Better Heisenberg Limits, Coherence Bounds, and Energy-Time Tradeoffs via Quantum Rényi Information.

An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form RMSE≥f(α)/(⟨N⟩+12), bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where f(α) is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as ΔΦ≥maxnpn, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems.

Experimental Validation of Quantum Steering Ellipsoids and Tests of Volume Monogamy Relations.

The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid-called the quantum steering ellipsoid-in the Bloch ball. This ellipsoid provides a simple visual characterization of the initial two-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarization-entangled photonic qubit states. Moreover, for pure three-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qubit are predicted to satisfy a tight monogamy relation, which is strictly stronger than the well-known monogamy of entanglement for concurrence. We experimentally verify these predictions, using polarization and path entanglement. We also show experimentally that this monogamy relation can be violated by a mixed entangled state, which nevertheless satisfies a weaker monogamy relation.

Strong Unitary and Overlap Uncertainty Relations: Theory and Experiment.

We derive and experimentally investigate a strong uncertainty relation valid for any n unitary operators, which implies the standard uncertainty relation and others as special cases, and which can be written in terms of geometric phases. It is saturated by every pure state of any n-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any n+1 pure states, and gives an upper bound for the out-of-time-order correlation function. We test these uncertainty relations experimentally for photonic polarization qubits, including the minimum uncertainty states of the overlap uncertainty relation, via interferometric measurements of generalized geometric phases.

Nonlocality in Bell's Theorem, in Bohm's Theory, and in Many Interacting Worlds Theorising.

"Locality" is a fraught word, even within the restricted context of Bell's theorem. As one of us has argued elsewhere, that is partly because Bell himself used the word with different meanings at different stages in his career. The original, weaker, meaning for locality was in his 1964 theorem: that the choice of setting by one party could never affect the outcome of a measurement performed by a distant second party. The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new approach to quantum mechanics, inspired by Bohmian mechanics, has been proposed: Many Interacting Worlds. While it is conceptually clear how the interaction between worlds can enable this strong nonlocality, technical problems in the theory have thus far prevented a proof by simulation. Here we report significant progress in tackling one of the most basic difficulties that needs to be overcome: correctly modelling wavefunctions with nodes.

Measurement-Device-Independent Approach to Entanglement Measures.

Within the context of semiquantum nonlocal games, the trust can be removed from the measurement devices in an entanglement-detection procedure. Here, we show that a similar approach can be taken to quantify the amount of entanglement. To be specific, first, we show that in this context, a small subset of semiquantum nonlocal games is necessary and sufficient for entanglement detection in the local operations and classical communication paradigm. Second, we prove that the maximum payoff for these games is a universal measure of entanglement which is convex and continuous. Third, we show that for the quantification of negative-partial-transpose entanglement, this subset can be further reduced down to a single arbitrary element. Importantly, our measure is measurement device independent by construction and operationally accessible. Finally, our approach straightforwardly extends to quantify the entanglement within any partitioning of multipartite quantum states.

Publisher's Note: Anisotropic Invariance and the Distribution of Quantum Correlations [Phys. Rev. Lett. 118, 010401 (2017)].

This corrects the article DOI: 10.1103/PhysRevLett.118.010401.

Anisotropic Invariance and the Distribution of Quantum Correlations.

We report the discovery of two new invariants for three-qubit states which, similarly to the three-tangle, are invariant under local unitary transformations and permutations of the parties. These quantities have a direct interpretation in terms of the anisotropy of pairwise spin correlations. Applications include a universal ordering of pairwise quantum correlation measures for pure three-qubit states; trade-off relations for anisotropy, three-tangle and Bell nonlocality; strong monogamy relations for Bell inequalities, Einstein-Podolsky-Rosen steering inequalities, geometric discord and fidelity of remote state preparation (including results for arbitrary three-party states); and a statistical and reference-frame-independent form of quantum secret sharing.

Erratum: Experimental Test of Entropic Noise-Disturbance Uncertainty Relations for Spin-1/2 Measurements [Phys. Rev. Lett. 115, 030401 (2015)].

This corrects the article DOI: 10.1103/PhysRevLett.115.030401.

Erratum: Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement Independence [Phys. Rev. Lett. 105, 250404 (2010)].

This corrects the article DOI: 10.1103/PhysRevLett.105.250404.

Experimental Test of Entropic Noise-Disturbance Uncertainty Relations for Spin-1/2 Measurements.

Information-theoretic definitions for noise and disturbance in quantum measurements were given in [Phys. Rev. Lett. 112, 050401 (2014)] and a state-independent noise-disturbance uncertainty relation was obtained. Here, we derive a tight noise-disturbance uncertainty relation for complementary qubit observables and carry out an experimental test. Successive projective measurements on the neutron's spin-1/2 system, together with a correction procedure which reduces the disturbance, are performed. Our experimental results saturate the tight noise-disturbance uncertainty relation for qubits when an optimal correction procedure is applied.

Experimental measurement-device-independent verification of quantum steering.

Bell non-locality between distant quantum systems--that is, joint correlations which violate a Bell inequality--can be verified without trusting the measurement devices used, nor those performing the measurements. This leads to unconditionally secure protocols for quantum information tasks such as cryptographic key distribution. However, complete verification of Bell non-locality requires high detection efficiencies, and is not robust to typical transmission losses over long distances. In contrast, quantum or Einstein-Podolsky-Rosen steering, a weaker form of quantum correlation, can be verified for arbitrarily low detection efficiencies and high losses. The cost is that current steering-verification protocols require complete trust in one of the measurement devices and its operator, allowing only one-sided secure key distribution. Here we present measurement-device-independent steering protocols that remove this need for trust, even when Bell non-locality is not present. We experimentally demonstrate this principle for singlet states and states that do not violate a Bell inequality.

Noise and disturbance in quantum measurements: an information-theoretic approach.

We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under the relabelling of outcomes and allow for the possibility of using quantum or classical operations to "correct" for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.

Stochastic Heisenberg limit: optimal estimation of a fluctuating phase.

The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramér-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ω(-p) with p>1, the minimum mean-square error in any (single-time) phase estimate scales as N(-2(p-1)/(p+1)), where N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p→∞) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2 (Brownian motion), this limit can be attained by phase tracking.

Experimental test of universal complementarity relations.

Complementarity restricts the accuracy with which incompatible quantum observables can be jointly measured. Despite popular conception, the Heisenberg uncertainty relation does not quantify this principle. We report the experimental verification of universally valid complementarity relations, including an improved relation derived here. We exploit Einstein-Poldolsky-Rosen correlations between two photonic qubits to jointly measure incompatible observables of one. The product of our measurement inaccuracies is low enough to violate the widely used, but not universally valid, Arthurs-Kelly relation.

Effects of reduced measurement independence on Bell-based randomness expansion.

With the advent of quantum information, the violation of a Bell inequality is used to witness the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell's theorem is the existence of experimental "free will," meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signaling model with reduced "free will" and bound the adversary's capabilities in the task of randomness expansion.

Local deterministic model of singlet state correlations based on relaxing measurement independence.

The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up just 14% of measurement independence. The underlying model is deterministic and no signaling. It may thus be favorably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signaling. A local deterministic model is also given that achieves the maximum possible violation of the well-known Bell-Clauser-Horne-Shimony-Holt inequality, at a cost of only 1/3 of measurement independence.