Browsing by Author "Sikora, Jamie"

Now showing 1 - 8 of 8
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    Bounding the Quantum and Classical Resources in Bell Experiments
    Koenig, Jonathan A. (Virginia Tech, 2022-05-23)
    Bell's theory of nonlocality in quantum mechanics allows for interesting correlations between separated parties. In this scenario, both parties share a quantum state and measure it to obtain a classical value. Through entanglement, the results of the measurement from one party can affect the results of the other party's measurement. Quantum correlations reflect this idea as a probability distribution p(ab|xy) based on the measurements used (x for Alice and y for Bob) and the respective results obtained (a and b). In this thesis, we introduce an expression that limits what quantum states could be used to generate a given quantum correlation. This, in turn, yields a lower bound on the dimension needed for this quantum state. For a quantum correlation p(ab|xy), the dimension of the quantum state acts as a resource needed to generate it. Thus, having a bound on the dimension helps one to quantify the resources needed to generate a given correlation. In addition to quantum correlations, we adjust the bound to work with classical correlations as well, which are correlations generated using a shared probability distribution instead of a quantum state. We apply our quantum and classical bounds to well-studied correlations to test them based on known results and also generate randomly generated correlations to better understand their behavior. Finally, we report on our numerical findings.
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    Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)
    Johnston, Nathaniel; Sikora, Jamie (Elsevier, 2022-11)
    In this work we examine quantum states which have non -negative amplitudes (in a fixed basis) and the channels which preserve them. These states include the ground states of stoquastic Hamiltonians and they are of interest since they avoid the Sign Problem and can thus be efficiently simulated. In optimization theory, the convex cone generated by such states is called the set of completely positive (CP) matrices (not be confused with completely positive superoperators). We introduce quantum channels which preserve these states and call them completely positive completely positive. To study these states and channels, we use the framework of resource theories and investigate how to measure and quantify this resource.(c) 2022 The Author(s). Published by Elsevier Inc.
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    Constant Lower Bounds on the Cryptographic Security of Quantum Two-Party Computations
    Osborn, Sarah Anne (Virginia Tech, 2022-05-24)
    In this thesis, we generate a lower bound on the security of quantum protocols for secure function evaluation. Central to our proof is the concept of gentle measurements of quantum states, which do not greatly disturb a quantum state if a certain outcome is obtained with high probability. We show how a cheating party can leverage gentle measurements to learn more information than should be allowable. To quantify our lower bound, we reduce a specific cryptographic task known as die-rolling to secure function evaluation and use the concept of gentle measurements to relate their security notions. Our lower bound is then obtained using a known security bound for die-rolling known as Kitaev's bound. Due to the generality of secure function evaluation, we are able to apply this lower bound to obtain lower bounds on the security of quantum protocols for many quantum tasks. In particular, we provide lower bounds for oblivious transfer, XOR oblivious transfer, the equality function, the inner product function, Yao's millionaires' problem, and the secret phrase problem. Note that many of these lower bounds are the first of their kind, which is a testament to the utility of our lower bound. As a consequence, these bounds prove that unconditional security for quantum protocols is impossible for these applications, and since these are constant lower bounds, this rules out any form of boosting toward perfect security. Our work lends itself to future research on designing optimal protocols for the above listed tasks, and potentially others, by providing constant lower bounds to approximate or improve.
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    Deciding if a Genus 1 Curve has a Rational Point
    Swanson, Nicolas J. Brennan (Virginia Tech, 2024-05-23)
    Many sources suggest a folklore procedure to determine if a smooth curve of genus 1 has a rational point. This procedure terminates conditionally on the Tate-Shafarevich conjecture. In this thesis, we provide an exposition for this procedure, making several steps explicit. In some instances, we also provide MAGMA implementations of the subroutines. In particular, we give an algorithm to determine if a smooth, genus 1 curve of arbitrary degree is locally soluble, we compute its Jacobian, and we give an exposition for descent in our context. Additionally, we prove there exists an algorithm to decide if smooth, genus 1 curve has a rational point if and only if there exists an algorithm to compute the Mordeil-Weil group of an elliptic curve.
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    Decomposing all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories
    Cavalcanti, Paulo J.; Selby, John H.; Sikora, Jamie; Sainz, Ana Belen (IOP Publishing, 2022-10-07)
    Non-signalling quantum channels-relevant in, e.g., the study of Bell and Einstein-Podolsky-Rosen scenarios-may be decomposed as an affine combinations of local operations in bipartite scenarios. Moreover, when these channels correspond to stochastic maps between classical variables, such a decomposition is possible even in multipartite scenarios. These two results have proven useful when studying the properties of these channels, such as their communication and information processing power, and even when defining measures of the non-classicality of physical phenomena (such as Bell non-classicality and steering). In this paper we show that such useful quasi-stochastic characterizations of channels may be unified and applied to the broader class of multipartite non-signalling channels. Moreover, we show that this holds for non-signalling channels in quantum theory, as well as in a larger family of generalised probabilistic theories. More precisely, we prove that channels are non-signalling if and only if they can be decomposed as an affine combinations of corresponding local operations, provided that the underlying physical theory is locally tomographic-a property that quantum theory satisfies. Our results then can be viewed as a generalisation of references (Phys. Rev. Lett. 111 170403) and (2013 Phys. Rev. A 88 022318) to the multipartite scenario for arbitrary tomographically local generalised probabilistic theories (including quantum theory). Our proof technique leverages Hardy's duotensor formalism, highlighting its utility in this line of research.
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    Impossibility of coin flipping in generalized probabilistic theories via discretizations of semi-infinite programs
    Sikora, Jamie; Selby, John H. (2020-10-23)
    Coin flipping is a fundamental cryptographic task where spatially separated Alice and Bob wish to generate a fair coin flip over a communication channel. It is known that ideal coin flipping is impossible in both classical and quantum theory. In this work, we give a short proof that it is also impossible in generalized probabilistic theories under the generalized no-restriction hypothesis. Our proof relies crucially on a formulation of cheating strategies as semi-infinite programs, i.e., cone programs with infinitely many constraints. This introduces a formalism which may be of independent interest to the quantum community.
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    Jordan products of quantum channels and their compatibility
    Girard, Mark; Plávala, Martin; Sikora, Jamie (Nature Research, 2021)
    Given two quantum channels, we examine the task of determining whether they are compatible— meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.
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    Post-quantum steering is a stronger-than-quantum resource for information processing
    Cavalcanti, Paulo J.; Selby, John H.; Sikora, Jamie; Galley, Thomas D.; Sainz, Ana Belen (Nature Portfolio, 2022-06-30)
    We present the first instance where post-quantum steering is a stronger-than-quantum resource for information processing - remote state preparation. In addition, we show that the phenomenon of post-quantum steering is not just a mere mathematical curiosity allowed by the no-signalling principle, but it may arise within compositional theories beyond quantum theory, hence making its study fundamentally relevant. We show these results by formulating a new compositional general probabilistic theory - which we call Witworld - with strong post-quantum features, which proves to be a intuitive and useful tool for exploring steering and its applications beyond the quantum realm.