Covariant influences for finite discrete dynamical systems.

We develop a rigorous theory of external influences on finite discrete dynamical systems, going beyond the perturbation paradigm, in that the external influence need not be a small contribution. Indeed, the covariance condition can be stated as follows: If we evolve the dynamical system for n time steps and then disturb it, then it is the same as first disturbing the system with the same influence and then letting the system evolve for n time steps. Applying the powerful machinery of resource theories, we develop a theory of covariant influences both when there is a purely deterministic evolution and when randomness is involved. Subsequently, we provide necessary and sufficient conditions for the transition between states under deterministic covariant influences and necessary conditions in the presence of stochastic covariant influences, predicting which transitions between states are forbidden. Our approach, for the first time, employs the framework of resource theories, borrowed from quantum information theory, to the study of finite discrete dynamical systems. The laws we articulate unify the behavior of different types of finite discrete dynamical systems, and their mathematical flavor makes them rigorous and checkable.

What is entropy? A perspective from games of chance.

The crucial role of channels in physics and information theory motivates the task of characterizing the entropy, or uncertainty, of a channel. Games of chance become a natural candidate for this task, as a system's performance in a gambling game depends solely on the uncertainty of its output. In this work, we construct families of games that induce preorders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all preorders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical channel entropy.

What Are the Minimal Conditions Required to Define a Symmetric Informationally Complete Generalized Measurement?

Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.

Dynamical Entanglement.

Unlike the entanglement of quantum states, very little is known about the entanglement of bipartite channels, called dynamical entanglement. Here we work with the partial transpose of a superchannel, and use it to define computable measures of dynamical entanglement, such as the negativity. We show that a version of it, the max-logarithmic negativity, represents the exact asymptotic dynamical entanglement cost. We discover a family of dynamical entanglement measures that provide necessary and sufficient conditions for bipartite channel simulation under local operations and classical communication and under operations with positive partial transpose.

Necessary and sufficient conditions on measurements of quantum channels.

Quantum supermaps are a higher-order genera- lization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive and trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.

How to Quantify a Dynamical Quantum Resource.

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. Then, we 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 max relative entropy distance to the set of quantum channels, and prove a variety of properties of all six generalizations of the relative entropy of a resource.

Quantifying Memory Capacity as a Quantum Thermodynamic Resource.

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 nonequilibrium 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.

Quantum majorization and a complete set of entropic conditions for quantum thermodynamics.

What does it mean for one quantum process to be more disordered than another? Interestingly, this apparently abstract question arises naturally in a wide range of areas such as information theory, thermodynamics, quantum reference frames, and the resource theory of asymmetry. Here we use a quantum-mechanical generalization of majorization to develop a framework for answering this question, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. We also investigate some of the applications of this framework, and remarkably find that, in the context of quantum thermodynamics it provides the first complete set of necessary and sufficient conditions for arbitrary quantum state transformations under thermodynamic processes, which rigorously accounts for quantum-mechanical properties, such as coherence. Our framework of generalized thermal processes extends thermal operations, and is based on natural physical principles, namely, energy conservation, the existence of equilibrium states, and the requirement that quantum coherence be accounted for thermodynamically.

Critical Examination of Incoherent Operations and a Physically Consistent Resource Theory of Quantum Coherence.

Considerable work has recently been directed toward developing resource theories of quantum coherence. In this Letter, we establish a criterion of physical consistency for any resource theory. This criterion requires that all free operations in a given resource theory be implementable by a unitary evolution and projective measurement that are both free operations in an extended resource theory. We show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations as a natural generalization of the physically consistent operations. Necessary and sufficient conditions are derived for the convertibility of qubit states using dephasing-covariant operations, and we show that these conditions also hold for other well-known classes of incoherent operations.

Erratum: Reversible Framework for Quantum Resource Theories [Phys. Rev. Lett. 115, 070503 (2015)].

This corrects the article DOI: 10.1103/PhysRevLett.115.070503.

Reversible Framework for Quantum Resource Theories.

In recent years it has been recognized that properties of physical systems such as entanglement, athermality, and asymmetry, can be viewed as resources for important tasks in quantum information, thermodynamics, and other areas of physics. This recognition was followed by the development of specific quantum resource theories (QRTs), such as entanglement theory, determining how quantum states that cannot be prepared under certain restrictions may be manipulated and used to circumvent the restrictions. Here we discuss the general structure of QRTs, and show that under a few assumptions (such as convexity of the set of free states), a QRT is asymptotically reversible if its set of allowed operations is maximal, that is, if the allowed operations are the set of all operations that do not generate (asymptotically) a resource. In this case, the asymptotic conversion rate is given in terms of the regularized relative entropy of a resource which is the unique measure or quantifier of the resource in the asymptotic limit of many copies of the state. This measure also equals the smoothed version of the logarithmic robustness of the resource.

Low-temperature thermodynamics with quantum coherence.

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.

Classification of multipartite entanglement of all finite dimensionality.

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.

Universal uncertainty relations.

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring noncommuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabeling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in M. H. Partovi, Phys. Rev. A 84, 052117 (2011). Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.

Evolution and symmetry of multipartite entanglement.

We discover a simple factorization law describing how multipartite entanglement of a composite quantum system evolves when one of the subsystems undergoes an arbitrary physical process. This multipartite entanglement decay is determined uniquely by a single factor we call the entanglement resilience factor. Since the entanglement resilience factor is a function of the quantum channel alone, we find that multipartite entanglement evolves in exactly the same way as bipartite (two qudits) entanglement. For the two qubits case, our factorization law reduces to the main result of [T. Konrad, Nature Phys. 4, 99 (2008)10.1038/nphys885]. In addition, for a permutation P, we provide an operational definition of P asymmetry of entanglement, and find the conditions when a permuted version of a state can be achieved by local means.

Remote preparation and distribution of bipartite entangled states.

We prove a powerful theorem for tripartite remote entanglement distribution protocols that establishes an upper bound on the amount of entanglement of formation that can be created between two single-qubit nodes of a quantum network. Our theorem also provides an operational interpretation of concurrence as a type of entanglement capacity.