Impossibility of comparing and sorting quantum states

Is there any point of principle that prohibits us from doing one or more forms of quantum information processing? It is now well known that an unknown quantum state can neither be copied nor deleted perfectly[1, 2]. Given a set of states which are not necessarily orthogonal, is it possible to compare any two states from the set, given some reasonable ordering of such states? Is it possible to sort them in some specific order? In the context of quantum computation, it is shown here that there is no quantum circuit implementing a unitary transformation, for comparing and sorting an unrestricted set of quantum states. Comparing and sorting are fundamental computational process like copying, deleting, storing etc. With recent advances in quantum information processing[3, 4] it is pertinent question to ask what computations are exactly permissible with a quantum model of computation compared to classical models of computation. For instance, in quantum computation while permutation and swap[4] operations of non-orthogonal states are allowed, unknown quantum states cannot be copied perfectly[1, 4], unless one chooses to work with, orthogonal set of quatum states. It was also shown recently [2], that a particular form of deletion is not allowed in the realm of quantum computation. The natural question to ask is, can one compare two non-orthogonal states using a quantum computer implementing some unitary transformation? Let S = {|ψk〉}(k = 1, 2..N) be some ordered set of quantum states defined such that |ψi〉 < |ψj〉 iff i < j and |ψ〉 are elements of Hilbert space H of dimension D. The comparator transformation U : H ′ → H ′ is defined as |ψi〉|ψj〉|Σ〉 → |1〉|Σ ′ 〉 |ψj〉|ψi〉|Σ〉 → |0〉|Σ ′′ 〉 (1) where |0〉 and |1〉 are orthogonal basis states, and |Σ〉 is ancilla state. |Σ ′ 〉 and |Σ ′′ 〉 are combination of ancilla and inputs after transformation. By looking at the first state (on rhs of equation 1 ) after the transformation, we should be able to infer whether the input state one is less than the second (on lhs of equation 1 ). The unitarity of the comparator transformation implies 〈ψj |ψi〉〈ψi|ψj〉 = 0 (2) which is true only if |ψi〉 and |ψj〉 are orthogonal. The inference is, non-orthogonal states can not be compared using a quantum comparator. In a classical model of computation, elements can be sorted by binary comparison [5], for example bubble sort, merge sort, quick sort etc, to name a few. As is well known, in classical computation one can sort even without doing comparison between the input elements, if we have some prior information about the elements, using count sort or bucket sort [5]. This leads us to the question whether quantum states can be sorted using a quantum circuit, although we cannot compare them? Let us see if the unrestricted