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This is the documentation for generating random samples from the quantum state space in accordance with a specified distribution, associated with this webpage: http://tinyurl.com/QSampling . Ready-made samples (each with at least a million points) from various distributions are available for download, or one can generate ones own samples from a chosen distribution using the provided source codes. The sampling relies on the Hamiltonian Monte Carlo algorithm as described in New J. Phys. 17, 043018 (2015). The random samples are reposited in the hope that they would be useful for a variety of tasks in quantum information and quantum computation. Constructing credible regions for tomographic data, optimizing a function over the quantum state space with a complicated landscape, testing the typicality of entanglement among states from a multipartite quantum system, or computing the average of some quantity of interest over a subset of quantum states are but some exemplary applications among many.
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Suppose we have many copies of an unknown $n$-qubit state $rho$. We measure some copies of $rho$ using a known two-outcome measurement $E_{1}$, then other copies using a measurement $E_{2}$, and so on. At each stage $t$, we generate a current hypothesis $sigma_{t}$ about the state $rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|operatorname{Tr}(E_{i} sigma_{t}) - operatorname{Tr}(E_{i}rho) |$, the error in our prediction for the next measurement, is at least $varepsilon$ at most $operatorname{O}!left(n / varepsilon^2 right) $ times. Even in the non-realizable setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that do significantly worse than the best possible states at most $operatorname{O}!left(sqrt {Tn}right) $ times on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.
Two-qubit Bell-diagonal states can be depicted as a tetrahedron in three dimensions. We investigate the structure of quantum resources, including coherence and quantum discord, in the tetrahedron. The ordering of different resources measures is a common problem in resource theories, and which measure should be chosen to investigate the structure of resources is still an open question. We consider the structure of quantum resources which is not affected by the choice of measure. Our work provides a complete structure of coherence and quantum discord for Bell-diagonal states. The pictorial approach also indicates how to explore the structure of resources even when we dont have consistent measure of a concrete quantum resource.
The study of properties of randomly chosen quantum states has in recent years led to many insights into quantum entanglement. In this work, we study private quantum states from this point of view. Private quantum states are bipartite quantum states characterised by the property that carrying out simple local measurements yields a secret bit. This feature is shared by the maximally entangled pair of quantum bits, yet private quantum states are more general and can in their most extreme form be almost bound entangled. In this work, we study the entanglement properties of random private quantum states and show that they are hardly distinguishable from separable states and thus have low repeatable key, despite containing one bit of key. The technical tools we develop are centred around the concept of locally restricted measurements and include a new operator ordering, bounds on norms under tensoring with entangled states and a continuity bound for a relative entropy measure.
Understanding the relation between the different forms of inseparability in quantum mechanics is a longstanding problem in the foundations of quantum theory and has implications for quantum information processing. Here we make progress in this direction by establishing a direct link between quantum teleportation and Bell nonlocality. In particular, we show that all entangled states which are useful for teleportation are nonlocal resources, i.e. lead to deterministic violation of Bells inequality. Our result exploits the phenomenon of super-activation of quantum nonlocality, recently proved by Palazuelos, and suggests that the latter might in fact be generic.
In this work we propose the generation of a hybrid entangled resource (HER) and its further application in a quantum teleportation scheme from an experimentally feasible point of view. The source for HER preparation is based on the four wave mixing process in a photonic crystal fiber, from which one party of its output bipartite state is used to herald a single photon or a single photon added coherent state. From the heralded state and linear optics the HER is created. In the proposed teleportation protocol Bob uses the HER to teleport qubits with different spectral properties. Bob makes a Bell measurement in the single photon basis and characterizes the scheme with its average quantum teleportation fidelity. Fidelities close to one are expected for qubits in a wide spectral range. The work also includes a discussion about the fidelity dependence on the geometrical properties of the medium through which the HER is generated. An important remark is that no spectral filtering is employed in the heralding process, which emphasizes the feasibility of this scheme without compromising photon flux.
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