Action-angle coordinates - Revision history

2601:2:4D00:27B:5582:2521:6AB:4186: /* Derivation */

2014-10-08T16:41:15Z

Derivation

← Older revision		Revision as of 18:41, 8 October 2014
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	~~In [[quantum mechanics]], in particular [[quantum information]], the '''Range criterion''' is a necessary condition that a state must satisfy in order to be [[separable states\|separable]]~~. ~~In other words, it is a ''separability criterion''.~~		Hi there. Let me start by introducing the writer, her title is Sophia. Her family life in Ohio. To play lacross is some thing he would by no means give up. Office supervising is where my primary income arrives from but I've always needed my own business.<br><br>Feel free to visit my web-site - [http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna psychic readings online]

	~~== The result ==~~

	Consider a quantum mechanical system composed of ''n'' subsystems. The state space ''H'' of such a system is the tensor product of those of the subsystems, i.e. <math>H = H_1 \otimes \cdots \otimes H_n</math>.

	~~For simplicity we will assume throughout that all relevant state spaces are finite dimensional.~~

	~~The criterion reads as follows: If ρ is a separable mixed state acting on ''H'', then the range of ρ is spanned~~ by ~~a set of product vectors.~~

	~~=== Proof ===~~

	~~In general, if a matrix ''M'' is of~~ the ~~form <math>M = \sum_i v_i v_i^*</math>~~, it is ~~obvious that the range of ''M'', ''Ran(M)'', is contained in the linear span of <math>\; \{ v_i \}</math>~~. ~~On the other hand, we can also show <math>v_i</math> lies~~ in ~~''Ran(M)'', for all ''i''. Assume without loss of generality ''i = 1''~~. ~~We can write~~
	~~<math>M = v_1 v_1 ^* + T</math>, where ''T''~~ is ~~Hermitian and positive semidefinite. There are two possibilities:~~

	~~1) ''span''<math>\{ v_1 \} \subset</math>''Ker(T)''. Clearly, in this case, <math>v_1 \in</math> ''Ran(M)''~~.

	~~2) Notice 1)~~ is ~~true if and only if ''Ker(T)''<math>\;^{\perp} \subset</math> ''span''<math>\{ v_1 \}^{\perp}</math>,~~ where ~~<math>\perp</math> denotes orthogonal compliment. By Hermiticity of ''T'', this is the same as ''Ran(T)''<math>\subset</math> ''span'~~'~~<math>\{ v_1 \}^{\perp}</math>~~. ~~So if 1) does not hold, the intersection ''Ran(T)''~~ <~~math~~>~~\cap~~<~~/math~~> ~~''span''<math>\{ v_1 \}</math> is nonempty, i.e. there exists some complex number α such that <math>\; T w = \alpha v_1</math>. So~~

	:~~<math>M w = \langle w, v_1 \rangle v_1 + T w = ( \langle w, v_1 \rangle + \alpha ) v_1.</math>~~

	~~Therefore <math>v_1<~~/~~math> lies in ''Ran(M)''.~~

	~~Thus ''Ran(M)'' coincides with the linear span of <math>\; \{ v_i \}<~~/~~math>. The range criterion is a special case of this fact~~.

	~~A density matrix ρ acting on ''H'' is separable if and only if it can be written as~~

	~~:<math>\rho = \sum_i \psi_{1,i} \psi_{1,i}^* \otimes \cdots \otimes \psi_{n,i} \psi_{n,i}^*</math>~~

	~~where <math>\psi_{j,i} \psi_{j,i}^*</math> is a (un-normalized) pure state on the ''j''-th subsystem~~. ~~This is also~~

	~~:<math>~~
	~~\rho = \sum_i ( \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} ) ( \psi_{1,i} ^* \otimes \cdots \otimes \psi_{n,i} ^* )~~.
	~~</math>~~

	~~But this is exactly the same form as ''M'' from above, with the vectorial product state <math>\psi_{1,i} \otimes \cdots \otimes \psi_{n,i}<~~/~~math> replacing <math>v_i<~~/~~math>. It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion~~.

	== ~~References~~ ==

	* P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", ''Physics Letters'' '''A 232''', (1997).

	~~[[Category:Quantum information science]~~]

en>Gsard: /* See also */ reference to articles "Integrable system" and "Superinterable Hamiltonian system" are added

2013-04-04T17:57:23Z

See also: reference to articles "Integrable system" and "Superinterable Hamiltonian system" are added

← Older revision		Revision as of 19:57, 4 April 2013
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	The ~~title~~ of the ~~author~~ is ~~Garland~~. ~~Managing people~~ is ~~what I do~~ in ~~my day job~~. ~~Alabama~~ is ~~exactly where he~~ and ~~his spouse reside~~ and ~~he has everything that he needs there~~. ~~Playing crochet~~ is ~~something that I~~'~~ve done for years~~.<br><br>~~Here~~ is ~~my webpage~~; ~~[http~~://~~bytmann~~.~~com~~/~~UserProfile~~/~~tabid~~/~~849~~/~~userId~~/~~131801~~/~~language/en-US/Default~~.~~aspx extended warranty auto~~]		In [[quantum mechanics]], in particular [[quantum information]], the '''Range criterion''' is a necessary condition that a state must satisfy in order to be [[separable states\|separable]]. In other words, it is a ''separability criterion''.

			== The result ==

			Consider a quantum mechanical system composed of ''n'' subsystems. The state space ''H'' of such a system is the tensor product of those of the subsystems, i.e. <math>H = H_1 \otimes \cdots \otimes H_n</math>.

			For simplicity we will assume throughout that all relevant state spaces are finite dimensional.

			The criterion reads as follows: If ρ is a separable mixed state acting on ''H'', then the range of ρ is spanned by a set of product vectors.

			=== Proof ===

			In general, if a matrix ''M'' is of the form <math>M = \sum_i v_i v_i^*</math>, it is obvious that the range of ''M'', ''Ran(M)'', is contained in the linear span of <math>\; \{ v_i \}</math>. On the other hand, we can also show <math>v_i</math> lies in ''Ran(M)'', for all ''i''. Assume without loss of generality ''i = 1''. We can write
			<math>M = v_1 v_1 ^* + T</math>, where ''T'' is Hermitian and positive semidefinite. There are two possibilities:

			1) ''span''<math>\{ v_1 \} \subset</math>''Ker(T)''. Clearly, in this case, <math>v_1 \in</math> ''Ran(M)''.

			2) Notice 1) is true if and only if ''Ker(T)''<math>\;^{\perp} \subset</math> ''span''<math>\{ v_1 \}^{\perp}</math>, where <math>\perp</math> denotes orthogonal compliment. By Hermiticity of ''T'', this is the same as ''Ran(T)''<math>\subset</math> ''span''<math>\{ v_1 \}^{\perp}</math>. So if 1) does not hold, the intersection ''Ran(T)'' <math>\cap</math> ''span''<math>\{ v_1 \}</math> is nonempty, i.e. there exists some complex number α such that <math>\; T w = \alpha v_1</math>. So

			:<math>M w = \langle w, v_1 \rangle v_1 + T w = ( \langle w, v_1 \rangle + \alpha ) v_1.</math>

			Therefore <math>v_1</math> lies in ''Ran(M)''.

			Thus ''Ran(M)'' coincides with the linear span of <math>\; \{ v_i \}</math>. The range criterion is a special case of this fact.

			A density matrix ρ acting on ''H'' is separable if and only if it can be written as

			:<math>\rho = \sum_i \psi_{1,i} \psi_{1,i}^* \otimes \cdots \otimes \psi_{n,i} \psi_{n,i}^*</math>

			where <math>\psi_{j,i} \psi_{j,i}^*</math> is a (un-normalized) pure state on the ''j''-th subsystem. This is also

			:<math>
			\rho = \sum_i ( \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} ) ( \psi_{1,i} ^* \otimes \cdots \otimes \psi_{n,i} ^* ).
			</math>

			But this is exactly the same form as ''M'' from above, with the vectorial product state <math>\psi_{1,i} \otimes \cdots \otimes \psi_{n,i}</math> replacing <math>v_i</math>. It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion.

			== References ==

			* P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", ''Physics Letters'' '''A 232''', (1997).

			[[Category:Quantum information science]]

en>RedBot: r2.5.2) (Robot: Modifying ru:Переменные действие — угол

2011-11-15T14:10:02Z

r2.5.2) (Robot: Modifying ru:Переменные действие — угол

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