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.
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.
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
3×3 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.
In this work, we reveal a type of impossibility discovered in our recent research which forbids comparing the closeness of multiple unknown quantum states with any nontrivial threshold in a perfect or unambiguous way. This impossibility is distinct from the existing impossibilities in that it is a ``collective'' impossibility on multiple quantum states; most other ``no-go'' theorems are concerned with only one single state each time, i.e., it is an impossibility on a nonlocal quantum operation. This impossibility may provide new insight into the nature of quantum mechanics, and it implies more limitations on quantum information tasks than the existing no-go theorems.
Hoping to simplify the classification of pure entangled states of multi $(m)\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ quantum systems, we study exactly and asymptotically (in
n) reversible transformations among
nth tensor powers of such states (i.e.,
n copies of the state shared among the same
m parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter---their partial entropy---which may be interpreted as the number of EPR pairs asymptotically interconvertible to the state in question by LOCC transformations. We show that $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states having an $m\ensuremath{-}\mathrm{w}\mathrm{a}\mathrm{y}$ Schmidt decomposition are similarly parametrizable, with the partial entropy across any nontrivial partition representing the number of standard quantum superposition or ``cat'' states $|{0}^{\ensuremath{\bigotimes}m}〉+|{1}^{\ensuremath{\bigotimes}m}〉$ asymptotically interconvertible to the state in question. For general $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the
m=4 cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the
m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically. For each number of parties
m we define a minimal reversible entanglement generating set (MREGS) as a set of states of minimal cardinality sufficient to generate all $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states by asymptotically reversible LOCC transformations. Partial entropy arguments provide lower bounds on the size of the MREGS, but for
m>2 we know no upper bounds. We briefly consider several generalizations of LOCC transformations, including transformations with some probability of failure, transformations with the catalytic assistance of states other than the states we are trying to transform, and asymptotic LOCC transformations supplemented by a negligible
[o(n)] amount of quantum communication.