# What are nontrivial examples of $n$-sharable bipartite states?

A bipartite state $$\newcommand{\ket}{\lvert #1\rangle}\rho_{AB}$$ is said to be $$n$$-sharable when it is possible to find an extended state $$\rho_{AB_1\cdots B_n}$$ such that partial tracing over any subset of $$n-1$$ systems $$B_i$$ we recover $$\rho_{AB}$$. More formally, if there exists a $$\rho_{AB_1\cdots B_n}$$ such that $$\rho_{AB_k}\equiv \mathrm{Tr}_{B_j\forall j\neq k}[\rho_{AB_1\cdots B_n}]=\rho_{AB},\quad \forall k.$$

In other words, it is a state shared by $$n+1$$ parties, that will look like $$\rho_{AB}$$ when looked from any pair of parties $$A$$ and $$B_k$$ (I took this form of the definition from Yang (2006)).

As proven in (I think) Doherty et al. (2003), a bipartite state is separable if and only if it is $$n$$-sharable for all $$n$$.

A trivial example of $$n$$-sharability is the maximally mixed state: if $$\rho_{AB}\equiv1/d^2\sum_{ij}\lvert ij\rangle\!\langle ij\rvert$$, then I can extend it by writing $$\rho_{AB_1\cdots B_n}=\frac{1}{d^{n+1}}\sum_{i_1,...,i_{n+1}}\left(\lvert i_1\rangle\!\langle i_1\rvert\otimes \dots\otimes \lvert i_{n+1}\rangle\!\langle i_{n+1}\rvert\right).$$ In this case, I don't even have to consider subsystems of the form $$A, B_k$$, as any pair of subsystems will give the same result.

What are nontrivial examples of $$n$$-sharability? In particular, what is an example of a non-separable state that is sharable up to some point, but cannot be shared by too many parties?

I've never thought about this before, and certainly not done any in-depth calculations, but see if this gets you started....

Obviously you want to do two different things: show that there are $$n$$-sharable states, and show that these states are not $$m$$-shareable for some $$m>n$$ (and hopefully get $$n$$ and $$m$$ as close as possible).

Let $$|\psi\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$$ and $$\rho(p)=p|\psi\rangle\langle\psi|+\frac{1-p}{2}(|00\rangle\langle 00|+|11\rangle\langle 11|).$$ I believe that we will be able to do this for this state, for a whole range of values of $$p$$ (yielding different $$n$$ values).

First, to prove $$n$$-sharability, and to relate $$p$$ and $$n$$. I would start with a state such as $$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\Psi^n_{(n-3)/2}\rangle+|\Psi^n_{(n+3)/2}\rangle)$$ where $$|\Psi^n_m\rangle$$ is the $$n$$-qubit symmetric state with $$m$$ 1s in it (so $$|\Psi^3_1\rangle$$ is the standard $$W$$-state), and I'm assuming $$n$$ is odd. Since this state is invariant under all permutations, every two-qubit reduced density matrix is the same, and I believe it's of the form $$\rho(p)$$, although I haven't calculated the value of $$p$$. (As a partial verification, note that $$|\Psi\rangle$$ in ivariant under $$X^{\otimes n}$$, which severely limits the possible structure of the reduced density matrix.)

Now, to prove that it isn't $$m$$-sharable, I point you towards my paper here. (If you want to be fussy, that paper contains a conjecture, which was only closed in the follow-up). Basically, it says that the amount of a single in a two-qubit density matrix (the parmeter $$p$$ here) obeys a monogamy relation, so if you want one party to share the same amount of single with $$m-1$$ other parties, there's a maximum size that $$p$$ can be.

If I get some time later, I'll try to fill in some of the concrete numbers.

• thanks, this sounds very interesting. I'll check out the paper
– glS
Jun 5, 2019 at 10:51