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arXiv:2210.13475 · Physical Review A

Finding maximal quantum resources

Jonathan Steinberg, Otfried Gühne

Published Dec 31, 2024• 3 citations
arXivHTMLPDF

Abstract

For many applications the presence of a quantum advantage crucially depends on the availability of resourceful states. Although the resource typically depends on the particular task, in the context of multipartite systems entangled quantum states are often regarded as resourceful. We propose an algorithmic method to find maximally resourceful states of several particles for various applications and quantifiers. We discuss in detail the case of the geometric measure, identifying physically interesting states and delivering insights to the problem of absolutely maximally entangled states. Moreover, we demonstrate the universality of our approach by applying it to maximally entangled subspaces, the Schmidt-rank, the stabilizer rank as well as the preparability in triangle networks.

DOI
10.1103/physreva.110.062428

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A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.
We investigate absolutely maximally entangled (AME) states, which are multipartite quantum states that are maximally entangled with respect to any possible bipartition. These strong entanglement properties make them a powerful resource for a variety of quantum information protocols. In this paper, we show the existence of AME states for any number of parties, given that the dimension of the involved systems is chosen appropriately. We prove the equivalence of AME states shared between an even number of parties and pure state threshold quantum secret sharing (QSS) schemes, and prove necessary and sufficient entanglement properties for a wider class of ramp QSS schemes. We further show how AME states can be used as a valuable resource for open-destination teleportation protocols and to what extend entanglement swapping generalizes to AME states.
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In the studies of quantum information and computation, the quantum correlations present in a non-classical system are seen as natural resources. In this thesis, we present a detailed study on quantification and characterization of multipartite quantum entanglement from a geometric perspective. Furthermore, we present protocols and algorithms where quantum correlations such as entanglement, quantum Fisher information, and quantum superposition have been used to provide advantages as well securities in quantum communication tasks, quantum speed-up, and protection against the decoherence in an open quantum system. First, we use a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices to obtain the minimum distance between the set of bipartite n- qudit density matrices with a positive partial transpose and the maximally mixed state. This minimum distance coincides with the radius of the largest separable ball. We present the interior geometry of the set of all positive semidefinite matrices and identify a particular class of Werner states for which the Peres-Horodecki criterion is both necessary and sufficient for separability even in the dimensions greater than six.
Abstract Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state—one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely used graph states, such as 1-D and 2-D cluster states and the Greenberger–Horne–Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglement-structure detection to benchmark multipartite quantum systems.
Quantum technologies are enjoying an unprecedented popularity, and some applications are already in the market. This thesis studies two phenomena that are behind a lot of quantum technologies: entanglement and nonlocality. We focus on multipartite systems, and ask what configurations of those systems are more useful than others. 'Usefulness' takes on different meanings depending on the context, but, roughly speaking, we aim for more entanglement or more nonlocality. Chapter 2 develops a resource theory of entanglement with a unique multipartite maximally entangled state. Chapter 3 shows that any connected network of bipartite pure entangled states is genuine multipartite nonlocal (GMNL). Chapter 4 shows that, unlike in the case of pure states, topology is crucial to determine whether networks of bipartite mixed entangled states are genuine multipartite entangled. We also obtain, to our knowledge, the first example of superactivation of GMNL. In a departure from the main ideas explored in previous chapters, Chapter 5 is devoted to developing a physical principle inspired by a seminal result in epistemics. We contend that the principle must hold for any theory of Nature and intend for it to rule out post-quantum theories which are consistent with experimentally observed results.