| Revision as of 00:48, 20 December 2020 editMegaBat (talk | contribs)313 editsm →Statement: Made the rho look the same as the other rho's.Tag: 2017 wikitext editor← Previous edit | Latest revision as of 09:41, 28 February 2022 edit undoChristian75 (talk | contribs)Extended confirmed users, New page reviewers, Pending changes reviewers, Rollbackers114,675 edits {{R with history}} | ||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| #REDIRECT ] | |||
| In ], especially ], '''purification''' refers to the fact that every ] acting on ] can be viewed as the ] of some pure state. | |||
| {{R with history}} | |||
| In purely linear algebraic terms, it can be viewed as a statement about ]. | |||
| == Statement == | |||
| Let <math>\rho</math> be a ] acting on a ] <math>H_A</math> of finite dimension ''n''. Then it is possible to construct a second Hilbert space <math>H_B</math> and a pure state <math>| \psi \rangle \in H_A \otimes H_B</math> such that <math>\rho</math> is the partial trace of <math>| \psi \rangle \langle \psi |</math> with respect to <math>H_B</math>. While the initial Hilbert space <math>H_A</math> might correspond to physically meaningful quantities, the second Hilbert space <math>H_B</math> needn't have any physical interpretation whatsoever. However, in physics the process of state purification is assumed to be physical, and so the second Hilbert space <math>H_B</math> should also correspond to a physical space, such as the environment. The exact form of <math>H_B</math> in such cases will depend on the problem. Here is a ], showing that at very least <math>H_B</math> has to have dimensions greater than or equal to <math>H_A</math>. | |||
| With these statements in mind, if, | |||
| :<math>\operatorname{tr_B} \left( | \psi \rangle \langle \psi | \right )= \rho,</math> | |||
| we say that <math>| \psi \rangle</math> purifies <math>\rho</math>. | |||
| === Proof === | |||
| A density matrix is by definition positive semidefinite. So ρ can be ] and written as <math>\rho = \sum_{i =1} ^n p_i | i \rangle \langle i |</math> for some ] <math>\{ | i \rangle \}</math>. Let <math>H_B</math> be another copy of the ''n''-dimensional Hilbert space with an ] <math>\{ | i' \rangle \}</math>. Define <math>| \psi \rangle \in H_A \otimes H_B</math> by | |||
| :<math>| \psi \rangle = \sum_{i} \sqrt{p_i} |i \rangle \otimes | i' \rangle.</math> | |||
| Direct calculation gives | |||
| :<math> | |||
| \begin{align} | |||
| \operatorname{tr_B} \left( | \psi \rangle \langle \psi | \right ) &= | |||
| \operatorname{tr_B} \left \\ | |||
| &=\operatorname{tr_B} \left( \sum_{i, j} \sqrt{p_ip_j} |i \rangle \langle j | \otimes | i' \rangle \langle j'| \right ) \\ | |||
| &= \sum_{i,j} \delta_{ij} \sqrt{p_i p_j}| i \rangle \langle j | \\ | |||
| &= \sum_{i} \sqrt{p_i}^2 |i\rangle \langle i| \\ | |||
| &= \rho. | |||
| \end{align} </math> | |||
| This proves the claim. | |||
| ==== Note ==== | |||
| * The purification is not unique, but if during the construction of <math> | \psi \rangle </math> in the proof above <math>H_B</math> is generated by only the <math>\{ | i' \rangle \}</math> for which <math> p_i </math> is non-zero, any other purification <math> | \varphi \rangle </math> on <math> H_A \otimes H_C </math> induces an ] <math> V: H_B \to H_C </math> such that <math> | \varphi \rangle = (I \otimes V ) | \psi \rangle </math>. | |||
| * The vectorial pure state <math>| \psi \rangle</math> is in the form specified by the ]. | |||
| * Since ] ] of a positive semidefinite matrix are not unique, neither are purifications. | |||
| * In linear algebraic terms, a square matrix is positive semidefinite ] it can be purified in the above sense. The ''if'' part of the implication follows immediately from the fact that the ] of a positive map remains a ]. | |||
| == An application: Stinespring's theorem == | |||
| {{Expand section|date=June 2008}} | |||
| By combining ] and purification of a mixed state, we can recover the ] for the finite-dimensional case. | |||
| {{DEFAULTSORT:Purification Of Quantum State}} | |||
| ] | |||
| ] | |||
Latest revision as of 09:41, 28 February 2022
Redirect to:
- With history: This is a redirect from a page containing substantive page history. This page is kept as a redirect to preserve its former content and attributions. Please do not remove the tag that generates this text (unless the need to recreate content on this page has been demonstrated), nor delete this page.
- This template should not be used for redirects having some edit history but no meaningful content in their previous versions, nor for redirects created as a result of a page merge (use {{R from merge}} instead), nor for redirects from a title that forms a historic part of Misplaced Pages (use {{R with old history}} instead).