弱测量及量子测量相关问题研究
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摘要
量子力学是现代物理学的理论基础,其中量子测量问题则是量子力学的基本问题。对量子测量问题的研究不仅能加深我们对物理世界的认识,推动物理学本身的发展,而且可以由此发展出新的实验方法,从而促进实验技术的提高。本论文主要讨论量子测量问题,具体针对弱测量问题、信息获取量与测量耦合强度的关系、量子系综的经典性和量子性问题进行研究。主要的研究内容有如下几方面：
1.简要描述量子测量的一般理论,并对弱测量理论的发展现状及应用做必要的介绍。现有的弱测量理论,由于在推导弱测量结论的时候使用了近似,使其适用范围受到了限制,例如当先选择态|Ψi)和后选择态|Ψi)接近或者正交的时候,现有的弱测量的结论将不再成立。所以需要比现有理论更加精确的弱测量理论。本文在研究弱测量问题的时候,没有使用任何近似,并抛弃了弱值的概念。我们给出的结论不仅对弱耦合强度的情况成立,而且对强耦合强度的情况也成立。我们可以通过变化耦合强度,来得到弱耦合和强耦合时的测量结果,并发现：当耦合强度弱的情况下,我们可以通过选取合适的先后选择态,来得到明显放大的仪器指针偏移量；当耦合强度强的时候,我们不能通过先后选择得到任何放大效应。对于测量系统为qubit的情况,我们解析地给出了测量结果的极值,并发现这些极值在不同的耦合强度下对仪器和系统的依赖关系是不一样的。
2.对在引入弱测量的方法下实验中信噪比的提高问题进行了研究,并给出了具体的计算方法。基于对Stern-Gerlach实验模型的研究,我们得到在考虑由后选择带来的实验数据数目降低的情况下,性噪比并不能得到明显提高的结论。在那些不会因后选择而带来实验数据降低的实验中,引入弱测量的方法能明显提高实验的性噪比。我们还考察了弱测量的方法对实验测量灵敏度的提高,得到与性噪比提高相似的结果：在考虑后选择引起实验数据降低的情况下,测量的灵敏度不会得到明显提高；在不考虑后选择引起实验数据降低的情况下,实验的测量灵敏度可以得到明显的提高。
3.研究了测量过程中的耦合强度和信息获取量之间的关系,基于量子信息理论中互信息的方法,我们给出了信息获取量以耦合强度为参量的具体计算方法。对于测量仪器的态为高斯波包类型的情况,我们给出了信息获取量具体的计算公式。并得到在耦合强度比较大的情况下,信息的获取量等于在进行正交投影测量下获取的信息量。在信息载体为qubit的情况下,我们证明信息的获取量随着耦合强度的增加单调递增。我们给出了沿互相垂直的方向上的进行测量获取的信息量之间的一个互补关系,即在两个互相垂直的方向上获取信息量的总和小于某一个上限。
4.利用正交投影测量的方法定义一个量J来衡量一个量子系综的经典性,J的大小告诉我们一个系综有多像一个经典系综。如果一个系综的经典性J=1,那么这个系综在某些情况下可以当成一个经典系综来处理；当一个系综的J<1,那么这个系综将不再能当成一个经典系综来处理。我们发现如果一个系综的经典性越大,那么来自于这个系综的未知态的近似克隆效果越好。一个系综的经典性的数值J给我一个衡量一些经典操纵(比如：态克隆,态删除,态区分)在该系综上能完成的最好程度。在此基础上我们定义了系综的量子性Q=1一J,令人惊奇的是Q刚好是用这个系综进行量子密钥分配时的误码率(error rate)的下限,因此这个量可以给我们在选择密钥分配的系综时提供一个重要参考。
Quantum mechanics is the base of the modern physical theories, and the problem of quantum measurement is one of foundational problems of quantum mechanics. The re-searches on quantum measurement can improve our understanding of the objective word and provide advanced technologies for our experiments. This paper is focused on the problems of weak measurements, the relationship between the coupling strength and the information gain in the measurement, the classicality and quantumness of a quantum en-semble. The main points are summarized as follows:
1. The theory of quantum measurement is briefly described, and the current the-ories and applications of weak measurements are introduced in this paper. Due to the approximations used in deriving the results of weak measurement, the existed measure-ment results in the previous literature are not valid when the pre-selected state|ψi> and the post-selected state|ψi> are approximatively orthogonal to each other. Thus, we need some more precise descriptions of weak measurements. In this paper, we calculate the shifts of the pointers without any approximation. Without using the concept of weak val-ue, the expressions we obtained are not only valid for the weak interaction cases, but also legitimate for the strong interaction cases. By changing the strength of the interaction, we find:when the coupling strength is weak, the shift of pointer with postselection can be can be much larger than the shift obtained without postselection, and the amplification effect disappears when the interaction strength is strong. We also give the maximum shifts of the measuring device for the two-dimensional systems by choosing appropriate PPS.
2. The improvement of the signal-to-noise ratio (SNR) by weak measurements is studied. Based on the researches of Stern-Gerlach experiment, we find:without consider-ing the probability of obtaining the outcomes decrease due to postselection, the SNR can be significantly improved by weak measurements; neither SNR cannot be effectively im-proved when the probability decrease is considered. We also study the the enhancement of the measure sensitivity (MS) by weak measurements, similarly with the SNR, the MS cannot be efficiently enhanced if the probability decrease due to postselection is taken into account; the MS can be efficiently enhanced if the probability should not be taken into account.
3. We investigate the relationships between the information gain and the interaction strength between the quantum systems and measuring apparatus. Based on the theories of quantum information, a general method is proposed to calculate the information gain of the measuring apparatus. The explicit expressions are obtain for the state of the mea-suring apparatus is a Gaussian type. For qubit systems, we prove that the information gain of the measuring apparatus increases monotonically with the coupling strength. A complementarity of information gain is given for the qubit systems in this paper.
4. In this paper, we propose a quantity J, the ensemble classicality based on projec-tive measurement strategy, to measure the classicality of a given ensemble. The quantity J can tell how classical an ensemble is. When J=1the ensemble behaves like a purely classical ensemble; and when J<1the ensemble cannot be considered as a classical en-semble anymore. We have revealed that the more classical an ensemble is, the better an unknown state from the ensemble can be cloned. The quantity of J provides us with a tool to evaluate how well classical tasks such as cloning, deleting, and distinguishing could be accomplished for quantum ensembles. We also define the quantumness Q of an ensemble and we surprisingly find that the quantumness Q of an ensemble used in quantum key distribution is exactly the attainable lower bound of the error rate.
1.简要描述量子测量的一般理论,并对弱测量理论的发展现状及应用做必要的介绍。现有的弱测量理论,由于在推导弱测量结论的时候使用了近似,使其适用范围受到了限制,例如当先选择态|Ψi)和后选择态|Ψi)接近或者正交的时候,现有的弱测量的结论将不再成立。所以需要比现有理论更加精确的弱测量理论。本文在研究弱测量问题的时候,没有使用任何近似,并抛弃了弱值的概念。我们给出的结论不仅对弱耦合强度的情况成立,而且对强耦合强度的情况也成立。我们可以通过变化耦合强度,来得到弱耦合和强耦合时的测量结果,并发现：当耦合强度弱的情况下,我们可以通过选取合适的先后选择态,来得到明显放大的仪器指针偏移量；当耦合强度强的时候,我们不能通过先后选择得到任何放大效应。对于测量系统为qubit的情况,我们解析地给出了测量结果的极值,并发现这些极值在不同的耦合强度下对仪器和系统的依赖关系是不一样的。
2.对在引入弱测量的方法下实验中信噪比的提高问题进行了研究,并给出了具体的计算方法。基于对Stern-Gerlach实验模型的研究,我们得到在考虑由后选择带来的实验数据数目降低的情况下,性噪比并不能得到明显提高的结论。在那些不会因后选择而带来实验数据降低的实验中,引入弱测量的方法能明显提高实验的性噪比。我们还考察了弱测量的方法对实验测量灵敏度的提高,得到与性噪比提高相似的结果：在考虑后选择引起实验数据降低的情况下,测量的灵敏度不会得到明显提高；在不考虑后选择引起实验数据降低的情况下,实验的测量灵敏度可以得到明显的提高。
3.研究了测量过程中的耦合强度和信息获取量之间的关系,基于量子信息理论中互信息的方法,我们给出了信息获取量以耦合强度为参量的具体计算方法。对于测量仪器的态为高斯波包类型的情况,我们给出了信息获取量具体的计算公式。并得到在耦合强度比较大的情况下,信息的获取量等于在进行正交投影测量下获取的信息量。在信息载体为qubit的情况下,我们证明信息的获取量随着耦合强度的增加单调递增。我们给出了沿互相垂直的方向上的进行测量获取的信息量之间的一个互补关系,即在两个互相垂直的方向上获取信息量的总和小于某一个上限。
4.利用正交投影测量的方法定义一个量J来衡量一个量子系综的经典性,J的大小告诉我们一个系综有多像一个经典系综。如果一个系综的经典性J=1,那么这个系综在某些情况下可以当成一个经典系综来处理；当一个系综的J<1,那么这个系综将不再能当成一个经典系综来处理。我们发现如果一个系综的经典性越大,那么来自于这个系综的未知态的近似克隆效果越好。一个系综的经典性的数值J给我一个衡量一些经典操纵(比如：态克隆,态删除,态区分)在该系综上能完成的最好程度。在此基础上我们定义了系综的量子性Q=1一J,令人惊奇的是Q刚好是用这个系综进行量子密钥分配时的误码率(error rate)的下限,因此这个量可以给我们在选择密钥分配的系综时提供一个重要参考。
Quantum mechanics is the base of the modern physical theories, and the problem of quantum measurement is one of foundational problems of quantum mechanics. The re-searches on quantum measurement can improve our understanding of the objective word and provide advanced technologies for our experiments. This paper is focused on the problems of weak measurements, the relationship between the coupling strength and the information gain in the measurement, the classicality and quantumness of a quantum en-semble. The main points are summarized as follows:
1. The theory of quantum measurement is briefly described, and the current the-ories and applications of weak measurements are introduced in this paper. Due to the approximations used in deriving the results of weak measurement, the existed measure-ment results in the previous literature are not valid when the pre-selected state|ψi> and the post-selected state|ψi> are approximatively orthogonal to each other. Thus, we need some more precise descriptions of weak measurements. In this paper, we calculate the shifts of the pointers without any approximation. Without using the concept of weak val-ue, the expressions we obtained are not only valid for the weak interaction cases, but also legitimate for the strong interaction cases. By changing the strength of the interaction, we find:when the coupling strength is weak, the shift of pointer with postselection can be can be much larger than the shift obtained without postselection, and the amplification effect disappears when the interaction strength is strong. We also give the maximum shifts of the measuring device for the two-dimensional systems by choosing appropriate PPS.
2. The improvement of the signal-to-noise ratio (SNR) by weak measurements is studied. Based on the researches of Stern-Gerlach experiment, we find:without consider-ing the probability of obtaining the outcomes decrease due to postselection, the SNR can be significantly improved by weak measurements; neither SNR cannot be effectively im-proved when the probability decrease is considered. We also study the the enhancement of the measure sensitivity (MS) by weak measurements, similarly with the SNR, the MS cannot be efficiently enhanced if the probability decrease due to postselection is taken into account; the MS can be efficiently enhanced if the probability should not be taken into account.
3. We investigate the relationships between the information gain and the interaction strength between the quantum systems and measuring apparatus. Based on the theories of quantum information, a general method is proposed to calculate the information gain of the measuring apparatus. The explicit expressions are obtain for the state of the mea-suring apparatus is a Gaussian type. For qubit systems, we prove that the information gain of the measuring apparatus increases monotonically with the coupling strength. A complementarity of information gain is given for the qubit systems in this paper.
4. In this paper, we propose a quantity J, the ensemble classicality based on projec-tive measurement strategy, to measure the classicality of a given ensemble. The quantity J can tell how classical an ensemble is. When J=1the ensemble behaves like a purely classical ensemble; and when J<1the ensemble cannot be considered as a classical en-semble anymore. We have revealed that the more classical an ensemble is, the better an unknown state from the ensemble can be cloned. The quantity of J provides us with a tool to evaluate how well classical tasks such as cloning, deleting, and distinguishing could be accomplished for quantum ensembles. We also define the quantumness Q of an ensemble and we surprisingly find that the quantumness Q of an ensemble used in quantum key distribution is exactly the attainable lower bound of the error rate.
引文
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