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arXiv:cond-mat/0607645 · Physical Review B

Entanglement detection for electrons via witness operators

Lara Faoro, Fabio Taddei

Published Apr 30, 2007• 14 citations
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Abstract

We discuss an implementation of the entanglement witness, a method to detect entanglement with few local measurements, in systems where entangled electrons are generated both in the spin and orbital degrees of freedom. We address the efficiency of this method in various setups, including two different particle-hole entanglement structures, and we demonstrate that it can also be used to infer information on the possible dephasing afflicting the devices.

DOI
10.1103/physrevb.75.165327

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We discuss an implementation of the entanglement witness, a method to detect entanglement with few local measurements, in systems where entangled electrons are generated both in the spin and orbital degrees of freedom. We address the efficiency of this method in various setups, including two different particle-hole entanglement structures, and we demonstrate that it can also be used to infer information on the possible dephasing afflicting the devices.
In this paper we address the problem of detection of entanglement using only few local measurements when some knowledge about the state is given. The idea is based on an optimized decomposition of witness operators into local operators. We discuss two possible ways of optimizing this local decomposition. We present several analytical results and estimates for optimized detection strategies for NPT states of 2 × 2 and N × M systems, entangled states in 3 qubit systems, and bound entangled states in 3 × 3 and 2 × 4 systems.
Entanglement, viz. the non‐separability of quantum states, is a fundamental prediction of quantum mechanics, which is at odds with the classical perception of reality. Furthermore, it constitutes a resource for quantum computation and quantum communication. Electronic degrees of freedom in nanostructures –in particular the spin –constitute promising candidates to implement quantum information architectures in scalable solid state circuits. In this topical review, we will summarize some efforts to create and detect entanglement in such structures. We concentrate first on entanglement in double quantum dots, since they promise to be viable candidates to produce entanglement by confining electrons to a small interaction region. The quantitative detection of the entanglement through transport measurements can be done via current and noise. Secondly, we concentrate on the creation of spin entanglement at quantum point contacts, which has the advantage that the two electrons are automatically spatially separated. We discuss the possibility of performing a Bell test of non‐local correlations. However, as we will point out, a reliable entanglement detection can be performed by current‐correlation measurements, although they require some trust in the experimental setup. Finally, we present a hierarchy of mesoscopic Bell tests, which could be useful to evaluate theoretical proposals and experimental setups.
We propose a scheme for detecting entanglement between two-electron spin qubits in a double quantum dot using an entanglement witness operator. We first calculate the optimal configuration of the two electron spins, defined as the position in the energy level spectrum where, averaged over the nuclear spin distribution, (1) the probability to have two separated electrons and (2) the degree of entanglement of the quantum state quantified by the concurrence are both large. Using a density matrix approach, we then calculate the evolution of the expectation value of the witness operator for the two-spin singlet state, taking into account the effect of decoherence due to quantum charge fluctuations modeled as a boson bath. We find that, for large interdot coupling, it is possible to obtain a highly entangled and robust ground state.
Entanglement witnesses are invaluable for efficient quantum entanglement certification without the need for expensive quantum state tomography. Yet, standard entanglement witnessing requires multiple measurements and its bounds can be elusive as a result of experimental imperfections. Here, we introduce and demonstrate a novel procedure for entanglement detection which simply and seamlessly improves any standard witnessing procedure by using additional available information to tighten the witnessing bounds. Moreover, by relaxing the requirements on the witness operators, our method removes the general need for the difficult task of witness decomposition into local observables. We experimentally demonstrate entanglement detection with our approach using a separable test operator and a simple fixed measurement device for each agent. Finally, we show that the method can be generalized to higher-dimensional and multipartite cases with a complexity that scales linearly with the number of parties.
Entanglement is a central concept in quantum information and a key resource for many quantum protocols. In this work we propose and analyze a class of entanglement witnesses that detect the presence of entanglement in subsystems of experimental multiqubit stabilizer states. The witnesses we propose can be decomposed into sums of Pauli operators and can be efficiently evaluated by either two measurement settings only or at most a number of measurements that only depends on the size of the subsystem of interest. We provide two constructive methods to design the local witness operators, the first one based on the local unitary equivalence between graph and stabilizer states, and the second one based on sufficient and necessary conditions that the respective set of constituent Pauli operators needs to fulfill. We theoretically establish the noise tolerance of the proposed witnesses and benchmark their practical performance by analyzing the local entanglement structure of an experimental seven-qubit quantum error correction code.