arXiv subjects

Antony Milne

Publications and source records attributed to Antony Milne.

9 recordsLinked to original sources

arXiv2018

Quantum steering with positive operator valued measures

We address the problem of quantum nonlocality with positive operator valued measures (POVM) in the context of Einstein-Podolsky-Rosen quantum steering. We show that, given a candidate for local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartite quantum state of finite dimension with POVMs can be formulated as a nesting problem of two convex objects. One consequence of this is the strengthening of the theorem that justifies choosing the LHS ensemble based on symmetry of the bipartite state. As a more practical application, we study the classic problem of the steerability of two-qubit Werner states with POVMs. We show strong numerical evidence that these states are unsteerable with POVMs up to a mixing probability of $\frac{1}{2}$ within an accuracy of $10^{-3}$.

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arXiv2017

Markovian evolution of quantum coherence under symmetric dynamics

Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of time-translation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalisation to higher-dimensional systems of the relation $T_2 \leq 2 T_1$ for qubit decoherence and relaxation times. It also enables us to witness and assess the role of non-Markovianity as a resource for coherence preservation and transfer. Moreover, we discuss the relationship between ergodicity and the ability of Markovian dynamics to indefinitely sustain a superposition of different energy states. Finally, we establish a formal connection between the resource-theoretic and the master equation approaches to thermodynamics, with the former being a non-Markovian generalisation of the latter. Our work thus brings the abstract study of quantum coherence as a resource towards the realm of actual physical applications.

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arXiv2017

Efficient detection of useful long-range entanglement in imperfect cluster states

Photonic cluster states are a crucial resource for optical quantum computing. Recently a quantum dot single photon source has been demonstrated to produce strings of photons in a linear cluster state, but high photon loss rates make it impossible to characterize the entanglement generated by conventional methods. We present a benchmarking method for such sources that can be used to demonstrate useful long-range entanglement with currently available collection/detection efficiencies below 1%. Measurement of the polarization state of single photons in different bases can provide an estimate for the three-qubit correlation function . This value constrains correlations spanning more than three qubits, which in turn provide a lower bound for the localizable entanglement between any two qubits in the large state produced by the source. Finite localizable entanglement can be established by demonstrating > 2/3. This result enables photonic experiments demonstrating computationally useful entanglement with currently available technology.

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arXiv2016

Volume monogamy of quantum steering ellipsoids for multi-qubit systems

The quantum steering ellipsoid can be used to visualise two-qubit states, and thus provides a generalisation of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters (CKW) inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication (SLOCC). We then show that this monogamy relation does not hold for general mixed 3-qubit states and derive a weaker monogamy relation that does hold for such states. We also prove a volume monogamy relation for pure 4-qubit states, and generalize our 3-qubit inequality to n qubits. Finally, we study the effect of noise on the quantum steering ellipsoid and find that the volume of any two-qubit state is non-increasing when the state is exposed to arbitrary local noise. This implies that any volume monogamy relation for a given class of multi-qubit states remains valid under the addition of local noise. We investigate this quantitatively for the experimentally relevant example of isotropic noise.

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arXiv2016

Quantum coherence of steered states

Lying at the heart of quantum mechanics, coherence has recently been studied as a key resource in quantum information theory. Quantum steering, a fundamental notion originally considered by Schr{ö}dinger, has also recently received much attention. When Alice and Bob share a correlated quantum system, Alice can perform a local measurement to `steer' Bob's reduced state. We introduce the maximal steered coherence as a measure describing the extent to which steering can remotely create coherence; more precisely, we find the maximal coherence of Bob's steered state in the eigenbasis of his original reduced state, where maximization is performed over all positive-operator valued measurements for Alice. We prove that maximal steered coherence vanishes for quantum-classical states whilst reaching a maximum for pure entangled states with full Schmidt rank. Although invariant under local unitary operations, maximal steered coherence may be increased when Bob performs a channel. For a two-qubit state we find that Bob's channel can increase maximal steered coherence if and only if it is neither unital nor semi-classical, which coincides with the condition for increasing discord. Our results show that the power of steering for coherence generation, though related to discord, is distinct from existing measures of quantum correlation.

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arXiv2015

The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory

We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a physically motivated derivation of very non-trivial geometric results, some of which are entirely new. We consider a sphere of radius $r$ contained inside another sphere of radius $R$, with the sphere centres separated by distance $d$. When does there exist a nested tetrahedron circumscribed about the smaller sphere and inscribed in the larger? We derive the Grace-Danielsson inequality $d^2 \leq (R+r)(R-3r)$ as the sole necessary and sufficient condition for the existence of a nested tetrahedron. Our method also gives the condition $d^2 \leq R(R-2r)$ for the existence of a nested triangle in the analogous 2-dimensional scenario. These results imply the Euler inequality in 2 and 3 dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids.

math.MG↗

arXiv2015

Geometric representation of two-qubit entanglement witnesses

Any two-qubit state can be represented geometrically by a steering ellipsoid inside the Bloch sphere. We extend this approach to represent any block positive two-qubit operator $B$. We derive a novel classification scheme based on the positivity of $\det B$ and $\det B^\mathrm{T_B}$; this shows that any ellipsoid inside the Bloch sphere must represent either a two-qubit state or a two-qubit entanglement witness. We focus on such witnesses and their corresponding ellipsoids, finding that properties such as witness optimality are naturally manifest in this geometric representation.

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arXiv2015

Quantum steering ellipsoids, extremal physical states and monogamy

Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman-Kundu-Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle.

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arXiv2014

Quantum correlations of two-qubit states with one maximally mixed marginal

We investigate the entanglement, CHSH nonlocality, fully entangled fraction and symmetric extendibility of two-qubit states that have a single maximally mixed marginal. Within this set of states, the steering ellipsoid formalism has recently highlighted an interesting family of so-called 'maximally obese' states. These are found to have extremal quantum correlation properties that are significant in the steering ellipsoid picture and for the study of two-qubit states in general.

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