The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of linear logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.
Abramsky and Coecke describe a realization of a model of multiplicative linear logic via the quantum processes of entangling and de-entangling by means of typed projectors. They briefly discuss how these processes can be represented as terms of an affine lambda calculus @cite_15 .
@cite_15
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We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations. We review some of our recent results (with collaborators) on information processing in an ordered linear spaces framework for probabilistic theories. These include demonstrations that many "inherently quantum" phenomena are in reality quite general characteristics of non-classical theories, quantum or otherwise. As an example, a set of states in such a theory is broadcastable if, and only if, it is contained in a simplex whose vertices are cloneable, and therefore distinguishable by a single measurement. As another example, information that can be obtained about a system in this framework without causing disturbance to the system state, must be inherently classical. We also review results on teleportation protocols in the framework, and the fact that any non-classical theory without entanglement allows exponentially secure bit commitment in this framework. Finally, we sketch some ways of formulating our framework in terms of categories, and in this light consider the relation of our work to that of Abramsky, Coecke, Selinger, Baez and others on information processing and other aspects of theories formulated categorically. We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input. Quantum information theory is a multidisciplinary field whose objective is to understand what happens when information is stored in the state of a quantum system. Quantum mechanics provides us with a new resource, called quantum entanglement, which can be exploited to achieve novel tasks such as teleportation and superdense coding. Current technologies allow the transmission of entangled photon pairs across distances up to roughly 100 kilometers. For longer distances, noise arising from various sources degrade the transmission of entanglement to the point that it becomes impossible to use the entanglement as a resource for future tasks. One strategy for dealing with this difficulty is to employ quantum repeaters, stations intermediate between the sender and receiver that can participate in the process of entanglement distillation, thereby improving on what the sender and receiver could do on their own. Motivated by the problem of designing quantum repeaters, we study entanglement distillation between two parties, Alice and Bob, starting from a mixed state and with the help of repeater stations. We extend the notion of entanglement of assistance to arbitrary tripartite states and exhibit a protocol, based on a random coding strategy, for extracting pure entanglement. We use these results to find achievable rates for the more general scenario, where many spatially separated repeaters help two recipients distill entanglement. We also study multiparty quantum communication protocols in a more general context. We give a new protocol for the task of multiparty state merging. The previous multiparty state merging protocol required the use of time-sharing, an impossible strategy when a single copy of the input state is available to the parties. Our protocol does not require time-sharing for distributed compression of two senders. In the one-shot regime, we can achieve multiparty state merging with entanglement costs not restricted to corner points of the entanglement cost region. Our analysis of the entanglement cost is performed using (smooth) min- and max-entropies. We illustrate the benefits of our approach by looking at different examples.