3
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.