Entanglement of two parts of a quantum system is a non-local property unaffected by local manipulations of these parts. It is described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of non-local properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-bit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete.
Similarly, non-local, entangling properties of two-qubit unitary gates are invariant under single-bit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-bit gates).
Quantum nonlocality can be revealed "via local contextuality" in qudit-qudit entangled systems with $d > 2$, that is, through the violation of inequalities containing Alice-Bob correlations that admit a local description, and Alice-Alice correlations (between the results of sequences of measurements on Alice's subsystem) that admit a local (but contextual) description. A fundamental question to understand the respective roles of entanglement and local contextuality is whether nonlocality via local contextuality exists when the parties have only qubit-qubit entanglement. Here we respond affirma…
Characterizing 3-qubit UPB states: violations of LHV models, preparation via nonlocal unitaries and PPT entangled nonlocal orbits
对于3量子位UPB状态，即，从Bennett等人的不可扩展的乘积基（Phys. Rev. Lett. 82：5385，1999），我们基于在该状态中编码的特定类型的反射对称性提供了一组局部隐藏变量（LHV）模型的违反。给出了从纯态的反射可分混合态中制备态所需的显式非局域幺正操作，以及一个具有正部分转置（PPT）的非局域单参数态轨道，该轨道在一个周期内将一个态与其反射之间的纠缠交换两次。
Nonlocality and entanglement are not only the fundamental characteristics of quantum mechanics but also important resources for quantum information and computation applications. Exploiting the quantitative relationship between the two different resources is of both theoretical and practical significance. The common choice for quantifying the nonlocality of a two-qubit state is the maximal violation of the Clauser-Horne-Shimony-Holt inequality. That for entanglement is entanglement of formation, which is a function of the concurrence. In this paper, we systematically investigate the quantitativ…
Entanglement, quantum steering and nonlocality are distinct quantum correlations which are the resources behind various of quantum information and quantum computation applications. However, a central question of determining the precise quantitative relation among them is still unresolved. Here, we present a mapping criterion between Bell nonlocality and quantum steering in bipartite qudit-qubit system, as well as a mapping criterion between quantum steering and quantum entanglement in bipartite qubit-qudit system, starting from the fundamental concepts of quantum correlations. Precise quantita…
Entanglement and Bell nonlocality are used to describe quantum inseparabilities. Bell-nonlocal states form a strict subset of entangled states. A natural question arises concerning how much territory Bell nonlocality occupies entanglement for a general two-qubit entangled state. In this work, we investigate the relation between entanglement and Bell nonlocality by using lots of randomly generated two-qubit states, and give out a constraint inequality relation between the two quantum resources. For studying the upper or lower boundary of the inequality relation, we discover maximally (minimally…
The direct measurement of nonlocal properties of entangled quantum systems has been the subject of several recent experimental investigations. Of particular interest is the implementation of nonlocal measurements via indirect measurement schemes, which allow for greater flexibility in the control of the measurement strength. Building on previous results established in the bipartite case, we present a scheme to implement genuine nonlocal measurements of N-qubit systems with variable strength, using GHZ-like entangled qubit meters. This method can be applied to the joint measurement of commuting…
Analysis of Quantum Correlation in Successive Spin Measurements, Classical Communication of Spin $S$ Singlet States and Quantum Nonlocality of two Qubit Entangled States
We analyze optimal schemes and also propose some practical schemes for the nonlocal conversion from two shared Bell pairs to four-qubit entangled states in optical quantum networks. In the analysis, we consider two-qubit operations as nonlocal operations and minimize the number of access to ancillary qubits as possible. First, we consider two-qubit unitary operations without using ancillary qubits and derive a necessary and sufficient condition for convertible states. Second, we consider nonlocal optical systems composed of passive linear optics and postselection. For the passive linear optica…
Mixed states appear naturally in experiment over pure states. So for studying different notions of nonlocality and their relation with entanglement in realistic scenarios, one needs to consider mixed states. In a recent article [Phys. Rev. Lett. \textbf{108}, 020502 (2012)], a complete characterization of entanglement of an entire class of mixed three qubit states with the same symmetry as Greenberger-Horne-Zeilinger state known as GHZ-symmetric states, has been achieved. In this paper we investigate different notions of nonlocality of the same class of states. By finding the analytical expres…
We study the relations between quantum coherence and quantum nonlocality, genuine quantum entanglement and genuine quantum nonlocality. We show that the coherence of a qubit state can be converted to the nonlocality of two-qubit states via incoherent operations. The results are also generalized to qudit case. Furthermore, rigorous relations between the quantum coherence of a single-partite state and the genuine multipartite quantum entanglement, as well as the genuine three-qubit quantum nonlocality are established.
Near-term quantum computers are limited by the decoherence of qubits to only being able to run low-depth quantum circuits with acceptable fidelity. This severely restricts what quantum algorithms can be compiled and implemented on such devices. One way to overcome these limitations is to expand the available gate set from single- and two-qubit gates to multi-qubit gates, which entangle three or more qubits in a single step. Here, we show that such multi-qubit gates can be realized by the simultaneous application of multiple two-qubit gates to a group of qubits where at least one qubit is invol…
We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality. We make use of the elegant parametrization in the canonical form of these states, proposed by Meill and Meyer (Phys. Rev. A, 96, 062310 (2017)) based on Majorana geometric representation. The reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state do not violate the CHSH inequality and hence they are CHSH-local. However, when Alice and Bob perform a CHSH test, after conditioning over measurement result…
We describe a direct method to experimentally determine local two-qubit invariants by performing interferometric measurements on multiple copies of a given two-qubit state. We use this framework to analyze two different kinds of two-qubit invariants of Makhlin and Jing et. al. These invariants allow to fully reconstruct any two-qubit state up to local unitaries. We demonstrate that measuring 3 invariants is sufficient to find, e.g., the optimal Bell inequality violation. These invariants can be measured with local or nonlocal measurements. We show that the nonlocal strategy that follows from M…
Quantum coherence and nonlocality capture nature of quantumness from different aspects. For the two-qubit states with diagonal correlation matrix, we prove strictly a hierarchy between the nonlocal advantage of quantum coherence (NAQC) and Bell nonlocality by showing geometrically that the NAQC created on one qubit by local measurement on another qubit captures quantum correlation which is stronger than Bell nonlocality. For general states, our numerical results present strong evidence that this hierarchy may still hold. So the NAQC states form a subset of the states that can exhibit Bell nonl…
Recently, the nonlocal optimal probabilistic cloning (NLOPC) of two non-orthogonal qubit states has been proposed [Phys. Rev. A 86, 052332 (2012)] by means of an experimental setup based on a pair of twin photons in a maximally entangled state. Here we study the performance of the NLOPC protocol when implemented via a partially entangled state. We show that the errors introduced by the use of partial entanglement can be undone by applying a quantum state discrimination process. Since quantum state discrimination is a probabilistic process the correction succeeds with a certain probability but …
