On the decomposition of the supersymmetric state

We prove that every symmetric separable state admits a convex decomposition into symmetric pure product states. While the result is not new in itself, here we focus on convex geometry. We discuss the decomposition in the context of numerical ranges and ground state problems of infinite bosonic systems.

arXiv:2005.11607May 23, 20200 citations

On the decomposition of the supersymmetric state

Lilong Qian, Delin Chu

Published Oct 7, 2018• 0 citations

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Abstract

|x,x\rangle\langle x,x|

, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or

N

. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to

4

N+1

. We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.

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Decomposition of symmetric separable states and ground state energy of bosonic systems

Stephan Weis

arXiv:2005.11607May 23, 20200 citations

|x,x\rangle\langle x,x|

, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or

N

. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to

4

N+1

|x,x\rangle\langle x,x|

, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or

N

. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to

4

N+1

N+1

. For the states of rank 6 or

N+2

, they are separable if and only if their range contains a product vector. We apply our results to a few widely useful states in quantum information, such as symmetric states, edge states, extreme states, and non-negative states. We also study the relation of completely symmetric states to Hankel and Toeplitz matrices.

We prove that every bipartite state

ρ

of a qubit coupled to an

n

-level system whose operator Schmidt rank equals three can be written as a mixture of

\text{rank}(ρ)

pure product states, and as a mixture of at most

n+1

mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.