A bipartite state $\newcommand{\ket}[1]{\lvert #1\rangle}\rho_{AB}$ 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?


1 Answer 1


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.

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

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