Asymptotic tensor rank, originally developed to characterize the complexity of matrix multiplication, is a parameter that plays a fundamental role in problems in mathematics, computer science and quantum information. This parameter is notoriously difficult to determine; indeed, determining its value for the 2× 2 matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. Strassen's asymptotic rank conjecture, on the other hand, makes the bold statement that asymptotic tensor rank equals the largest dimension of the tensor and is thus as easy to compute as matrix rank. Recent works have proved strong consequences of Strassen's asymptotic rank conjecture in computational complexity theory. Despite tremendous interest, much is still unknown about the structural and computational properties of asymptotic rank; for instance whether it is computable. We prove that asymptotic tensor rank is "computable from above", that is, for any real number r there is an (efficient) algorithm that determines, given a tensor T, if the asymptotic tensor rank of T is at most r. The algorithm has a simple structure; it consists of evaluating a finite list of polynomials on the tensor. Indeed, we prove that the sublevel sets of asymptotic rank are Zariski-closed (just like matrix rank). While we do not exhibit these polynomials explicitly, their mere existence has strong implications on the structure of asymptotic rank. As one such implication, we find that the values that asymptotic tensor rank takes, on all tensors, is a well-ordered set. In other words, any non-increasing sequence of asymptotic ranks stabilizes ("discreteness from above"). In particular, for the matrix multiplication exponent (which is the base-2 logarithm of an asymptotic rank) there is no sequence of exponents of bilinear maps that approximates it arbitrarily closely from above without being eventually constant. In other words, any such upper bound on the matrix multiplication exponent that is close enough, will "snap"to it. Previously such discreteness results were only known for finite fields or for other tensor parameters (e.g., asymptotic slice rank). We obtain them for infinite fields like the complex numbers. We prove our result more generally for a large class of functions on tensors, and in particular obtain similar properties for all functions in Strassen's asymptotic spectrum of tensors. We prove a variety of related structural results on the way. For instance, we prove that for any converging sequence of asymptotic ranks, the limit is also an asymptotic rank for some tensor. We leave open whether asymptotic rank is also discrete from below (which would be implied by Strassen's asymptotic rank conjecture).

This paper explores the trade-off relation between the rate and the strong converse exponent for asymptotic LOCC transformations between pure multipartite states. Any single-copy probabilistic transformation between a pair of states implies that an asymptotic transformation at rate 1 is possible with an exponentially decreasing success probability. However, it is possible that an asymptotic transformation is feasible with nonzero probability, but there is no transformation between any finite number of copies with the same rate, even probabilistically. In such cases it is not known if the optimal success probability decreases exponentially or faster. A fundamental tool for showing the feasibility of an asymptotic transformation is degeneration. Any degeneration gives rise to a sequence of stochastic LOCC transformations from copies of the initial state plus a sublinear number of GHZ states to the same number of copies of the target state. These protocols involve parameters that can be freely chosen, but the choice affects the success probability. In this paper, we characterize an asymptotically optimal choice of the parameters and derive a single-letter expression for the error exponent of the resulting protocol. In particular, this implies an exponential lower bound on the success probability when the stochastic transformation arises from a degeneration.

The trade-off relation between the rate and the strong converse exponent for probabilistic asymptotic entanglement transformations between pure multipartite states can in principle be characterised in terms of a class of entanglement measures determined implicitly by a set of strong axioms. A nontrivial family of such functionals has recently been constructed, but their previously known characterisations have so far only made it possible to evaluate them in very simple cases. In this paper we derive a new regularised formula for these functionals in terms of a subadditive upper bound, complementing the previously known superadditive lower bound. The upper and lower bounds evaluated on tensor powers differ by a logarithmically bounded term, which provides a bound on the convergence rate. In addition, we find that on states satisfying a certain sparsity constraint, the upper bound is equal to the value of the corresponding additive entanglement measure, therefore the regularisation is not needed for such states, and the evaluation is possible via a single-letter formula. Our results provide explicit bounds on the success probability of transformations by local operations and classical communication and, due to the additivity of the entanglement measures, also on the strong converse exponent for asymptotic transformations.

We study a generalization of relative submajorization that compares pairs of positive operators on representation spaces of some fixed group. A pair equivariantly relatively submajorizes another if there is an equivariant subnormalized channel that takes the components of the first pair to a pair satisfying similar positivity constraints as in the definition of relative submajorization. In the context of the resource theory approach to thermodynamics, this generalization allows one to study transformations by Gibbs-preserving maps that are in addition time-translation symmetric. We find a sufficient condition for the existence of catalytic transformations and a characterization of an asymptotic relaxation of the relation. For classical and certain quantum pairs the characterization is in terms of explicit monotone quantities related to the sandwiched quantum Rényi divergences. In the general quantum case the relevant quantities are given only implicitly. Nevertheless, we find a large collection of monotones that provide necessary conditions for asymptotic or catalytic transformations. When applied to time-translation symmetric maps, these give rise to second laws that constrain state transformations allowed by thermal operations even in the presence of catalysts.

We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen’s theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum. These strong constraints allow us to classify and explicitly describe all such monotones, leading to a rate formula expressed as an optimization involving sandwiched Rényi divergences. As an application we give a new derivation of the strong converse error exponent in quantum hypothesis testing.

We study rates of asymptotic transformations between entangled states by local operations and classical communication and a sublinear amount of quantum communication. It is known that additive asymptotically continuous entanglement measures provide upper bounds on the rates that are achievable with asymptotically vanishing error. We show that for transformations between pure states, the optimal rate between any pair of states can be characterized as the infimum of such upper bounds provided by fully additive asymptotically continuous entanglement measures.

We study the exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability, known as the converse error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterization of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [1], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent, the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case, we prove that this bound holds with equality.

Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent ω, is a central problem in algebraic complexity theory. The best upper bounds on ω, leading to the state-of-the-art ω ≤ 2.37.., have been obtained via the laser method of Strassen and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on ω. We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of “irreversibility” of a tensor and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e.g., as a starting tensor in the laser method) cannot give ω = 2. In quantitative terms, we prove that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor. The quantum functionals and Strassen support functionals give (so far, the best) lower bounds on irreversibility. We provide lower bounds on the irreversibility of key intermediate tensors, including the small and big Coppersmith–Winograd tensors, that improve limitations shown in previous work. Finally, we discuss barriers on the group-theoretic approach in terms of “monomial” irreversibility.

We prove a lower bound on the rate of Greenberger-Horne-Zeilinger states distillable from pure multipartite states by local operations and classical communication (LOCC). Our proof is based on a modification of a combinatorial argument used in the fast matrix multiplication algorithm of Coppersmith and Winograd. Previous use of methods from algebraic complexity in quantum information theory concerned transformations with stochastic LOCC (SLOCC), resulting in an asymptotically vanishing success probability. In contrast, our new protocol works with an asymptotically vanishing error.