
Modewise entanglement of Gaussian states
Alonso Botero and Benni ReznikWe address the decomposition of a multimode pure Gaussian state with respect to a bipartite division of the modes. For any such division the state can always be expressed as a product state involving entangled twomode squeezed states and singlemode local states at each side. The character of entanglement of the state can therefore be understood modewise; that is, a given mode on one side is entangled with only one corresponding mode of the other, and therefore the total bipartite entanglement is the sum of the modewise entanglement. This decomposition is generally not applicable to all mixed Gaussian states. However, the result can be extended to a special family of “isotropic” states, characterized by a phase space covariance matrix with a completely degenerate symplectic spectrum.It is well known that, despite the misleading imagery conjured by the name, entanglement in a multipartite system cannot be understood in terms of pairwise entanglement of the parts. Indeed, there are only pairs of systems, but the number of qualitatively distinct types of entanglement scales exponentially in . A good way to think about this is to recognize that a quantum state of a multipartite system is, in terms of parameters, much more akin to a classical probability distribution than a classical state. When we ask about the information stored in a probability distributions, there are lots and lots of “types” of information, and correlations can be much more complex than just knowing all the pairwise correlations. (“It’s not just that A knows something about B, it’s that A knows something about B conditional on a state of C, and that information can only be unlocked by knowing information from either D or E, depending on the state of F…”).
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 How to think about Quantum Mechanics—Part 1: Measurements are about bases (6)
 Riedel I'm pretty sure that the eigenvalues of $chi_{[a,b]}(hat A)times 2+chi_{[a,b]}(hat B)$ are not in general... – Jul 28, 10:40 AM
 Riedel Yea, I had to use a raw unicode character for the lessthan signs to keep... – Jul 28, 10:39 AM
 Peter Morgan Apparently my last paragraph "rant"..."/rant" was converted to just ... . Hey ho. – Jul 27, 12:49 PM
 Peter Morgan [latexpage] FWIW, your answers seem OK to me, though I suppose different materials are needed... – Jul 27, 12:44 PM
 Riedel Thanks Peter, these are great questions for refining my position. Yea, I'm very comfortable with... – Jul 27, 10:51 AM
 Peter Morgan [latexpage] I wonder whether you're OK with one particular consequence of allowing normal operators, that... – Jul 27, 9:01 AM
 How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution (2)
 Riedel Yea, you can use a KolmogorovSinai (KS) entropy to define classical chaos as follows: Fix... – Jul 24, 3:31 PM
 Yuan Wan Jess, nice write up as usual. Regarding the classical chaos, is it possible to define... – Jul 24, 2:55 PM
 Comments on Cotler, Penington, & Ranard (1)
 Peter Morgan For the QFT case, I note also that insofar as the Hamiltonian, the infinitesimal generator... – Jul 04, 2:41 PM
 Legendre transform (2)
 Kuas Thank you for this! Poor explanations survive because nobody wants to admit they found the... – Jun 28, 6:47 PM
 Francesco Thank you for this post! Making use of slightly more abstract mathematical concepts to truly... – Jun 28, 2:22 PM
 How to think about Quantum Mechanics—Part 1: Measurements are about bases (6)
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