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Chinese Physics Letters

Fisher Information of Wavefunctions： Classical and Quantum

骆顺龙

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

参量的量机械波浪功能自然地由拿绝对平方导致参量的概率分布，并且我们能考虑它的古典 Fisherinformation。在另一方面，当密度操作员，和我们能谈论它的量菲希尔信息，它也导致可以被看的参量的等级一个设计。在量菲希尔信息的许多版本之中，有二突出的。第一，经由量定义 20 功能，被 Helstrom 在 1967 介绍并且是众所周知的。第二，经由密度操作员的方形的根定义，它在斜信息的起源由 Wigner 和 Yanasein 1963 和遗体介绍了相对不被注意。这研究被奉献给调查在古典菲希尔信息和量菲希尔信息 forwavefunctions 的这二个版本之间的关系。量菲希尔信息的二个版本由一个因素不同，这被显示出第 2a 他们统治古典菲希尔信息。量菲希尔信息的这二个版本的非巧合可以作为量不一致的表明被解释。我们进一步计算 Helstrom 量菲希尔信息和古典 Fisherinformation 之间的差别，并且证明它正是波浪功能的即时阶段变化。

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³ÑÆÇÓÉÂÐËÇ 1. £ÄÇÆÇÐËÇ (651).2. ¯ÇÑÃÓÂÕËÏÑÔÕß (652).3. £ÑÎÐÑÄÞÇ ×ÖÐÍÙËË ÂÕÑÏÑÄ ÅÂÊ (652).4. ¤ÂÊ ÍÂÍ ËÊÏÇÓËÕÇÎßÐÞÌ ÒÓËÃÑÓ (654). 5. ¬ÄÂÐÕÑÄÂâ ÍÑÏÏÖÐËÍÂÙËâ (655).6. ©ÂÍÎáÚÇÐËÇ (658).³ÒËÔÑÍ ÎËÕÇÓÂÕÖÓÞ (659).1. £ÄÇÆÇÐËÇ ±ÓË ÑÃÝâÔÐÇÐËË ÒÓËÐÙËÒÑÄ ÍÄÂÐÕÑÄÑÌ ÕÇÑÓËË Ë ÇÇ ÔÕÂÕËÔÕËÚÇÔÍÑÅÑ ØÂÓÂÍÕÇÓ (ÔÏ., ÐÂÒÓËÏÇÓ, [1]) ÐÇÓÇÆÍÑ ËÔÒÑÎßÊÖÇÕÔâ ÔÎÇÆÖáÜËÌ ÒÓÑÔÕÑÌ ÒÓËÏÇÓ.±ÖÔÕß ËÏÇÇÕÔâ ÒÑÔÇÓÇÃÓÇÐÐÂâ ÔÕÇÍÎâÐÐÂâ ÒÎÂÔÕËÐÍÂ, ÍÑÕÑÓÂâ ÒÓË ÒÂÆÇÐËË Ð ÐÇÇ ÔÄÇÕÑÄÑÅÑ ÒÖÚÍ ÒÓÑÒÖÔÍÂÇÕ Ë ÑÕÓÂÉÂÇÕ ÓÑÄÐÑ ÒÑÎÑÄËÐÖ ËÔØÑÆÐÑÌ ËÐÕÇÐÔËÄÐÑÔÕË.¥ÑÒÖÔÕËÏ ÕÇÒÇÓß, ÚÕÑ Ð ÕÂÍÖá ÒÎÂÔÕËÐÍÖ ÒÂÆÂÇÕ ÑÆËÐ ÇÆËÐÔÕÄÇÐÐÞÌ ×ÑÕÑÐ.¦ÅÑ ÄÑÎÐÑÄÂâ ×ÖÐÍÙËâ ÇÔÕÇÔÕÄÇÐÐÑ ÓÂÔÜÇÒÎâÇÕÔâ РÑÕÓÂÉÇÐÐÖá Ë ÒÓÑØÑÆâÜÖá ÄÑÎÐÞ.¯Ñ ÇÔÎË Ð ÒÖÕË àÕËØ ÄÑÎÐ ÖÔÕÂÐÑÄËÕß ÆÄ ×ÑÕÑÆÇÕÇÍÕÑÓÂ, ÕÑ ÔÓÂÃÑÕÂÇÕ ÕÑÎßÍÑ ÑÆËÐ ËÊ ÐËØ: ×ÑÕÑÐ ÑÍÂÉÇÕÔâ ÎËÃÑ ÔÒÓÂÄÂ, ÎËÃÑ ÔÎÇÄ ÑÕ ÒÎÂÔÕËÐÍË, Õ.Ç.ÑÐ ÎËÃÑ ÑÕÓÂÊËÕÔâ, ÎËÃÑ ÒÓÑÌÆÇÕ ÔÍÄÑÊß ÒÎÂÔÕËÐÍÖ.²ÇÅËÔÕÓÂÙËâ ×ÑÕÑРÒÓÇÆÔÕÂÄÎâÇÕ ÔÑÃÑÌ ÔÎÖÚÂÌÐÞÌ ÒÓÑÙÇÔÔ Ô ÄÇÓÑâÕÐÑÔÕßá ÓÇÅËÔÕÓÂÙËË Ä ÍÂÉÆÑÏ ËÊ ÆÇÕÇÍÕÑÓÑÄ, ÓÂÄÐÑÌ 1/2.©ÆÇÔß ÏÞ ÄÔÕÓÇÚÂÇÏÔâ Ô ÕËÒËÚÐÞÏ ÒÓËÏÇÓÑÏ ÍÑÎ-ÎÂÒÔ ÄÑÎÐÑÄÑÌ ×ÖÐÍÙËË.£ ÏÑÏÇÐÕ ÓÇÅËÔÕÓÂÙËË ×ÑÕÑÐÂ, ÕÑÚÐÇÇ, Ä ÕÇÚÇÐËÇ ÑÚÇÐß ÍÑÓÑÕÍÑÅÑ ÒÓÑÏÇÉÖÕÍ ÄÓÇÏÇÐË ÔÓÂÃÂÕÞÄÂÐËâ ×ÑÕÑÆÇÕÇÍÕÑÓ ÄÑÎÐÑÄÂâ ×ÖÐÍÙËâ ×ÑÕÑРÖÐËÚÕÑÉÂÇÕÔâ ÄÑ ÄÔÇÏ ÒÓÑÔÕÓÂÐÔÕÄÇ ÄÐÇ ÆÇÕÇÍ-ÕÑÓÂ,  ÄÐÖÕÓË ÆÇÕÇÍÕÑÓ ×ÑÕÑÐ ÕÂÍÉÇ ËÔÚÇÊÂÇÕ

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参量的量机械波浪功能自然地由拿绝对平方导致参量的概率分布，并且我们能考虑它的古典 Fisherinformation。在另一方面，当密度操作员，和我们能谈论它的量菲希尔信息，它也导致可以被看的参量的等级一个设计。在量菲希尔信息的许多版本之中，有二突出的。第一，经由量定义 20 功能，被 Helstrom 在 1967 介绍并且是众所周知的。第二，经由密度操作员的方形的根定义，它在斜信息的起源由 Wigner 和 Yanasein 1963 和遗体介绍了相对不被注意。这研究被奉献给调查在古典菲希尔信息和量菲希尔信息 forwavefunctions 的这二个版本之间的关系。量菲希尔信息的二个版本由一个因素不同，这被显示出第 2a 他们统治古典菲希尔信息。量菲希尔信息的这二个版本的非巧合可以作为量不一致的表明被解释。我们进一步计算 Helstrom 量菲希尔信息和古典 Fisherinformation 之间的差别，并且证明它正是波浪功能的即时阶段变化。
由在几何光学和古典统计力学的运动的方程之间的比较，这篇论文发现应该在几何光学和古典统计力学之间有类比而不是几何力学和古典力学。由在量力学和古典统计力学的古典限制之间的比较，而且，它发现量力学的那古典限制是古典统计力学不古典的力学，因此，它证明量力学是古典统计力学的自然 generahzation 而不是古典力学。在它的真外观的 Thencequantum 力学是一个波浪统计力学而不是波浪力学。
A parametric quantum mechanical wavefunction naturally induces parametric probability distributions by taking absolute square, and we can consider its classical Fisher information. On the other hand, it also induces parametric rank-one projections which may be viewed as density operators, and we can talk about its quantum Fisher information. Among many versions of quantum Fisher information, there are two prominent ones. The first, defined via a quantum score function, was introduced by Helstrom in 1967 and is well known. The second, defined via the square root of the density operator, has its origin in the skew information introduced by Wigner and Yanase in 1963 and remains relatively unnoticed. This study is devoted to investigating the relationships between the classical Fisher information and these two versions of quantum Fisher information for wavefunctions. It is shown that the two versions of quantum Fisher information differ by a factor 2 and that they dominate the classical Fisher information. The non-coincidence of these two versions of quantum Fisher information may be interpreted as a manifestation of quantum discord. We further calculate the difference between the Helstrom quantum Fisher information and the classical Fisher information, and show that it is precisely the instantaneous phase fluctuation of the wavefunctions.
A closed-form expression for Wigner–Yanase skew information in mixed-state quantum systems is derived. It is shown that limit values of the mixing coefficients exist such that Wigner–Yanase information is equal to Helstrom information. The latter constitutes an upper bound for the classical expected Fisher information, hence the inverse Wigner–Yanase information provides an approximate lower bound to the variance of an unbiased estimator of the parameter of interest. The advantage of approximating Helstrom’s sharp bound lies in the fact that Wigner–Yanase information is straightforward to compute, while it is often very difficult to obtain a feasible expression for Helstrom information. In fact, the latter requires the solution of an implicit second order matrix differential equation, while the former requires just scalar differentiation.
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In classical statistics,the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities.In quantum theory,this uniqueness breaks down,and there are many natural quantum analogues of the Fisher information,among which two particular versions distinguish themselves by their intuitive and informational significance:The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement,and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967,and is defined via the symmetric logarithmic derivative.The aim of this paper is to compare these two versions of quantum Fisher information,and to establish two informational inequalities relating them.