Entropy - Revision history

Nathaniel: Fix latex-isms

2023-08-04T16:59:44Z

Fix latex-isms

← Older revision		Revision as of 16:59, 4 August 2023
Line 12:		Line 12:
	: <math>S_\alpha(\rho) := \frac{1}{1-\alpha}\log_2\big(\mathrm{Tr}(\rho^\alpha)\big)</math>		: <math>S_\alpha(\rho) := \frac{1}{1-\alpha}\log_2\big(\mathrm{Tr}(\rho^\alpha)\big)</math>

	(i.e., the Rényi-$\alpha$ entropy).		(i.e., the Rényi-<math>\alpha</math> entropy).

	==Syntax==		==Syntax==
Line 35:		Line 35:
	</syntaxhighlight>		</syntaxhighlight>

	A d-by-d maximally-mixed state has entropy $\log_2(d)$:		A d-by-d maximally-mixed state has entropy <math>\log_2(d)</math>:
	<syntaxhighlight>		<syntaxhighlight>
	>> Entropy(eye(4)/4)		>> Entropy(eye(4)/4)
Line 54:		Line 54:

	==Notes==		==Notes==
	The Rényi-$\alpha$ entropy approaches the von Neumann entropy as $\alpha \rightarrow 1$.		The Rényi-<math>\alpha</math> entropy approaches the von Neumann entropy as <math>\alpha \rightarrow 1</math>.

	{{SourceCode\|name=Entropy}}		{{SourceCode\|name=Entropy}}

Nathaniel: Updated to include Renyi entropy functionality

2014-11-27T21:46:15Z

Updated to include Renyi entropy functionality

← Older revision		Revision as of 21:46, 27 November 2014
Line 1:		Line 1:
	{{Function		{{Function
	\|name=Entropy		\|name=Entropy
	\|desc=Computes the von Neumann entropy of a density matrix		\|desc=Computes the von Neumann or Rényi entropy of a density matrix
	\|cat=[[List of functions#Information_theory\|Information theory]]		\|cat=[[List of functions#Information_theory\|Information theory]]
	\|upd=~~September 9~~, 2014		\|upd=November 27, 2014}}
	~~\|v=0.50~~}}		<tt>'''Entropy'''</tt> is a [[List of functions\|function]] that computes the von Neumann entropy or Rényi entropy of a density matrix. That is, given a density matrix $\rho$, it computes the following quantity:
	<tt>'''Entropy'''</tt> is a [[List of functions\|function]] that computes the von Neumann entropy of a density matrix. That is, given a density matrix $\rho$, it computes the following quantity:

	: <math>S(\rho) := -\mathrm{Tr}\big(\rho\log_2(\rho)\big).</math>		: <math>S(\rho) := -\mathrm{Tr}\big(\rho\log_2(\rho)\big)</math>

			(i.e., the von Neumann entropy) or the following quantity:

			: <math>S_\alpha(\rho) := \frac{1}{1-\alpha}\log_2\big(\mathrm{Tr}(\rho^\alpha)\big)</math>

			(i.e., the Rényi-$\alpha$ entropy).

	==Syntax==		==Syntax==
	* <tt>ENT = Entropy(RHO)</tt>		* <tt>ENT = Entropy(RHO)</tt>
	* <tt>ENT = Entropy(RHO,BASE)</tt>		* <tt>ENT = Entropy(RHO,BASE)</tt>
			* <tt>ENT = Entropy(RHO,BASE,ALPHA)</tt>

	==Argument descriptions==		==Argument descriptions==
	* <tt>RHO</tt>: A density matrix to have its entropy computed.		* <tt>RHO</tt>: A density matrix to have its entropy computed.
	* <tt>BASE</tt> (optional, default 2): The base of the logarithm used in the entropy calculation.		* <tt>BASE</tt> (optional, default 2): The base of the logarithm used in the entropy calculation.
			* <tt>ALPHA</tt> (optional, default 1): A non-negative real parameter that determines which entropy is computed (<tt>ALPHA = 1</tt> corresponds to the von Neumann entropy, otherwise the Rényi-<tt>ALPHA</tt> entropy is computed).

	==Examples==		==Examples==
Line 45:		Line 52:
	1.6157		1.6157
	</syntaxhighlight>		</syntaxhighlight>

			==Notes==
			The Rényi-$\alpha$ entropy approaches the von Neumann entropy as $\alpha \rightarrow 1$.

	{{SourceCode\|name=Entropy}}		{{SourceCode\|name=Entropy}}

Nathaniel: Created page with "{{Function |name=Entropy |desc=Computes the von Neumann entropy of a density matrix |cat=Information theory |upd=September 9, 2014 |v=..."

2014-10-15T16:25:02Z

Created page with "{{Function |name=Entropy |desc=Computes the von Neumann entropy of a density matrix |cat=Information theory |upd=September 9, 2014 |v=..."

New page

{{Function
|name=Entropy
|desc=Computes the von Neumann entropy of a density matrix
|cat=[[List of functions#Information_theory|Information theory]]
|upd=September 9, 2014
|v=0.50}}
<tt>'''Entropy'''</tt> is a [[List of functions|function]] that computes the von Neumann entropy of a density matrix. That is, given a density matrix $\rho$, it computes the following quantity:

: <math>S(\rho) := -\mathrm{Tr}\big(\rho\log_2(\rho)\big).</math>

==Syntax==
* <tt>ENT = Entropy(RHO)</tt>
* <tt>ENT = Entropy(RHO,BASE)</tt>

==Argument descriptions==
* <tt>RHO</tt>: A density matrix to have its entropy computed.
* <tt>BASE</tt> (optional, default 2): The base of the logarithm used in the entropy calculation.

==Examples==
===The extreme cases: pure states and maximally-mixed states===
A pure state has entropy zero:
<syntaxhighlight>
>> Entropy(RandomDensityMatrix(4,0,1)) % entropy of a random 4-by-4 rank-1 density matrix

ans =

7.3396e-15 % silly numerical errors: this is effectively zero
</syntaxhighlight>

A d-by-d maximally-mixed state has entropy $\log_2(d)$:
<syntaxhighlight>
>> Entropy(eye(4)/4)

ans =

2
</syntaxhighlight>

All other states have entropy somewhere between these two extremes:
<syntaxhighlight>
>> Entropy(RandomDensityMatrix(4))

ans =

1.6157
</syntaxhighlight>

{{SourceCode|name=Entropy}}