Relational quantum computing using only maximally mixed initial qubit states

We disprove the conjecture of [1], namely that it would require smarter authors to find a way of making the two-qubit singlet/triplet measurement quantum computationally universal given an ensemble of initial single qubit states with less than three linearly independent Bloch vectors. We show, in fact, that an initial ensemble of maximally mixed single qubits suffices.Comment: 2 + epsilon page

Highly efficient estimation of entanglement measures for large experimentally created graph states via simple measurements

Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor structure of the Hilbert space, this approach becomes unfeasible even for medium-sized systems. Here we present methods for quantifying the entanglement of arbitrarily large two-colorable graph states from simple measurements. The presented methods provide non-trivial bounds on the entanglement for any state as long as there is sufficient fidelity with such a graph state. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement

Generalised versions of separable decompositions applicable to bipartite entangled quantum states

The investigation of separable states in quantum theory has been driven by the notion that they are highly classical, in that they do not demonstrate nonlocality, and are in some contexts unable to support non-classical computation. The converse question, the extent to which entangled states do or do not support non-classical information processing, is less well understood. Motivated by this question we extend the notion of quantum separability into the entangled quantum states, by constructing separable decompositions that describe them with the 'smallest' possible sets of non-physical local operators. We consider a few ways to define the word 'smallest' and present techniques for obtaining them. The methods involve calculating certain forms of cross norm. The results generalise significantly the results obtained in our previous work on this topic (2015 New J. Phys. 17 093047), and can be be used to construct classical simulation methods and local hidden variable models for subsets of local measurements on entangled quantum states

Classically efficient regimes in measurement based quantum computation performed using diagonal two qubit gates and cluster measurements

In a recent work arXiv:2201.07655v2 we showed that there is a constant $\lambda >0$ such that it is possible to efficiently classically simulate a quantum system in which (i) qudits are placed on the nodes of a graph, (ii) each qudit undergoes at most $D$ diagonal gates, (iii) each qudit is destructively measured in the computational basis or bases unbiased to it, and (iv) each qudit is initialised within $\lambda^{-D}$ of a diagonal state according to a particular distance measure. In this work we explicitly compute $\lambda$ for any two qubit diagonal gate, thereby extending the computation of arXiv:2201.07655v2 beyond CZ gates. For any finite degree graph this allows us to describe a two parameter family of pure entangled quantum states (or three parameter family of thermal states) which have a non-trivial classically efficiently simulatable "phase" for the permitted measurements, even though other values of the parameters may enable ideal cluster state quantum computation. The main the technical tool involves considering separability in terms of "cylindrical" sets of operators. We also consider whether a different choice of set can strengthen the algorithm, and prove that they are optimal among a broad class of sets, but also show numerically that outside this class there are choices that can increase the size of the classically efficient regime.Comment: 12 pages, 3 figure

The geometric measure of entanglement for a symmetric pure state with positive amplitudes

In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: 'for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state'. We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis. The more general conjecture is still open.Comment: Similar results have been obtained independently and with different methods by T-C. Wei and S. Severini, see arXiv:0905.0012v

A relational quantum computer using only two-qubit total spin measurement and an initial supply of highly mixed single qubit states

We prove that universal quantum computation is possible using only (i) the physically natural measurement on two qubits which distinguishes the singlet from the triplet subspace, and (ii) qubits prepared in almost any three different (potentially highly mixed) states. In some sense this measurement is a `more universal' dynamical element than a universal 2-qubit unitary gate, since the latter must be supplemented by measurement. Because of the rotational invariance of the measurement used, our scheme is robust to collective decoherence in a manner very different to previous proposals - in effect it is only ever sensitive to the relational properties of the qubits.Comment: TR apologises for yet again finding a coauthor with a ridiculous middle name [12

Smallest disentangling state spaces for general entangled bipartite quantum states

PACS numbers: 03.67.-a, 03.65.-w, 03.65.Ta, 03.65.Ud.Entangled quantum states can be given a separable decomposition if we relax the restriction that the local operators be quantum states. Motivated by the construction of classical simulations and local hidden variable models, we construct `smallest' local sets of operators that achieve this. In other words, given an arbitrary bipartite quantum state we construct convex sets of local operators that allow for a separable decomposition, but that cannot be made smaller while continuing to do so. We then consider two further variants of the problem where the local state spaces are required to contain the local quantum states, and obtain solutions for a variety of cases including a region of pure states around the maximally entangled state. The methods involve calculating certain forms of cross norm. Two of the variants of the problem have a strong relationship to theorems on ensemble decompositions of positive operators, and our results thereby give those theorems an added interpretation. The results generalise those obtained in our previous work on this topic [New J. Phys. 17, 093047 (2015)].EP/K022512/1/Engineering and Physical Sciences Research Counci

Entanglement and local information access for graph states

We exactly evaluate a number of multipartite entanglement measures for a class of graph states, including d-dimensional cluster states (d = 1,2,3), the Greenberger-Horne-Zeilinger states, and some related mixed states. The entanglement measures that we consider are continuous, 'distance from separable states' measures, including the relative entropy, the so-called geometric measure, and robustness of entanglement. We also show that for our class of graph states these entanglement values give an operational interpretation as the maximal number of graph states distinguishable by local operations and classical communication (LOCC), as well as supplying a tight bound on the fixed letter classical capacity under LOCC decoding

Entanglement quantification and local discrimination

SIGLEAvailable from British Library Document Supply Centre- DSC:DXN062555 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Erratum: Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states (vol 77, 012104, 2008)

This is an erratum of 'Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states' published in the Physical Review A, 2008, 77, 012104