximate the optimal qubit thermal information capacity arbitrarily well., Comment: 6 main + 7 appendix pages; 5 main + 2 appendix figures

Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P-representation -- a state lacking a positive P-function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed-forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multi-mode non-Gaussian states. Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single mode Gaussian states., Comment: 21 pages, 7 figures; comments welcome; close to published version

In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition., Comment: 12 pages; 9 figures; Comments welcome

The "thermal operations" framework developed in past works is used to model the evolution of microscopic quantum systems in contact with thermal baths. Here we extend this model to bipartite devices with one part acting as a control external to the system--bath setup. We define the operations of such hybrid devices as conditioned thermal operations. We examine the resource under these operations, which we call conditional athermality. In the quasiclassical limit, we quantify this resource and find the conditions for its conversion between different forms., Comment: Published version incorporating referee feedback. 6.5 pages main (2 figures) + 6.5 pages supplement (2 figures)

We develop a general operational framework that formalizes the concept of conditional uncertainty in a measure-independent fashion. Our formalism is built upon a mathematical relation which we call conditional majorization. We define conditional majorization and, for the case of classical memory, we provide its thorough characterization in terms of monotones, i.e., functions that preserve the partial order under conditional majorization. We demonstrate the application of this framework by deriving two types of memory-assisted uncertainty relations: (1) a monotone-based conditional uncertainty relation, (2) a universal measure-independent conditional uncertainty relation, both of which set a lower bound on the minimal uncertainty that Bob has about Alice's pair of incompatible measurements, conditioned on arbitrary measurement that Bob makes on his own system. We next compare the obtained relations with their existing entropic counterparts and find that they are at least independent., Comment: 5 pages main + 10 pages appendix. Changes since v1: new results demonstrating independence of our results from existing conditional uncertainty relations; significant revision in notation and presentation; expanded bibliography

Historically, the element of uncertainty in quantum mechanics has been expressed through mathematical identities called uncertainty relations, a great many of which continue to be discovered. These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this paper we use operational information-theoretic principles to identify the common essence of all such measures, thereby defining measure-independent notions of uncertainty and joint uncertainty. We find that most existing entropic uncertainty relations use measures of joint uncertainty that yield themselves to a small class of operational interpretations. Our notion relaxes this restriction, revealing previously unexplored joint uncertainty measures. To illustrate the utility of our formalism, we derive an uncertainty relation based on one such new measure. We also use our formalism to gain insight into the conditions under which measure-independent uncertainty relations can be found., Comment: Published version with major revisions; 8 pages main + bibliography + 1 page appendix; 3 figures

Thermal operations are an operational model of non-equilibrium quantum thermodynamics. In the absence of coherence between energy levels, exact state transition conditions under thermal operations are known in terms of a mathematical relation called thermo-majorization. But incorporating coherence has turned out to be challenging, even under the relatively tractable model wherein all Gibbs state-preserving quantum channels are included. Here we find a mathematical generalization of thermal operations at low temperatures, "cooling maps", for which we derive the necessary and sufficient state transition condition. Cooling maps that saturate recently-discovered bounds on coherence transfer are realizable as thermal operations, motivating us to conjecture that all cooling maps are thermal operations. Cooling maps, though a less conservative generalization to thermal operations, are more tractable than Gibbs-preserving operations, suggesting that cooling map-like models at general temperatures could be of use in gaining insight about thermal operations., Comment: Main matter (7 pages, 2 figures) + Supplemental matter (19 pages, 2 figures). Published version

We study the quantumness of correlations for ensembles of bi- and multi-partite systems and relate it to the task of quantum data hiding. Quantumness is here intended in the sense of minimum average disturbance under local measurements. We consider a very general framework, but focus on local complete von Neumann measurements as cause of the disturbance, and, later on, on the trace-distance as quantifier of the disturbance. We discuss connections with entanglement and previously defined notions of quantumness of correlations. We prove that a large class of quantifiers of the quantumness of correlations are entanglement monotones for pure bipartite states. In particular, we define an entanglement of disturbance for pure states, for which we give an analytical expression. Such a measure coincides with negativity and concurrence for the case of two qubits. We compute general bounds on disturbance for both single states and ensembles, and consider several examples, including the uniform Haar ensemble of pure states, and pairs of qubit states. Finally, we show that the notion of ensemble quantumness of correlations is most relevant in quantum data hiding. Indeed, while it is known that entanglement is not necessary for a good quantum data hiding scheme, we prove that ensemble quantumness of correlations is., Comment: 35 pages, 2 figures, close to published version

Practical implementations of quantum information-theoretic applications rely on phase coherences between systems. These coherences are disturbed by misalignment between phase references. Quantum states carrying phase-asymmetry act as resources to counteract this misalignment. In this paper, we construct protocols for interconverting these resources by first estimating the misalignment. Our method achieves sublinear, but otherwise arbitrarily high, asymptotic rate of conversion of pure resource states to mixed ones. No other method is known for asymptotic pure-to-mixed state conversion. Our method also has the advantage of achieving phase reference alignment in addition to resource conversion., Comment: 8 pages; published version

We develop a framework that allows a description of measurements in Hilbert spaces that are smaller than their natural representation. This description, which we call a "squashing model", consists of a squashing map that maps the input states of the measurement from the original Hilbert space to the smaller one, followed by a targeted prescribed measurement on the smaller Hilbert space. This framework has applications in quantum key distribution, but also in other cryptographic tasks, as it greatly simplifies the theoretical analysis under adversarial conditions., Comment: 24 pages main text, 12 figures, 4 pages appendix, 1 figure

We review recent work on the foundations of thermodynamics in the light of quantum information theory. We adopt a resource-theoretic perspective, wherein thermodynamics is formulated as a theory of what agents can achieve under a particular restriction, namely, that the only state preparations and transformations that they can implement for free are those that are thermal at some fixed temperature. States that are out of thermal equilibrium are the resources. We consider the special case of this theory wherein all systems have trivial Hamiltonians (that is, all of their energy levels are degenerate). In this case, the only free operations are those that add noise to the system (or implement a reversible evolution) and the only nonequilibrium states are states of informational nonequilibrium, that is, states that deviate from the maximally mixed state. The degree of this deviation we call the state's nonuniformity; it is the resource of interest here, the fuel that is consumed, for instance, in an erasure operation. We consider the different types of state conversion: exact and approximate, single-shot and asymptotic, catalytic and noncatalytic. In each case, we present the necessary and sufficient conditions for the conversion to be possible for any pair of states, emphasizing a geometrical representation of the conditions in terms of Lorenz curves. We also review the problem of quantifying the nonuniformity of a state, in particular through the use of generalized entropies. Quantum state conversion problems in this resource theory can be shown to be always reducible to their classical counterparts, so that there are no inherently quantum-mechanical features arising in such problems. This body of work also demonstrates that the standard formulation of the second law of thermodynamics is inadequate as a criterion for deciding whether or not a given state transition is possible., Comment: 51 pages, 9 figures, Revised Version

In optical implementations of the phase-encoded BB84 protocol, the bit information is usually encoded in the phase of two consecutive photon pulses generated in a Mach-Zehnder interferometer. In the actual experimental realization, the loss in the arms of the Mach-Zehnder interferometer is not balanced, for example because only one arm contains a lossy phase modulator. Therefore, the amplitudes of the pulses is not balanced, and the structure of the signals and measurements no longer corresponds to the (balanced) ideal BB84 protocol. Hence, the BB84 security analysis no longer applies in this scenario. We provide a security proof of the unbalanced phase-encoded BB84. The resulting key rate turns out to be lower than the key rate of the ideal BB84 protocol. Therefore, in order to guarantee security, the loss due to the phase modulator cannot be ignored.