|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in the area of quantum [[information geometry]], the '''Bures metric'''<ref>D. Bures, (1969) ''Trans. Am. Math. Soc.'' '''135''', p.199.</ref> or '''Helstrom metric'''<ref>C.W. Helstrom, (1967) "Minimum mean-squared error of estimates in quantum statistics", ''Phys. Lett.'' '''A 25''' pp.101-102.</ref> defines the infinitesimal distance between [[density matrix]] operators defining [[quantum state]]s. It is a quantum generalization of the [[Fisher information metric]], and is identical to the [[Fubini-Study metric]]<ref>Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia "[http://arxiv.org/abs/1009.5219 Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics]" (2010), ''Physics Letters'' '''A 374''' pp. 4801. DOI: 10.1016/j.physleta.2010.10.005</ref> (the Fubini-Study metric normally being written for pure states, not mixed states).
| | The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. A Wordpress plugin is a software that you can install into your Wordpress site. In case you liked this information and you wish to be given more details regarding [http://GET7.pw/wordpress_dropbox_backup_457390 wordpress backup plugin] i implore you to go to our site. You can up your site's rank with the search engines by simply taking a bit of time with your site. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs. Article Source: Stevens works in Internet and Network Marketing. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. It's now become a great place to sell it thanks to Woo - Commerce. Make sure you have the latest versions of all your plugins are updated. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. I don't want that for my visitors and I'm quite sure they don't either. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. You should stay away from plugins that are full of flaws and bugs. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals. |
| | |
| ==Definition==
| |
| The metric may be defined as
| |
| :<math>
| |
| [d(\rho, \rho+d\rho)]^2 = \frac{1}{2}\mbox{tr}( d \rho G ),
| |
| </math>
| |
| where <math>G</math> is Hermitian 1-form operator implicitly given by
| |
| :<math>
| |
| \rho G + G \rho = d \rho^{\,}
| |
| </math>
| |
| | |
| Some of the applications of the Bures metric include that given a target error, it allows the calculation of the minimum number of measurements to distinguish two different states {{ref|distinct}} and the use of the volume element as a candidate for the [[Jeffreys prior]] probability density {{ref|priorprob}} for mixed quantum states.
| |
| | |
| ==Bures distance==
| |
| The Bures distance is the finite version of the infinitesimal square distance described above and is given
| |
| by | |
| :<math>
| |
| D_B(\rho_1,\rho_2)^2 = 2(1-\sqrt{F(\rho_1,\rho_2)}),
| |
| </math>
| |
| where the fidelity function is defined
| |
| as {{ref|fidelityconvention}}
| |
| | |
| :<math>
| |
| F(\rho_1,\rho_2) = \left[ \mbox{tr}( \sqrt{ \sqrt{\rho_1}\rho_2\sqrt{\rho_1}})\right]^2
| |
| </math> | |
| Another associated function is the Bures arc also known as Bures angle, Bures length or [[quantum angle]], defined as
| |
| :<math>
| |
| D_A(\rho_1,\rho_2) = \arccos \sqrt{F(\rho_1,\rho_2)},
| |
| </math>
| |
| which is a measure of the '''statistical distance'''{{ref|statdist}} between the quantum states.
| |
| | |
| ==Quantum Fisher information==
| |
| The Bures metric can be seen as the quantum equivalent of the Fisher information metric and can be rewritten in terms of the variation of coordinate parameters as
| |
| :<math> | |
| [d(\rho, \rho+d\rho)]^2 = \frac{1}{2}
| |
| \mbox{tr}\left( \frac{d \rho}{d \theta^{\mu}} L_{\nu} \right) d \theta^{\mu} d\theta^{\nu},
| |
| </math>
| |
| where <math>L_\mu</math> is the Symmetric Logarithmic Derivative operator (SLD) defined from
| |
| :<math>
| |
| \frac{\rho L_{\mu} + L_{\mu} \rho}{2} = \frac{d \rho^{\,}}{d \theta^{\mu}}.
| |
| </math>
| |
| | |
| In this way, one has
| |
| :<math>
| |
| [d(\rho, \rho+d\rho)]^2 =
| |
| \frac{1}{2} \mbox{tr}\left[ \rho \frac{L_{\mu} L_{\nu} + L_{\nu} L_{\mu}}{2} \right] d \theta^{\mu} d\theta^{\nu}
| |
| </math>
| |
| where the quantum Fisher metric (tensor components) is identified as
| |
| :<math>
| |
| J_{\mu \nu} = \mbox{tr}\left[ \rho \frac{L_{\mu} L_{\nu} + L_{\nu} L_{\mu}}{2}\right].
| |
| </math>
| |
| | |
| The definition of the SLD implies that the quantum Fisher metric is 4 times the Bures metric. In other words, given that <math>g_{\mu\nu}</math> are components of the Bures metric tensor, one has
| |
| :<math>
| |
| J_{\mu\nu}^{ } = 4 g_{\mu \nu}
| |
| </math>
| |
| | |
| As it happens with the classical Fisher information metric, the quantum Fisher metric can be used to find the [[Cramér–Rao bound]] of the [[covariance]].
| |
| | |
| ==Explicit formulas==
| |
| The actual computation of the Bures metric is not evident from the definition, so, some formulas were developed for that purpose. Dittmann obtained the following formulas for the quadratic form of the Bures metric, valid for 2x2 and 3x3 systems, respectively
| |
| :<math>
| |
| [d(\rho, \rho+d\rho)]^2 =
| |
| \frac{1}{4}\mbox{tr}\left[ d \rho d \rho + \frac{1}{\det(\rho)}(\mathbf{1}-\rho)d\rho (\mathbf{1}-\rho)d\rho \right]
| |
| </math>
| |
| | |
| :<math>
| |
| [d(\rho, \rho+d\rho)]^2 =
| |
| \frac{1}{4}\mbox{tr}\left[ d \rho d \rho + \frac{3}{1-\mbox{tr} \rho^3} (\mathbf{1}-\rho)d\rho (\mathbf{1}-\rho)d\rho
| |
| + \frac{3 \det{\rho} }{1-\mbox{tr} \rho^3} (\mathbf{1}-\rho^{-1})d\rho (\mathbf{1}-\rho^{-1})d\rho
| |
| \right]
| |
| </math>
| |
| | |
| Another important formula is the one found by Hübner. This formula is written in terms of the eigenvectors and eigenvalues of the density matrix and reads
| |
| :<math>
| |
| [d(\rho, \rho+d\rho)]^2 = \frac{1}{2} \sum_{j,k=1}^{n} \frac{|\langle j| d\rho | k\rangle |^2}{\lambda_j+\lambda_k}.
| |
| </math>
| |
| | |
| ==Two-level system==
| |
| The state of a two-level system can be parametrized with three variables as
| |
| :<math> | |
| \rho = \frac{1}{2}( \sigma_0 + x^1 \sigma_1 + x^2 \sigma_2 + x^3 \sigma_3 )
| |
| </math>
| |
| with <math> (x^1)^2 + (x^2)^2 + (x^3)^2 \le 1 </math>.
| |
| The components of the Bures metric in this parametrization can be calculated as
| |
| :<math>
| |
| g_{jk} = \frac{1}{4(1 - (x^1)^2 - (x^2)^2 - (x^3)^2)}
| |
| \begin{pmatrix}
| |
| 1 - (x^2)^2 - (x^3)^2 & x^1 x^2 & x^1 x^3 \\
| |
| x^1 x^2 & 1 - (x^1)^2 - (x^3)^2 & x^2 x^3 \\
| |
| x^1 x^3 & x^2 x^3 & 1 - (x^1)^2 - (x^2)^2
| |
| \end{pmatrix}
| |
| </math>.
| |
| The Bures measure can be calculated by taking the square root of the determinant to find
| |
| :<math>
| |
| dV_B = \frac{dx^1 dx^2 dx^3}{8\sqrt{ 1 - (x^1)^2 - (x^2)^2 - (x^3)^2 }},
| |
| </math> | |
| | |
| which can be used to calculate the Bures volume as
| |
| :<math>
| |
| V_B = \int_{-1}^{1}dx^1 \int_{-\sqrt{1-(x^1)^2}}^{ \sqrt{1-(x^1)^2} }dx^2
| |
| \int_{-\sqrt{1-(x^1)^2-(x^2)^2}}^{\sqrt{1-(x^1)^2-(x^2)^2}}dx^3
| |
| \frac{1}{8\sqrt{ 1 - (x^1)^2 - (x^2)^2 - (x^3)^2 }} = \frac{\pi^2}{8}
| |
| </math>
| |
| | |
| ==See also==
| |
| * [[Fubini-Study metric]]
| |
| * [[Fidelity of quantum states]]
| |
| * [[Fisher information]]
| |
| * [[Fisher information metric]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| * Uhlmann, A., [http://www.math.utexas.edu/mp_arc/c/92/92-44.ps.gz ''The Metric of Bures and the Geometric Phase''], Quantum Groups and Related Topics: Proceedings of the First Max Born Symposium, Gielerak, R.; Lukierski, J.; Popowicz, Z. (Eds.), pp 267, 1992.
| |
| | |
| * {{note|distinct}} Braunstein, Samuel L. and Caves, Carlton M., ''Statistical distance and the geometry of quantum states'', Phys. Rev. Lett.,'''72''', 22, 1994.
| |
| | |
| * Sommers, H.J. and Zyczkowski, K., [http://chaos.if.uj.edu.pl/~karol/pdf/SZ03.pdf ''Bures volume of the set of mixed quantum states''], Journal of Physics A,'''36''', 39, 2003.
| |
| | |
| * Dittmann, J., [http://www.heldermann-verlag.de/jlt/jlt03/DITTLAT.PDF ''On the Riemannian Geometry of Finite Dimensional Mixed States''], Seminar Sophus Lie, '''73''', 1993.
| |
| | |
| * Dittmann J., ''Explicit formulae for the Bures metric'', Journal of Physics A, '''32''', 14, 1999.
| |
| | |
| * Hübner, M., '' Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space'', Phys. Lett. A, '''179''', 4-5, 1993.
| |
| | |
| * {{note|priorprob}} Paul B Slater, ''Applications of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems '', J. Math. Phys. 37, 2682, 1996.
| |
| | |
| * Paul B Slater, ''Quantum Fisher-Bures information of two-level systems and a three-level extension'', J. Phys. A: Math. Gen., '''29''', No 10, 1995.
| |
| | |
| * M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
| |
| | |
| * {{note|fidelityconvention}} Unfortunately, some authors use a different definition as <math>
| |
| F(\rho_1,\rho_2) = \mbox{tr}( \sqrt{ \sqrt{\rho_1}\rho_2\sqrt{\rho_1}}) </math>
| |
| | |
| * {{note|statdist}} Wootters WK., [http://link.aps.org/doi/10.1103/PhysRevD.23.357 ''Statistical distance and Hilbert space''], Phys. Rev. D, '''23''', 357 (1981)
| |
| | |
| [[Category:Quantum mechanics]]
| |
| [[Category:Quantum information science]]
| |
The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. It is used by around 25% of all new websites, and there are more than 27 thousand plugins currently available. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster.
Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. A Wordpress plugin is a software that you can install into your Wordpress site. In case you liked this information and you wish to be given more details regarding wordpress backup plugin i implore you to go to our site. You can up your site's rank with the search engines by simply taking a bit of time with your site. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links.
Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs. Article Source: Stevens works in Internet and Network Marketing.
Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. It's now become a great place to sell it thanks to Woo - Commerce. Make sure you have the latest versions of all your plugins are updated.
A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. I don't want that for my visitors and I'm quite sure they don't either. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. You should stay away from plugins that are full of flaws and bugs. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.