Quantum entanglement for systems of identical bosons: II. Spin squeezing and other entanglement tests
Quantum entanglement for systems of identical bosons: II. Spin squeezing and other entanglement tests
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DOI：
10.1088/1402-4896/92/2/023005
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发表时间：
2017-02-01
期刊：
影响因子：
2.9
通讯作者：
Reid, M. D.
中科院分区：
文献类型：
--
作者：
Dalton, B. J.;Goold, J.;Reid, M. D.
这两篇论文讨论了全同质量玻色子系统的纠缠以及纠缠与自旋压缩和其他量子关联效应的关系。研究的重点是双模纠缠，但也考虑了多模纠缠。玻色子可以是原子或分子，就像在冷量子气体中一样。在前一篇论文中，我讨论了量子纠缠的一般特征及其在全同玻色子系统中的具体定义。纠缠是两个（或多个）量子子系统之间共享的属性。在定义相同质量粒子系统的纠缠时，得出的结论是，单粒子态或模式是可区分的子系统的最合适的选择，一般量子态必须符合对称化原理和超级选择规则（SSR），禁止具有不同总粒子数的状态的量子叠加（全局SSR遵守）。此外，得出的结论是，（在可分离的状态）量子叠加的子系统状态与不同的子系统粒子数（本地SSR顺应性）也不会发生。本论文II根据论文I中对纠缠的处理确定了可能的纠缠检验。先前已经提出了几个不等式，涉及方差和平均值的运营商作为测试两个子系统之间的纠缠。这些不等式通常涉及模式湮灭和创建运营商，并包括定义自旋压缩的不等式。本文研究了双模系统的自旋压缩准则，并考虑了协方差矩阵为对角阵的主自旋算符分量的自旋压缩。证明，这是基于我们的SSR兼容的方法表明，在任何一个自旋分量的自旋压缩的存在下，需要相关的模式对纠缠。一个简单的布洛赫矢量测试纠缠也被推导出来。因此，我们表明，自旋压缩成为一个严格的测试纠缠在一个系统的大质量玻色子，当被视为一个测试两个模式之间的纠缠。此外，其他先前提出的测试纠缠涉及自旋运营商被认为是，包括那些基于两个自旋组件的方差之和。当应用基于对称化和SSR准则的纠缠概念时，所有的测试仍然有效。这些测试也适用于多模纠缠的情况下，虽然在每个子系统由成对的模式组成的情况下有限制。涉及量子相关函数的测试也被认为是全球SSR兼容状态，这些被证明是等价的测试涉及自旋算子。一个新的弱相关性测试可分离态的本地SSR遵守的基础上得出的纠缠，补充了以前得到的强相关性测试时，这被忽略。Bloch向量检验等价于这种弱相关性检验的一种情况。正交压缩单模也检查，但没有发现产生一个有用的纠缠测试，而两个模式的正交压缩被证明是一个有效的纠缠测试，虽然没有作为有用的布洛赫矢量测试。各种纠缠测试被认为是众所周知的纠缠态，如二项式态，相对相位本征态和NOON态-本文第二部分概述了一个简单的双模干涉仪的理论，指出这种干涉仪可以用来测量涉及到的自旋算符的平均值和协方差矩阵，并指出这种干涉仪可以用来测量自旋算符的平均值和协方差矩阵。双模玻色子系统的纠缠实验。治疗也广义覆盖多模干涉。该干涉仪涉及一个脉冲的经典场，其特征在于由一个相位变量和一个面积变量定义的场振幅的时间积分，并导致两个模式之间的耦合。为了简单起见，中心频率被选择为与模间过渡频率谐振。测量的平均值和方差的两种模式之间的人口差异的干涉仪的输出状态的各种选择的干涉仪变量示出，使平均值和协方差矩阵的自旋算符的输入量子态的两个模式系统被确定。文章最后讨论了几篇关于自旋压缩的重要实验论文。
These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum subsystems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states-sometimes the new tests are satisfied while than those obtained in other papers are not.The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing.