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 $2\times 2$ pure entangled states only by deterministic LOCC. We provide general theory for the case of $3\times3$ system of incomparable states by the above processes where incomparability seems to be the most hardest one.
Using graphs and hypergraphs to systematically model collections of arbitrary subsets of parties representing {\it ensembles (or collections)} of shared multipartite CAT states, we study transformations between such {\it ensembles} under {\it local operations and classical communication (LOCC)}. We show using partial entropic criteria, that any two such distinct ensembles represented by {\it $r$-uniform hypergraphs} with the same number of hyperedges (CAT states), are LOCC incomparable for even integers $r\geq 2$, generalizing results in \cite{mscthesis,sin:pal:kum:sri}. We show that the cardi…
No-Cloning and No-Deleting theorems are verified with the constraint on local state transformations via the existence of incomparable states. Assuming the existence of exact cloning or deleting operation defined on a minimum number of two arbitrary states, an incomparable pair of states of the joint system between two parties can be made to compare under deterministic LOCC. We have restricted our proof with the assumption that the machine states of the cloning or deleting operations do not keep any information about the input states. We use the same setting to establish the no-cloning and no-d…
Let (\{| ψ> ,| ϕ>}) be an incomparable pair of states ((| ψ\nleftrightarrow | ϕ>)), \emph, i.e., (| ψ>) and (| ϕ>) cannot be transformed to each other with probability one by local transformations and classical communication (LOCC). We show that incomparable states can be multiple-copy transformable, \emph, i.e., there can exist a \emph{k}, such that (| ψ> ^{\otimes k+1}\to | ϕ> ^{\otimes k+1}), i.e., (k+1) copies of (| ψ>) can be transformed to (k+1) copies of (| ϕ>) with probability one by LOCC but (| ψ> ^{\otimes n}\nleftrightarrow | ϕ> ^{\otimes n} \forall n\leq k). We call such states \em…
In this paper we investigate the effect of superposition of states on local conversion of pure bipartite states under deterministic LOCC. We are able to form a bridge between comparable and incomparable classes of states through the linear superposition of states. For example, if we consider two pairs of incomparable states, then their superposition may result into a comparable pair of states. We investigate many such cases and provide some of the results in tabular form. We also investigate the entanglement behavior of such classes of states, specifically their monotone nature. Finally we pro…
We provide a unifying framework for exact, probabilistic, and approximate conversions by local operations and classical communication (LOCC) between bipartite states. This framework allows us to formulate necessary and sufficient conditions for LOCC conversions from pure states to mixed states and it provides necessary conditions for LOCC conversions between mixed states. The central idea is the introduction of convex sets for exact, probabilistic, and approximate conversions, which are closed under LOCC operations and which are largely characterized by simple properties of pure states.
The distribution of typical bipartite pure states is studied within the framework of state transformation via local operation and classical communication (LOCC). We report the statistics of comparable and incomparable states in different dimensions for single- and multi-copy regimes and establish a connection between state transformation and the difference between the entanglement contents of the initial and the target states. From the analysis of catalyst resources, required to further otherwise impossible LOCC transformations between pairs, we demonstrate a universal pattern in the average a…
In an effort to simplify the classification of pure entangled states of multi (m) -partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among n'th tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). With regard to exact transformations, we show that two states whose 1-party entropies agree are either locally-unitarily (LU) equivalent or else LOCC-incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-…
We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of first- and second- order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, w…
The most basic local conversion is local operations and classical communications (LOCC), which is also the most natural restriction in quantum information processing. We investigate the conversions between the ground states in quantum critical systems via LOCC and propose an novel method to reveal the different convertibility via majorization relation when a quantum phase transition occurs. The ground-state local convertibility in the one-dimensional transverse field Ising model is studied. It is shown that the LOCC convertibility changes nearly at the phase transition point. The relation betw…
Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by LOCC when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies…
In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of 'incomparable entanglement'--whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)--is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.
Entanglement theory is formulated as a quantum resource theory in which the free operations are local operations and classical communication (LOCC). This defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almos…
We consider the transformation of multi-partite states in the single copy setting under positive-partial-transpose-preserving operations (PPT-operations) and obtain both qualitative and quantitative results. Firstly, for some pure state transformations that are impossible under local operations and classical communication (LOCC), we demonstrate that they become possible with a surprisingly large success probability under PPT-operations. Furthermore, we clarify the convertibility of arbitrary multipartite pure states under PPT-operations, and show that a drastic simplification in the classifica…
The class of quantum operations known as Local Operations and Classical Communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations related to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimensi…
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically moti…
University of Geneva（日内瓦大学）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）；Sorbonne Université, CNRS, LIP6（索邦大学、法国国家科学研究中心、LIP6实验室）
In order to cope with the fact that there exists no single maximally entangled state (up to local unitaries) in the multipartite setting, we introduced in [J. I. de Vicente, C. Spee and B. Kraus, Phys. Rev. Lett. 111, 110502 (2013)] the maximally entangled set of n-partite quantum states. This set consists of the states that are most useful under conversion of pure states via Local Operations assisted by Classical Communication (LOCC). We will review our results here on the maximally entangled set of three- and generic four-qubit states. Moreover, we will discuss the preparation of arbitrary (…
We discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second-order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space. To derive the second-order conversion rate, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case. Using this family of probability distributions, we represent …
We investigate how much amount of Greenberger-Horne-Zeilinger (GHZ) entanglement is required in order to prepare a given multipartite state by local operations and classical communication (LOCC).We present a LOCC procedure that asymptotically converts GHZ states into an arbitrary multipartite pure state, whose conversion rate is given by the multipartite discord of the state. This reveals that the GHZ-entanglement cost of preparing a pure state is not higher than the multipartite discord of the state. It also provides an operational interpretation of multipartite discord for pure states, namel…
We study exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in success probability, known as the converese error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full under…
Two remote parties, Alice and Bob initially share some non-maximally entangled states. Through the entanglement concentration by local operation and classical communication(LOCC), they may obtain an outcome of maximally entangled states in the price of decreasing the number of pairs shared by them. Recently, entanglement concentration have been experimently demonstrated by post-selection (T. Yamamoto et al, Nature, 421, 343(2003) and Z. Zhao et al, quant-ph/0301118). Here we give a modified scheme which can be used for the entanglement concentration without any post selection by using only pra…
Conversion of entangled states under (Stochastic) Local Operations and Classical Communication admits the phenomenon of catalysis. Here we explore the possibility of a copy of the initial state itself to perform as a catalyst, which we call a self-catalytic process. We show explicit examples of self-catalysis. Necessary and sufficient conditions for the phenomenon to take place are discussed. We numerically estimate how frequent it is and we show that increasing the number of copies used as catalyst can increases the probability of conversion, but do not make the process deterministic. By the …
Quantifying entanglement is one of the most important tasks in the entanglement theory. In this paper, we establish entanglement monotones in terms of an operational approach, which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications (LOCC). It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices. We especially show that our entanglement monotone is the maximal one among all that have the same form for pure states. In some particu…
Suppose several parties jointly possess a pure multipartite state, |ψ>. Using local operations on their respective systems and classical communication (i.e. LOCC) it may be possible for the parties to transform deterministically |ψ> into another joint state |ϕ>. In the bipartite case, Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states in the same stochastic LOCC (SLOCC) class. Here we generalize Nielsen m…
