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arXiv:quant-ph/0609226 · Quantum Information and Computation

General classes of impossible operations through the existence of incomparable states

Indrani Chattopadhyay, D. Sarkar

Published May 1, 2007• 3 citations
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Abstract

In this work we show that the most general class of anti-unitary operators are nonphysical in nature through the existence of incomparable pure bipartite entangled states. It is also shown that a large class of inner-product-preserving operations defined only on the three qubits having spin-directions along x,y and z are impossible. If we perform such an operation locally on a particular pure bipartite state then it will exactly transform to another pure bipartite state that is incomparable with the original one. As subcases of the above results we find the nonphysical nature of universal exact flipping operation and existence of universal Hadamard gate. Beyond the information conservation in terms of entanglement, this work shows how an impossible local operation evolve with the joint system in a nonphysical way.

DOI
10.26421/qic7.4-8

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In this work we show that the most general class of anti-unitary operators are nonphysical in nature through the existence of incomparable pure bipartite entangled states. It is also shown that a large class of inner-product-preserving operations defined only on the three qubits having spin-directions along x,y and z are impossible. If we perform such an operation locally on a particular pure bipartite state then it will exactly transform to another pure bipartite state that is incomparable with the original one. As subcases of the above results we find the nonphysical nature of universal exact flipping operation and existence of universal Hadamard gate. Beyond the information conservation in terms of entanglement, this work shows how an impossible local operation evolve with the joint system in a nonphysical way.
We present here a scheme that relates seemingly two different kinds of physical impossibilities of quantum information processing. We derive, exact flipping of three arbitrary states not lying in one great circle is not possible with certainty, by using the existence of incomparable states which are not interconvertible deterministically by local operations and classical communications. In contrast, considering the nonexistence of exact universal flipper, the incomparability of a pair of bipartite pure entangled states can be established.
The work is intended to represent some interesting and apparently peculiar features of entangled system in both pure as well as mixed states level. In the pure state level, we are largely concerned about the existence and characteristics of bipartite incomparable pair of states. We have found some schemes for local transformation of one pure bipartite state to another, incomparable in nature, by exploiting free unpreserved entanglement. Also, we observed that different incomparable pair of states in 3×33\times 33×3 may co-operate with each other to cross the barrier of incomparability. Our next goal was to probe critically the roots of the existence of incomparable states. We have succeeded in connecting incomparability with some famous no-go theorems of quantum mechanics. We have established it with no-flipping principle, general anti-unitary operators and lastly with inner product preserving operations. Thus we are able to pose incomparability as a good detector of some non-physical operations. In the next part, we provide a new class of multipartite bound entanglement in mixed state level. The class of states are activable bound entangled. There are exactly four orthogonal activable bound entangled states in any even number of qubit system. They have some interesting features like Bell correlation, permutation symmetry, local Pauli connection, non-compatibility, etc. Apart from those properties, the class of bound entangled states have the remarkable feature of local indistinguishability in quantum systems.
We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We discuss why it is impossible to design a controlled-NOT gate for two unknown qubits and discuss the implications of these limitations. 03.67.Hk, 03.65.Ta
Incomparability of pure bipartite entangled states under deterministic LOCC is a very strange phenomena. We find two possible ways of getting our desired pure entangled state which is incomparable with the given input state, by collective LOCC with certainty. The first one is by providing some pure entanglement through the lower dimensional maximally-entangled states or using further less amount of entanglement and the next one is by collective operation on two pairs which are individually incomparable. It is quite surprising that we are able to achieve maximally entangled states of any Schmidt rank from a finite number of 2x2 pure entangled states only by deterministic LOCC. We provide general theory for the case of 3x3 system of incomparable states by the above processes where incomparability seems to be the most hardest one.
We introduce the general catalysts for pure entanglement transformations under local operations and classical communications in such a way that we disregard the profit and loss of entanglement of the catalysts per se. As such, the possibilities of pure entanglement transformations are greatly expanded. Remarkably, we find an interesting phenomenon that, in some situations, incomparable pairs ${| ψ> ,| ϕ> \} $ and ${| χ> ,| χ^{\prime}> \} $ can assist each other mutually so as to realize the transformation ∣ψ>∣χ>→∣ϕ>∣χ′>| ψ> | χ> \to| ϕ> | χ^{\prime}>∣ψ>∣χ>→∣ϕ>∣χ′>. We also design an efficient algorithm to detect whether a k×kk\times kk×k general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.
A bipartite state, which is secretly chosen from a finite set of known entangled pure states, cannot immediately be useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whether orthogonal or not, can be locally distinguished in this way, which further implies that pure entanglement can be deterministically extracted from unknown entanglement. These results make it clear why a large class of entangled bipartite quantum operations including unitary operations and measurements that are globally distinguishable can also be locally distinguishable: They can generate pure entanglement consistently.