Local quantum transformations requiring infinite rounds of classical communication

In this Letter, we investigate the number of measurement and communication rounds needed to implement certain tasks by local quantum operations and classical communication (LOCC), a relatively unexplored topic. To demonstrate the possible strong dependence on the round number, we consider the problem of converting three-qubit entanglement into two-qubit form, specifically in the random distillation setting of [Phys. Rev. Lett. 98, 260501 (2007)]. We find that the number of LOCC rounds needed for a transformation can depend on the amount of entanglement distilled. In fact, for a wide range of transformations, the required number of rounds is infinite (unbounded). This represents the first concrete example of a task needing an infinite number of rounds to implement.     

On the Chernoff Distance for Asymptotic LOCC Discrimination of Bipartite Quantum States

Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d × d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations.     

Local Copying of <Emphasis Type="Italic">d</Emphasis>×<Emphasis Type="Italic">d</Emphasis>-dimensional Partially Entangled Pure States

We examine the problem of copying a set of orthogonal, entangled partially (non-maximally) bipartite pure states with an entangled blank state under the restriction to local operations and classical communication (LOCC), and show a protocol for copying these states by LOCC. The necessary and sufficient condition for locally copying partially entangled pure states is then represented. As a result, we find that the problem of local copying these entangled states can be regarded to some extent as that of catalytic transformation between them by LOCC.     

Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication

We show that an arbitrary basis of a multipartite quantum state space consisting of K distant parties such that the kth party has local dimension dk always contains at least N= Sigma(k=1)(K) (dk-1)+1 members that are unambiguously distinguishable using local operations and classical communication (LOCC). We further show that this lower bound is optimal by analytically constructing a special product basis having only N members unambiguously distinguishable by LOCC. Interestingly, such a special product basis not only gives a stronger form of the weird phenomenon "nonlocality without entanglement," but also implies the existence of a locally distinguishable entangled basis.     

A Framework for Bounding Nonlocality of State Discrimination

We consider the class of protocols that can be implemented by local quantum operations and classical communication (LOCC) between two parties. In particular, we focus on the task of discriminating a known set of quantum states by LOCC. Building on the work in the paper Quantum nonlocality without entanglement (Bennett et al., Phys Rev A 59:1070–1091, 1999), we provide a framework for bounding the amount of nonlocality in a given set of bipartite quantum states in terms of a lower bound on the probability of error in any LOCC discrimination protocol. We apply our framework to an orthonormal product basis known as the domino states and obtain an alternative and simplified proof that quantifies its nonlocality. We generalize this result for similar bases in larger dimensions, as well as the “rotated” domino states, resolving a long-standing open question (Bennett et al., Phys Rev A 59:1070–1091, 1999).     

Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding

We consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements. Let the bias of a POVM be defined as the total variational distance between the outcome distributions for the two states to be distinguished. The performance of a set of measurements can then be defined as the ratio of the bias of this POVM and the largest bias achievable by any measurements. We first provide lower bounds on the performance of various POVMs acting on a single system such as the isotropic POVM, and spherical 2 and 4-designs, and show how these bounds can lead to certainty relations. Furthermore, we prove lower bounds for several interesting POVMs acting on multipartite systems, such as the set of local POVMS, POVMs which can be implemented using local operations and classical communication (LOCC), separable POVMs, and finally POVMs for which every bipartition results in a measurement having positive partial transpose (PPT). In particular, our results show that a scheme of Terhal et. al. for hiding data against local operations and classical communication [31] has the best possible dimensional dependence.     

No-flipping as a consequence of no-signalling and non-increase of entanglement under LOCC

Non-existence of universal flipper for arbitrary quantum states is a fundamental constraint on the allowed operations performed on physical systems. The largest set of qubits that can be flipped by a single machine is a great circle of the Bloch-sphere. In this Letter, we show the impossibility of universal exact-flipping operation, first by using the fact that no faster than light communication is possible and then by using the principle of “non-increase of entanglement under LOCC”. Interestingly, in both the cases, there is no violation of the two principles if and only if the set of states to be flipped, form a great circle.     

Entanglement cost of implementing controlled-unitary operations

We investigate the minimum entanglement cost of the deterministic implementation of two-qubit controlled-unitary operations using local operations and classical communication (LOCC). We show that any such operation can be implemented by a three-turn LOCC protocol, which requires at least 1 ebit of entanglement when the resource is given by a bipartite entangled state with Schmidt number 2. Our result implies that there is a gap between the minimum entanglement cost and the entangling power of controlled-unitary operations. This gap arises due to the requirement of implementing the operations while oblivious to the identity of the inputs.     

Multi-party d-Level Quantum Secret Sharing Scheme

We develop a multiparty quantum secret sharing (QSS) scheme of classical messages based on arbitrary dimensional multi-particle Greenberger-Horne-Zeilinger (GHZ) states. This scheme can be implemented using only local operations, e.g. generalized Z gate and Hadamard gate, and classical communication (LOCC) between participants. The security of the present scheme against exterior eavesdropping and interior dishonest party has been analyzed and confirmed. Moreover, we discuss the possibility of successful sharing of classical messages in the realistic situation where our QSS scheme is carried out in generalized Pauli channels.     

Optimal and covariant single-copy LOCC transformation between two two-qubit states

Given two two-qubit pure states characterized by their Schmidt numbers we investigate an optimal strategy to convert the states between themselves with respect to their local unitary invariance. We discuss the efficiency of this transformation and its connection to LOCC convertibility properties between two single-copy quantum states. As an illustration of the investigated transformations we present a communication protocol where in spite of all expectations a shared maximally entangled pair between two participants is the worst quantum resource.     

Local distinguishability of multipartite unitary operations

We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguished by local operations and classical communication when a finite number of runs is allowed. Intuitively, this result indicates that the lost identity of a nonlocal unitary operation can be recovered locally. No entanglement between distant parties is required.     

Entanglement transformation between two-qubit mixed states by LOCC

Based on a new set of entanglement monotones of two-qubit pure states, we give sufficient and necessary conditions that one two-qubit mixed state is transformed into another one by local operations and classical communication (LOCC). This result can be viewed as a generalization of Nielsen's theorem Nielsen (1999) [1]. However, we find that it is more difficult to manipulate the entanglement transformation between single copy of two-qubit mixed states than to do between single copy of two-qubit pure ones.     

Distillation of GHZ State from Multiple Copies of Arbitrary W-Class State

W. Dur et al. have shown that it is impossible to obtain a GHZ state from one copy of arbitrary W-class state via local operations and classical communication （LOCC） [W. Dur, G. Vidal, and J.I. Cirac, Phys. Rev. A 62 （2000） 062314]. In our paper, the more general case is carefully investigated. We first show that, with a supply of two copies of arbitrary W-class state, we can always construct an explicit procedure to distill a GHZ state with a nonzero probability. Then based on this result, a simple procedure for distilling GHZ state from n copies of arbitrary W-class state is presented. Finally, we briefly discuss the applications.     

Bidirectional Quantum States Sharing

With the help of the shared entanglement and LOCC, multidirectional quantum states sharing is considered. We first put forward a protocol for implementing four-party bidirectional states sharing (BQSS) by using eight-qubit cluster state as quantum channel. In order to extend BQSS, we generalize this protocol from four sharers to multi-sharers utilizing two multi-qubit GHZ-type states as channel, and propose two multi-party BQSS schemes. On the other hand, we generalize the three schemes from two senders to multi-senders with multi GHZ-type states of multi-qubit as quantum channel, and give a multidirectional quantum states sharing protocol. In our schemes, all receivers can reconstruct the original unknown single-qubit state if and only if all sharers can cooperate. Only Pauli operations, Bell-state measurement and single-qubit measurement are used in our schemes, so these schemes are easily realized in physical experiment and their successful probabilities are all one.     

Sufficient and Necessary Conditions of Entanglement Transformations Between Mixed States

Based on a new measure of entanglement for finite-dimensional bi-particle pure states, we give sufficient and necessary conditions that a bi-particle mixed state ρ can be transformed into another mixed state σ by local operations and classical communication (LOCC). This result can be regarded as a generalization of Nielsen’s theorem (Nielsen, Phys. Rev. Lett. 83:436, 1999). However, we find that it is more difficult to determine the entanglement transformations between mixed states than to do between pure ones.     

Entanglement transformation between ensembles of three-qubit GHZ-type states by LOCC

We present a novel method to verify whether an ensemble of three-qubit GHZ-type state could be transformed to another one by local operations and classical communication (LOCC). This result highlights intriguing similarity compared with the case in the transformation between two ensembles of two-qubit states. Our approach may provide a splendid insight into the transformations between two general multipartite states.     

Simple Procedure for Entanglement Transformation of Bipartite Pure States

An explicit procedure for transforming one bipartite entangled state into another via local operations and classical communication （LOCC） is presented. Our procedure is much simper than the previous ones in the sense that, it only involves two steps and the explicit expression of local general measurement used in the procedure can be obtained by solving a set of linear equations. Furthermore, this procedure is still applicable in high dimensional case.     

Distinguishing Quantum States with Holevo Bound and Its Application to Spatially Separated Bell States

Consider a system prepared in one of the quantum states of the ensemble {ρ _i } with a priori probability p _i . We prove that the quantum state can be deterministically distinguished if and only if the information gain from the measurement reaches the Holevo bound. We find it can be applied to distinguish spatially separated Bell states. A single copy of spatially separated Bell state cannot be distinguished under local operation and classical communication (LOCC), but it can be identified with an ancillary qubit. When two ancillary qubits are used, a spatially separated Bell state can be identified without demolition.     

Quasienergy anholonomy and its application to adiabatic quantum state manipulation

The parametric dependence of a quantum map under the influence of a rank-1 perturbation is investigated. While the Floquet operator of the map and its spectrum have a common period with respect to the perturbation strength lambda, we show an example in which none of the quasienergies nor the eigenvectors obey the same period: After a periodic increment of lambda, the quasienergy arrives at the nearest higher one, instead of the initial one, exhibiting an anholonomy, which governs another anholonomy of the eigenvectors. An application to quantum state manipulations is outlined.