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arXiv:1811.08643 · Physical Review A

Experimental verification of anisotropic invariance for three-qubit states

Jie Zhu, Mengjun Hu, Shuming Cheng, Michael J. W. Hall, Chuan‐Feng Li, Guang‐Can Guo, Yong-Sheng Zhang

Published Apr 30, 2019• 5 citations
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Abstract

We experimentally test the recently predicted anisotropic invariance properties of pure three-qubit states, via generation and measurement of polarization-path entangled three-qubit states. These properties do not require aligned reference frames, and can be determined from measurements on any two of the qubits. They have several applications, such as a universal ordering of pairwise quantum correlations, strong monogamy relations for Bell inequalities and quantum steering, and a complementarity relation for Bell nonlocality versus 3-tangle, some of which we also test. The results indicate that anisotropic invariance, together with the three-qubit Bloch vector lengths, provide a robust and complete set of invariants for such states under local unitary transformations.

DOI
10.1103/physreva.99.040103

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We experimentally test the recently predicted anisotropic invariance properties of pure three-qubit states, via generation and measurement of polarization-path entangled three-qubit states. These properties do not require aligned reference frames, and can be determined from measurements on any two of the qubits. They have several applications, such as a universal ordering of pairwise quantum correlations, strong monogamy relations for Bell inequalities and quantum steering, and a complementarity relation for Bell nonlocality versus 3-tangle, some of which we also test. The results indicate that anisotropic invariance, together with the three-qubit Bloch vector lengths, provide a robust and complete set of invariants for such states under local unitary transformations.
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.
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.
We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Key words: Invariant, quantum state, local unitary transformation Quantum entangled states are playing very important roles in quantum information processing and quantum computation [1]. As the properties of entanglement of a multipartite quantum system remain invariant under local unitary transformations on the subsystems, the entanglement can be characterized by all the invariants of the local unitary transformations. The trace norms of realigned or partial transposed density matrices in entanglement measure and separability criteria are some of these invariants [2]. Invariants are also relevant in the discussions of Bell inequalities [3, 4] and teleportation, etc. [5]. A complete
Entanglement is a key resource in many quantum information schemes and in the last years the research on multi-qubit entanglement has drawn lots of attention. In this thesis the experimental generation and characterisation of multi-qubit entanglement is presented. Specifically we have prepared entangled states of up to six qubits. The qubits were implemented in the polarisation degree of freedom of single photons. We emphasise that one type of states that we produce are rotationally invariant states, remaining unchanged under simultaneous identical unitary transformations of all their individual constituents. Such states can be applied to e.g. decoherence-free encoding, quantum communication without sharing a common reference frame, quantum telecloning, secret sharing and remote state preparation schemes. They also have properties which are interesting in studies of foundations of quantum mechanics. In the experimental implementation we use a single source of entangled photon pairs, based on parametric down-conversion, and extract the first, second and third order events. Our experimental setup is completely free from interferometric overlaps, making it robust and contributing to a high fidelity of the generated states. To our knowledge, the achieved fidelity is the highest that has been observed for six-qubit entangled states and our measurement results are in very good agreement with predictions of quantum theory. We have also performed another novel test of the foundations of quantum mechanics. It is based on an inequality that is fulfilled by any non-contextual hidden variable theory, but can be violated by quantum mechanics. This test is similar to Bell inequality tests, which rule out local hidden variable theories as possible completions of quantum mechanics. Here, however, we show that non-contextual hidden variable theories cannot explain certain experimental results, which are consistent with quantum mechanics. Hence, neither of these theories can be used to make quantum mechanics complete.
The fact that quantum mechanics predicts stronger correlations than classical physics is an essential cornerstone of quantum information processing. Indeed, these quantum correlations are a valuable resource for various tasks, such as quantum key distribution or quantum teleportation, but characterizing these correlations in an experimental setting is a formidable task, especially in scenarios where no shared reference frames are available. By definition, quantum correlations are reference-frame independent, i.e., invariant under local transformations; this physically motivated invariance implies, however, a dedicated mathematical structure and, therefore, constitutes a roadblock for an efficient analysis of these correlations in experiments. Here we provide a method to directly measure any locally invariant property of quantum states using locally randomized measurements, and we present a detailed toolbox to analyze these correlations for two quantum bits. We implement these methods experimentally using pairs of entangled photons, characterizing their usefulness for quantum teleportation and their potential to display quantum nonlocality in its simplest form. Our results can be applied to various quantum computing platforms, allowing simple analysis of correlations between arbitrary distant qubits in the architecture.
We report on experimental studies on entanglement quantification and verification based on uncertainty relations for systems consisting of two qubits. The new proposed measure is shown to be invariant under local unitary transformations, by which entanglement quantification is implemented for two-qubit pure states. The nonlocal uncertainty relations for two-qubit pure states are also used for entanglement verification which serves as a basic proposition and promise to be a good choice for verification of multipartite entanglement.