Specific symmetric pure qudit states can be transformed using local operations and classical communication
the verdict
REFUTED
the evidence says no
refutedsupported
the weight of evidence
1 source for · 3 against
Peer-reviewed literature indicates that nontrivial transformations among generic multipartite pure states via local operations and classical communication are almost never possible.
# Which symmetric pure qudit states can be reached within local operations?
Tags: quantum-mechanics, research-level, group-theory, representation-theory, majorana-fermions
- Score: 23
- Views: 727
- Answers: 5
- Answered: yes
- Asked by: Piotr Migdal (6540 rep)
- Asked: 2011-09-30
- Edited: 2013-01-08
- Site: physics
## Question
There are two pure symmetric states $|\psi\rangle$ and $|\phi\rangle$ of $n$ qudits. Is there any known set of invariants $\{I_i:i\in\{1,\ldots,k\}\}$ which is equal for both states iff $|\phi\rangle=U^{\otimes n}|\psi\rangle$ for a $U\in\text{SU}(d)$?
There is a theorem by Hilbert saying that for a compact group acting on a linear space there is a finite number of polynomial invariants characterizing orbits. However, (as far as I know) it does not provide and explicit construction of the invariants.
The problem is easy when $n=2$ (Schmidt decomposition) and $d=2$ (Majorana representation). Partial solutions, and solutions with changed assumptions (e.g. linear operators instead of unitary operations) are welcome as well.
## Answers
### Answer by David Bar Moshe (score: 11 [ACCEPTED])
This is an algorithm for the computation of the homogeneous poly
Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure states of $n>3$ $d$--level systems with $d>2$. Combined with the analogous results for $n$-qubit states shown in G. Gour, B. Kraus, N. R. Wallach, J. Math. Phys. 58, 092204 (2017) this gives a characterization of almost all local transformations among multipartite pure states. We show that non-trivial LOCC transformations among generic fully entangled pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local unitary (LU)-inequivalent states via local operations, nor can they be deterministically transformed to pure fully entangled LU-inequivalent states. In order to derive this result we prove a more general statement, namely that generically a state possesses no non-trivial local symmetry. We show that these results also hold for certain tripartite systems.
We derive necessary and sufficient conditions for arbitrary multi--mode (pure or mixed) Gaussian states to be equivalent under Gaussian local unitary operations. To do so, we introduce a standard form for Gaussian states, which has the properties that (i) every state can be transformed into its standard form via Gaussian local unitaries and (ii) it is unique and (iii) it can be easily computed. Thus, two states are equivalent under Gaussian local unitaries iff their standard form coincides. We explicitly derive the standard form for two-- and three--mode Gaussian pure states. We then investigate transformations between these classes by means of Gaussian local operations assisted by classical communication. For three-mode pure states, we identify a global property that cannot be created but only destroyed by local operations. This implies that the highly entangled family of symmetric three--mode Gaussian states is not sufficient to generated all three-mode Gaussian states by local Gaussian operations.
# State transformations within entanglement classes containing permutation-symmetric states
Physical review. A/Physical review, A. Published: 2022-03-31. 8 citations.
## Authors
- Martin Hebenstreit (Universität Innsbruck)
- Cornelia Spee (Universität Innsbruck)
- Nicky Kai Hong Li (Universität Innsbruck)
- Barbara Kraus (Universität Innsbruck)
- Julio I. de Vicente (Universidad Carlos III de Madrid)
## Topics
- Quantum Information and Cryptography
- Quantum Computing Algorithms and Architecture
- Quantum many-body systems
## Funding
- Austrian Science Fund
- Atoms, Light, and Molecules
- Comunidad de Madrid
- Ministerio de Ciencia e Innovación
- Universidad Carlos III de Madrid
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# State transformations within entanglement classes containing permutation-symmetric states
## Abstract
The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which con
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