Locally Unextendible Non-Maximally Entangled Basis

We introduce the concept of the locally unextendible non-maximally entangled basis (LUNMEB) in $H^d \bigotimes H^d$. It is shown that such a basis consists of $d$ orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of $(d-1)^2$ orthogonal vectors for non-maximally entangled state if it is maximally entangled in $(d-1)$ dimensional subspace. Such a basis plays an important role in determining the number of classical bits that one can send in a superdense coding protocol using a non-maximally entangled state as a resource. By constructing appropriate POVM operators, we find that the number of classical bits one can transmit using a non-maximally entangled state as a resource is $(1+p_0\frac{d}{d-1})\log d$, where $p_0$ is the smallest Schmidt coefficient. However, when the state is maximally entangled in its subspace then one can send up to $2\log (d-1)$ bits. We also find that for $d= 3$, former may be more suitable for the superdense coding.

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