Classifying locally distinguishable sets: No activation across bipartitions

TL;DR

This work provides different structures of locally distinguishable product and entangled states which do not allow the local indistinguishable states under orthogonality-preserving LOCC (OP-LOCC) to be converted to locally indistinguishable sets by OP-LOCC across any bipartition.

Abstract

A set of orthogonal quantum states is said to be locally indistinguishable if they cannot be perfectly distinguished by local operations and classical communication (LOCC). Otherwise, the states are locally distinguishable. Interestingly, locally indistinguishable states can have productive applications in quantum information processing protocols. In this sense, locally indistinguishable states are useful. On the other hand, it is usual to consider that locally distinguishable states are useless. Nevertheless, recent works suggest that locally distinguishable states should be given due consideration as in certain situations these states can be converted to locally indistinguishable states under orthogonality-preserving LOCC (OP-LOCC). Such a counterintuitive phenomenon motivates us to ask when the aforesaid conversion is possible and when it is not. In this work, we provide different structures of locally distinguishable product and entangled states which do not allow the aforesaid conversion. We also provide certain structures of locally distinguishable states which allow the aforesaid conversion. In this way, we classify the locally distinguishable sets by introducing hierarchies among them. In a multipartite system, this study becomes more involved as there exist multipartite locally distinguishable sets which cannot be converted to locally indistinguishable sets by OP-LOCC across any bipartition. We say this as ``no activation across bi-partitions".

Figures (5)

• Figure 1: (Color online) Representation of product states in ${\mathbb{C}}^{4}\otimes{\mathbb{C}}^{4}$. The bottom side represents Alice's side and top left side represents Bob's side (this is also maintained in other figures unless explicitly stated). We represent quantum states $\mathbf{\ket{i\pm\overline{i+1} }\ket{j}}$ or, $\mathbf{\ket{j}\ket{i\pm\overline{i+1} }}$ by rectangular tiles where $\mathbf{\ket{i\pm\overline{i+1} }=\frac{1}{\sqrt{2}}(\ket{i}\pm\ket{i+1})},$ for integer '$i$'. Each of the square tiles represents a state of the form $\mathbf{\ket{j}\ket{k}}$. Tile indices correspond to consecutive ordered basis states of set $\mathcal{S}_{1}$, while tile colours indicate compatible measurement setups for both parties.
• Figure 2: (Color online) Representation of product states in ${\mathbb{C}}^{4}\otimes{\mathbb{C}}^{4}$. Tile indices correspond to consecutively ordered basis states of set $\mathcal{S}_{1}$, while tile colors indicate compatible measurement setups for both parties. $\hbox{M}_i^j = {\mathcal{M}_i^j}^\dagger\mathcal{M}_i^j$; $i=\hbox{A, B}$, $j=1,2$ (this is also maintained in other figures unless explicitly stated).
• Figure 3: (Color online) Tiling diagram for the states in $\mathbf{\mathcal{S}_3}$. The outlined region indicates the support of Alice's and Bob's measurement outcomes, resulting in post-measurement states, contained in a UPB subspace. $\hbox{M}_i^j = {\mathcal{K}_j^i}$; $i=\hbox{A, B}$, $j=1,2$
• Figure 4: (Color online) Tile structure of the states in ${\mathbb{C}}^{4}\otimes{\mathbb{C}}^{4}$, given in (\ref{['3']}). The indices of the tiles follow the ordering of the states in $\mathcal{S}_5$. Tile colors represent the measurement configuration for both parties.
• Figure 5: (Color online) Tiles representation of states in $\mathbb{C}^6 \otimes \mathbb{C}^6$, indexed by states of $\mathcal{S}_6$ in order and colored according to the possibility of simultaneous local measurements by both parties.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 1
• proof
• Proposition 2
• proof