Timeline for Expansion of multi-particle state vector as a sum of n-entangled states
Current License: CC BY-SA 3.0
17 events
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| Apr 17, 2011 at 18:51 | comment | added | lurscher | @Deepak, very interesting stuff.. | |
| Apr 17, 2011 at 18:32 | comment | added | user346 | former: $CP^{k_1} \times CP^{k_2} \in CP^{(k_1+1)(k_2+1)-1}$ known as the Segre embedding. Understanding the structure of this embedding allows us to classify all multipartite entangled states in terms of hyperdeterminants which are generalization of determinant to the case of objects with more than two indices. For a very pedagogical and intuitive description of this geometric picture see the paper by Bengtsson et al.. The end result of all this is that one needs to go beyond slater determinants to understand multipartite entanglement! | |
| Apr 17, 2011 at 18:27 | comment | added | user346 | @Lurscher the question of whether or not a state is separable is neatly answered in Miyake's paper on the classification of multipartite entanglement. One has to realize that the state space for a vector living in a $k+1$ dimensional Hilbert space is the complex projective space $CP^k$. The space of the composite system of two "particles" in a $k_1+1$ and $k_2+1$ dimensional hilbert spaces respectively is $CP^{(k_1+1)(k_2+1)-1}$. The subspace consisting of separable states is of the form $CP^{k_1} \times CP^{k_2}$. The latter has an embedding in the | |
| Apr 17, 2011 at 16:33 | history | edited | lurscher | CC BY-SA 3.0 |
added 2nd edit with arguments that unitary transformations don't change the decomposition and reference to QSP; added 6 characters in body
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| Apr 17, 2011 at 16:09 | history | edited | lurscher | CC BY-SA 3.0 |
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| Apr 17, 2011 at 16:03 | history | edited | lurscher | CC BY-SA 3.0 |
added 2nd edit with proof that unitary transformations don't change the decomposition; added 65 characters in body; added 5 characters in body
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| Apr 17, 2011 at 13:48 | history | edited | lurscher | CC BY-SA 3.0 |
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| Apr 17, 2011 at 5:12 | history | edited | lurscher | CC BY-SA 3.0 |
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| Apr 17, 2011 at 5:06 | history | edited | lurscher | CC BY-SA 3.0 |
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| Apr 17, 2011 at 4:57 | history | edited | lurscher | CC BY-SA 3.0 |
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| Apr 16, 2011 at 1:22 | answer | added | wsc | timeline score: 5 | |
| Apr 15, 2011 at 18:09 | history | tweeted | twitter.com/#!/StackPhysics/status/58955119336763392 | ||
| Apr 15, 2011 at 17:41 | comment | added | lurscher | @Johannes, exactly! this is what i'm looking for; i didn't knew the term, a quantum version of the virial expansion; however i think this becomes subtle because interactions and entanglement are mutually related, but have a not very clear overlap | |
| Apr 15, 2011 at 17:19 | comment | added | Johannes | In classical statistical mechanics you use the cluster expansion to derive the Van der Waals equation. This is a expansion in powers of the number of interacting particles. I know nothing about its application to quantum statistical mechanics. | |
| Apr 15, 2011 at 16:54 | comment | added | Lagerbaer | I like the idea. But are you talking about the wave-function, or about an ensemble represented by a density matrix? For the wave-function, the (Anti-)symmetrized products $\prod_i |\phi_i\rangle$ already form a complete set of basis states, so every wavefunction can be written as a linear combination of these. For Fermions, e.g., it would be a linear combination of Slater Determinants. | |
| Apr 15, 2011 at 16:15 | comment | added | user346 | I don't think there is any such general method to decompose a general many-body state in this manner. If you were to discover one that would be huge. But awesome idea. | |
| Apr 15, 2011 at 16:03 | history | asked | lurscher | CC BY-SA 3.0 |