No-splitting Theorem: Principles of Quantum Information

In the context of quantum information there arise a number of interesting questions regarding the domains of operations which are feasible in classical context to be applicable in quantum information domain or not. The criterion of feasibility and non feasibility of various physical processes in quantum regime creates a set of impossible operations[1-6]. This may be probably due to the linear structure or may be due to the unitary evolution in quantum mechanics. Regardless of their origin, these impossible operations are making quantum information processing much more restricted than it’s classical counterpart. On the other hand this restriction on many quantum information processing tasks is making quantum information more secure. In the famous land mark paper of Wootters and Zurek it was shown that a single quantum cannot be cloned [1]. Later it was also shown by Pati and Braunstein that we cannot delete either of the two quantum states when we are provided with two quantum states at our input port[2]. Inspite of these two famous ’no-cloning’ and ’no-deletion’ theorem there are many other ’no-go’ theorems like ’no-flipping’,’nodisturbance’,’no-self replication’,’no-partial erasure’,’nosplitting’ theorems which have come up. Recent research has revealed that these theorems are consistent with different principles like principle of no-signalling and conservation of entanglement under LOCC[7-9]. If we put it in a different way it means that if we violate these ’nogo’ theorems we will violate the principle of no-signalling and conservation of entanglement under LOCC. In this paper we will show that ’no-splitting’ theorem for separable state can be proved from the unitarity of quantum mechanics. We will also see that this theorem is consistent with the principle of no-signalling and conservation of entanglement under LOCC.