d | |

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1 1

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d cos2 d cos sin

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d cos sin d sin2

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(313)

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This matrix is obviously positive since, for any nonzero vector |v in the plane, we 1 have v |P ( )|v = d | v | |2 > 0 POVMs are important partners of resolutions of the unity, as will be seen at different places in this book

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324 Analytical Bridge

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The resolution of the identity (34) provided by the coherent states is the key for transforming any abstract or concrete realization of the quantum states into the Fock Bargmann analytical one Let | =

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n |n

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(314)

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be a vector in the Hilbert space H of quantum states in some of its realizations Its scalar product with a Schr dinger coherent state (chosen preferably to be a standard coherent state to avoid the Gaussian weights) reads as the power series

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z | =

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s n=0

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z n def n = s (z) , n!

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(315)

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32 Properties in the Hilbertian Framework

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with in nite convergence radius The series (315) de nes an entire analytical function, s (z), which is square-integrable with respect to the Fock Bargmann measure s (dz) Hence, this function is an element of the Fock Bargmann space F B and we have established a linear map from H into F B This map is an isometry: s

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2 FB

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| s (z)|2 s (dz) | z | |2 s (dz)

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|z z | s (dz) =

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(316)

where we have used the fact that s (dz) = s (d z ) and the resolution of the identity (34)

325 Overcompleteness and Reproducing Properties

The coherent states form an overcomplete family of states in the sense that (i) it is total, which is equivalent to stating that if there exists H such that |z = 0 for all z C, then = 0 Now |z = 0 implies s ( ) = 0 for all z z C So, by analyticity, n = 0 for all n and nally = 0, (ii) at least two of them in the family are not linearly independent Thus, the coherent states do not form a Hilbertian basis of H, but they form a dense family in the Hilbert space H and they resolve the unity An immediate consequence of the latter is their reproducing action on the elements of F B that emerge from the map H s F B: s (z) = z | = z |

s s s s

|z z | s (dz )

z |z s (z ) s (dz )

(317)

Hence, the scalar product z |z = e z z = z |z plays the role of the reproducing kernel in the Fock Bargmann space F B The latter is a reproducing kernel space Note that this object indicates to what extent the coherent states are linearly dependent: |z =

s s s

e z z |z s (dz ) ,

(318) d2z (319)

|z =

e iz z e 2 |z z | |z

This precisely shows that a coherent state is, by itself, a kind of average over all the coherent state family weighted with a Gaussian distribution (up to a phase)

3 The Standard Coherent States: the (Elementary) Mathematics

33 Coherent States in the Quantum Mechanical Context 331 Symbols

As shown in 325, the coherent states form an overcomplete system (often abusively termed overcomplete basis ) or frame that allows one to analyze quantum states or observables from a coherent state point of view Indeed, when we decompose a state | in H as | = z| |z s (dz) =

s ( ) |z s (dz) , z

(320)

z the continuous expansion component s ( ) is an entire (anti-) holomorphic function called symbol On the same footing, the nonanalytical function (z) = e

|z|2 2

s (z) = z |

(321)

is the symbol of | in its decomposition over the family of standard coherent states: | = z| |z d2z = ( ) |z z d2z (322)

Note that the following upper bound, readily derived from the Cauchy Schwarz inequality, | (z)| = | z | | u z|z = , (323)