Local quantum state marking

We propose the task of local state marking (LSM), where some multipartite quantum states chosen randomly from a known set of states are distributed among spatially separated parties without revealing the identities of the individual states. The collaborative aim of the parties is to correctly mark the identities of states under the restriction that they can perform only local quantum operations (LOs) on their respective subsystems and can communicate with each other classically (CC)—popularly known as the operational paradigm of LOCC. While mutually orthogonal states can always be marked exactly under global operations, this is in general not the case under LOCC. We show that the LSM task is distinct from the vastly explored task of local state distinguishability (LSD)—perfect LSD always implies perfect LSM, whereas we establish that the converse does not hold in general. We also explore entanglement-assisted marking of states that are otherwise locally unmarkable, and we report intriguing entanglement-assisted catalytic LSM phenomena.

Figure 1

The task of $m$-LSM is illustrated for the bipartite scenario. $m$ states chosen randomly from a set of $K$ states are distributed between spatially separated Alice and Bob without revealing the identities of the individual states. They have to identify the indices $i_1,\cdots ,i_m$ using LOCC. In this particular example, the indices are identified to be $(i_1=3,i_2=5,\cdots ,i_m=1$). The special case of $m=1$ corresponds to the task of LSD.

Figure 2

Flow chart of the protocol if correlated outcomes are obtained in Step-1. The number written in each branch indicates the probability of occurrence of that branch. The average amount of entanglement left in this case is $[\frac{1}{2}\times 3+\frac{1}{2}\{\frac{1}{3}\times 4+\frac{2}{3}(\frac{1}{2}\times 3+\frac{1}{2}\times 2)\}]=3$ ebits.

Figure 3

Flow chart of the protocol if anticorrelated outcomes are obtained in Step-1. The average amount of entanglement left in this case is $[\frac{1}{3}\times 4+\frac{2}{3}\{\frac{1}{2}\times 2+\frac{1}{2}\times 3\}]=3$ ebits.