Schedule for: 19w5120 - Algebraic and Statistical ways into Quantum Resource Theories

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

To describe certain facets of non-classicality, it is necessary to quantify properties of operations instead of states. This is the case if one wants to quantify how well an operation detects non-classicality, which is a necessary prerequisite for its use in quantum technologies. To do so rigorously, we build resource theories on the level of operations, exploiting the concept of resource destroying maps. We discuss the two basic ingredients of these resource theories, the free operations and the free super-operations, which are sequential and parallel concatenations with free operations. This leads to defining properties of functionals that are well suited to quantify the resources of operations. We introduce these concepts at the example of coherence. In particular, we present two measures quantifying the ability of an operation to detect, i.e. to use, coherence, one of them with an operational interpretation, and provide methods to evaluate them. We also present an application of these concepts to an experiment in quantum optics.

We discuss the standard and generalized robustness of $k$-coherence, which are shown to be equal when restricted to pure states. The resulting expression can be used to analyze the Schmidt rank $k$-entanglement for pure states.

The range of a quantum measurement is the set of output probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of output distributions, infers the quantum measurement which is, i) consistent with the data, in the sense that its range contains all the distributions observed, and, ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of output distributions. We show that data-driven inference is able to return a measurement up to symmetries of the state space---as it is solely based on observed distributions---and that such limit accuracy is achieved for any data set if and only if the inference adopts a (hyper)-spherical state space (for example, the classical or the quantum bit). When using data-driven inference as a protocol to reconstruct an unknown quantum measurement, we show that a crucial property to consider is that of observational completeness, which is defined, in analogy to the property of informational completeness in quantum tomography, as the property of any set of states that, when fed into any given measurement, produces a set of output distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)-spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The most general (probabilistic) transformation of a quantum state is described by a quantum oper- ation. Quantum operations can be axiomatically defined as the most general map which are compatible with the probabilistic structure of the theory, and produce a legitimate output when applied locally on one side of a bipartite input. These admissibility requirements characterise quantum operations as com- pletely positive trace non increasing linear maps. What happens if we now consider maps from quantum operations to quantum operations? Can we give an axiomatic characterisation of these objects according to some generalised notion of admissibility? What happens if we recursively iterate the construction and we define a full hierarchy of higher order maps? Some special cases of higher order maps have been already studied in the literature. Causally ordered Quantum Networks, which encompass all conceivable quantum protocols, form a sub-hierarchy of maps which are endowed with a well ordered causal structure and they can be realised as quantum circuits. However, more general higher order maps may exhibit an indefinite causal structure which prevents a physical implementation as a quantum circuit. Non circuital higher order maps allow to accomplishing certain tasks that cannot be achieved by circuital maps, like the violation of causal inequalities, and can outperform circuital maps in certain quantum information pro- cessing tasks. The experimental realisation of non-circuital higher order maps has also been considered. Notwithstanding many results on the subject, a general mathematical framework is still missing. The aim of this contribution is to fill this critical gap by providing an axiomatic framework for higher order quantum theory. Higher order quantum theory is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. We will see that a complete mathematical characterisation of admissible maps is possible and that the set of admissible maps of a given type is in correspondence with a convex subset of the cone of positive operators. This result encompasses the analysis existing in the literature and gives them an axiomatic operational foundation. The present axioms for higher order quantum theory have an operational nature and do not refer to the specific mathematical structure of quantum theory. Therefore, with due care, our framework can be applied to general operational probabilistic theories.

"Although it is believed that quantum computation cannot be classically efficiently simulated in general, there exist certain restricted classes of quantum circuits for which classical simulation is indeed possible. The most prominent example are the Clifford circuits. Here, we consider another such class, the so-called matchgate circuits (MGCs) [1,2]. MGCs can be classically efficiently simulated and moreover, performed as a compressed quantum computation, i.e., the computation can be performed on an quantum computer using exponentially fewer qubits and only polynomial overhead in runtime [3]. We elaborate on and extend recent results [4] on classical simulability of MGCs. To this end, we discuss the notion of magic states in this context. [1] L. Valiant, SIAM J. Computing 31, 1229 (2002), B. Terhal and D. DiVincenzo, Phys. Rev. A 65, 032325 (2002) [2] R. Jozsa and A. Miyake, Proc. R. Soc. A 464, 3089 (2008) [3] R. Jozsa, B. Kraus, A. Miyake, J. Watrous, Proc. R. Soc. A 466, 809 (2010) [4] D. J. Brod, Phys. Rev. A 93, 062332 (2016)"

In this work, we develop resource-theoretic approaches to study the non-stabilizer resources in fault-tolerant quantum computation. First, we introduce a family of magic measures to quantify the amount of magic in a quantum state, several of which can be efficiently computed via convex optimization. Second, we show that two classes of states with maximal mana, a previously established magic measure, cannot be interconverted asymptotically at a rate equal to one. This reveals the fundamental difference between the resource theory of magic states and other resource theories such as entanglement and coherence. Third, we establish efficiently computable benchmarks for the rate and efficiency of magic state distillation via our magic measures. Fourth, we introduce efficiently computable magic measures to quantify the magic of quantum channels, which can be applied to evaluate the magic generating capability and gate synthesis. Finally, we propose a classical algorithm for simulating noisy quantum circuits whose sample complexity is quantified by our channel measure. We further show by concrete examples that our algorithm can outperform previous approaches in simulating noisy quantum circuits.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

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.

Entanglement theory is usually formulated as a quantum resource theory in which the free operations are local operations and classical communication (LOCC). This defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almost all pure entangled multipartite states are incomparable (i.e. LOCC transformations among them are almost never possible). In order to cope with this problem I will consider alternative resource theories in which I relax the class of LOCC to operations that do not create entanglement. In more detail, I will consider two possible theories depending on whether resources correspond to multipartite entangled or genuinely multipartite entangled (GME) states and I will show that they are both non-trivial: they induce a meaningful partial order since every pure state is deterministically transformable to more weakly entangled pure states. Moreover, I will also show that no inequivalent forms of entanglement exist in these theories (i.e. all resource states are interconvertible with non-zero probability). Last but not least, I will prove that the resource theory of GME that I formulate here has a unique maximally entangled state, the generalized GHZ state, which can be transformed deterministically to any other state by the allowed free operations.

We may regard quantum entanglement as individual subsystems having mixture not found in the whole, and coherence as individual bases having entropy beyond that of the complete quantum system. In each case a particular von Neumann subalgebra of the system's observables appears more mixed than the full system. Both the partial trace and the complete dephasing are conditional expectations onto different subalgebras, which are invariant under subgroups of the full system's unitary group. Hence we may view these as quantum asymmetries in noncommutative observable algebras. We examine monotones and operation sets that connect these resources from an algebraic perspective. Going beyond the subalgebra case, we consider replacing conditional expectations by general quantum channels and how one might build noise into a resource theory. In particular, we derive generalizations of strong subadditivity beyond subsystems, subsuming some uncertainty relations and bounds on coherence and asymmetry. Key to this analysis is complementarity of noncommutative observables, which connects the quantumness of a resource to the impossibility of a copy existing in any environment.

Resource theorists have racked up scores of theorems over the past decade. But can these abstract theories inform science beyond quantum information and quantum thermodynamics? Can resource theories answer other scientists’ questions about specific systems in the real physical world? We argue affirmatively, illustrating with photoisomers, or molecular switches. Photoisomers surface across nature and technologies, from our eyes to solar-fuel cells. How effectively can these switches switch? This question defies standard tools, because photoisomers are small, quantum and far from equilibrium. We answer by modeling a photoisomer within a thermodynamic resource theory. Using thermomajorization, we upper-bound the switching probability. Then, we compare the bound with detailed balance and Lindbladian evolution. Thermomajorization constrains the yield tightly if a laser barely excites the molecule, such that thermal fluctuations drive switching. We also quantify the coherence in the molecule’s postswitching electronic state. Electronic coherence cannot boost the yield in the absence of extra resources, we argue, because modes of coherence transform independently via thermal operations. This work illustrates how thermodynamic resource theories can illuminate nature, experiments, and synthetics. This work appears in https://arxiv.org/abs/1811.06551 and was undertaken with David Limmer.

We identify and explore the intriguing property of resource resonance arising within resource theories of entanglement, coherence and thermodynamics. While the theories considered are reversible asymptotically, the same is generally not true in realistic scenarios where the available resources are bounded. The finite-size effects responsible for this irreversibility could potentially prohibit small quantum information processors or thermal machines from achieving their full potential. Nevertheless, we show here that by carefully engineering the resource interconversion process any such losses can be greatly suppressed. Our results are predicted by higher order expansions of the trade-off between the rate of resource interconversion and the achieved fidelity, and are verified by exact numerical optimizations of the appropriate underlying approximate majorization conditions.

One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exact quantifier for the maximal advantage enabled by the given resource state in a class of subchannel discrimination problems, providing a universal operational interpretation to this fundamental resource quantifier. We also consider a wider range of subchannel discrimination tasks and show that the generalized robustness still serves as the operational advantage quantifier for several well-known theories such as entanglement, coherence, and magic.

How fast do infinite-dimensional quantum systems evolve? Do entropies of infinite-dimensional quantum systems satisfy continuity bounds? If so, what are the corresponding convergence rates? These are the questions that will addressed in this talk. By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite-dimensions, which are stronger than previously known bounds. Our results have interesting applications including quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. We also obtain explicit log-Lipschitz continuity bounds for entropies of infinite-dimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl’s law. This is joint work with Simon Becker.

We address the question of efficient implementation of quantum protocols, with small communication and entanglement, and short depth circuit for encoding or decoding. We introduce two new methods to achieve this, the first method involving two new versions of the convex-split lemma that use much smaller amount of additional resource (in comparison to previous version) and the second method being inspired by the technique of classical correlated sampling in computer science literature. These lead to a series of new consequences, as follows. First, we consider the task of quantum decoupling, where the aim is to apply an operation on a n-qubit register so as to make it independent of an inaccessible quantum system. Many previous works achieve decoupling with the aid of a random unitary. It is known that random unitaries can be replaced by random circuits of size O(nlog n) and depth poly(log n), or unitary 2-designs based on Clifford circuits of similar size and depth. We show that given any choice of basis such as the computational basis, decoupling can be achieved by a unitary that takes basis vectors to basis vectors. Thus, the circuit acts in a `classical' manner and additionally uses O(n) catalytic qubits in maximally mixed quantum state. Our unitary performs addition and multiplication modulo a prime and hence achieves a circuit size of O(n\log n) and logarithmic depth. This shows that the circuit complexity of integer multiplication (modulo a prime) is lower bounded by the optimal circuit complexity of decoupling. Next, we construct a new one-shot entanglement-assisted protocol for quantum channel coding that achieves near-optimal communication through a given channel. Furthermore, the number of qubits of pre-shared entanglement is exponentially smaller than that used in the previous protocol that was near-optimal in communication. We also achieve similar results for the one-shot quantum state redistribution. Joint work with Rahul Jain. https://arxiv.org/abs/1809.07056

TBA

Given two sets of quantum states, is it possible to transform one to the other by a quantum channel? This question has been studied by many authors and recently fully settled by a set of entropic inequalities. We will consider an extension of this problem that can be formulated as follows: given two bipartite quantum channels, how precisely can one be approximated by applying a suitable supermap to a part of the other? Taking some inspiration from the classical randomization criterion by Le Cam, we study this question it the case when the precision of the approximation is measured by distance in the diamond norm.

Partially based on arXiv:1810.12197 (joint work with Francesco Borderi, Omar Fawzi, Volkher Scholz)

We present a general framework of a resource theory based on the assumption of a) convexity and b) that the overlap of free states with maximally resourceful state is bounded. Using this structure, we derive bounds on the one-shot distillation rate for such a resource theory, thereby reproducing known bounds in coherence and entanglement. We use the free robustness and introduce a function $G_min$ related to overlap between states to express our bounds. To deal with resource theories where the free robustness in not finite we introduce the notion of imperfect free operations which we call $\epsilon$-resource generating operations and generalize the free robustness to$\epsilon$-free robustness. We construct an $\epsilon$-resource generating map that achieves pure state transformations and derive the conditions for such a transformation in terms of the $\epsilon$-free robustness.

Embezzlement of entanglement was studied by van Dam and Hayden who showed that it was imposible to catylyically produce entanglement in the tensor product model, but that one could "nearly" acheive this goal. However, in the commuting operator model one can accomplish this task. We show that there is a parallel between this theory and Tsirelson's problems about the different models for conditional probability densities. In analogy there is a theory of "unitary correlations" that yields a set of matrices that allows one to study quantum input--classical output games.

We systematically develop the resource theory of asymmetric distinguishability, as initiated roughly a decade ago [K. Matsumoto, arXiv:1006.0302 (2010)]. The key constituents of this resource theory are quantum boxes, consisting of a pair of quantum states, which can be manipulated for free by means of an arbitrary quantum channel. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. The distillable distinguishability is the optimal rate at which a quantum box consisting of independent and identically distributed (i.i.d.) states can be converted to bits of asymmetric distinguishability, and the distinguishability cost is the optimal rate for the reverse transformation. Both of these quantities are equal to the quantum relative entropy. The exact one-shot distillable distinguishability is equal to the min-relative entropy, and the exact one-shot distinguishability cost is equal to the max-relative entropy. Generalizing these results, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. As a notable application of the former results, we prove that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum states to another pair of i.i.d. quantum states is fully characterized by the ratio of their quantum relative entropies. We also generalize the theory to quantum channels and quantum strategies (combs), with one key result being the solution of the Stein's lemma for quantum channels in the sequential setting in terms of the amortized channel relative entropy. This is joint work with Xin Wang (Univ. Maryland & Baidu, Inc.) and is available at https://arxiv.org/abs/1905.11629 and https://arxiv.org/abs/1907.06306

Bell inequalities were derived as conditions satisfied by local hidden-variable models. However, in recent years they have acquired a new status as device-independent certificates of relevant quantum properties. We provide several constructions in which Bell inequalities are constructed from the quantum states and/or measurements to certify, rather than from considerations concerning the geometry of classical correlations. In particular, we present Bell inequalities maximally violated by maximally entangled states or arbitrary dimension and also graph states.

Quantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number N of ports, the error of the optimal protocol is proportional to the inverse square of N . We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on connecting recently-derived representation-theoretic formulas to random matrix theory. Along the way, we refine a convergence result for the fluctuations of the Schur-Weyl distribution by Johansson, which might be of independent interest. arXiv:1809.10751, joint work with M. Christandl, C. Majenz, G. Smith, F. Speelman, M. Walter

In this talk I discuss coherence distillation under Time translation Invariant operations. I show that although for a generic mixed state the distillation rate is zero, it is still possible to distill a sub-linear number of a pure coherent state, with fidelity approaching one, provided that we can consume asymptotically many copies of the mixed state. Furthermore, for a generic mixed input state, there is a tradeoff between the maximum achievable yield and the fidelity with pure coherent states. Interestingly, it turns out that Petz-Renyi relative entropy for alpha=2 gives a tight bound on the maximum achievable fidelity. Furthermore, coherence distillation provides an operational explanation for the violation of the monotonicity of Petz-Renyi relative entropy for the parameter range alpha>2. Finally, I talk about the limitations of measure-and-prepare (entanglement-breaking) processes for coherence distillation. If time allows, I also briefly discuss a new no-broadcasting theorem for coherence and asymmetry. The no-go theorem states that if two initially uncorrelated systems interact by symmetric dynamics and asymmetry is created at one subsystem, then the asymmetry of the other subsystem must be reduced. I also present a quantitative relation describing the tradeoff between the subsystems.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.