Two types of results are presented for distinguishing pure bipartite quantum states using Local Operations and Classical Communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C 3 ⊗C 3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in C n ⊗ C m . In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.
We show that there exist sets of three mutually orthogonal d-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of d. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal d-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability.These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.
We define and study the properties of channels which are analogous to unital qubit channels in several ways. A full treatment can be given only when the dimension d = p m a prime power, in which case each of the d + 1 mutually unbiased bases (MUB) defines an axis. Along each axis the channel looks like a depolarizing channel, but the degree of depolarization depends on the axis. When d is not a prime power, some of our results still hold, particularly in the case of channels with one symmetry axis. We describe the convex structure of this class of channels and the subclass of entanglement breaking channels. We find new bound entangled states for d = 3.For these channels, we show that the multiplicativity conjecture for maximal output p-norm holds for p = 2. We also find channels with behavior not exhibited by unital qubit channels, including two pairs of orthogonal bases with equal output entropy in the absence of symmetry. This provides new numerical evidence for the additivity of minimal output entropy. * Partially supported by by the National Science Foundation under Grants DMS-0314228 and DMS-0604900 and by the National Security Agency and Advanced Research and Development Activity under Army Research Office contract number DAAD19-02-1-0065. IntroductionThe results presented here are motivated by the desire to find channels for dimension d > 2 whose properties are similar to those of the unital qubit channels, particularly with respect to optimal output purity. A channel is described by a completely positive, trace-preserving (CPT) map. The channels we construct are similar to unital qubit channels in the sense that their effect on a density matrix can be defined in terms of multipliers of components along different "axes" defined in terms of mutually unbiased bases (MUB). When all multipliers are positive, these channels are very much like unital qubit channels with positive multipliers. However, when some of the multipliers are negative the new channels can exhibit behavior not encountered for unital qubit channels.For a fixed orthonormal basis B = {|ψ k }, the quantum-classical (QC) channelprojects a density matrix ρ onto the corresponding diagonal matrix in this basis. A convex combination J t J Ψ QC J (ρ) of QC channels in a collection of orthonormal bases B J = {|ψ J k } is also a channel; in fact, it is an entanglement breaking (EB) channel. We consider channels which are a linear combination of the identity map I(ρ) = ρ and a convex combination of QC channels whose bases are mutually unbiased, i.e., satisfySuch channels can be written in the form 3The first condition ensures that Φ is trace-preserving (TP), and the pair that it is completely positive (CP), as will be shown in Section 2.It is well-known that C d can have at most d + 1 MUB and that this is always possible when d = p m is a prime power. We are primarily interested in channels of the form (3) when such a full set of d + 1 MUB exist. In that case, it is natural to generalize the Bloch sphere representation so that a density matri...
Multiplicativity of certain maximal p → q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity (CP) is either augmented by, or replaced by, the requirement that the entries of a matrix representative of the map are non-negative (EP). In particular, for integer t, multiplicativity holds for the maximal 2 → 2t norm of a product of two maps, whenever one of the pair is EP; for the maximal 1 → t norm for pairs of CP maps when one of them is also EP; and for the maximal 1 → 2t norm for the product of an EP and a 2-positive map. Similar results are shown in the infinitedimensional setting of convolution operators on L 2 (R), with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix. These results apply in particular to Gaussian bosonic channels.
We give examples of qubit channels that require three input states in order to achieve the Holevo capacity.The Holevo capacity C(Φ) of a channel Φ is defined as the supremum over all possible ensembles E = {π j , ρ j } (consisting of a probability distribution π j and set of density matrices ρ j ), of the quantitywhere ρ = j π j ρ j is the average input, and S(γ) = −Trγ log γ denotes the von Neumann entropy. Thus, C(Φ) = sup E χ(E). It has been shown [4,10] that C(Φ) is the maximum information carrying capacity of a channel restricted to product inputs, but permitting entangled measurements. C(Φ) ≥ C Shan (Φ), where the classical Shannon capacity C Shan (Φ) describes the information carrying capacity of a channel when output measurements (as well as input ensembles) are restricted to products. (See, e.g., [2,5,7] for precise definitions.) We will sometimes use subscripts to denote the supremum of (1) restricted to a particular class of ensembles; in particular, we write C n (Φ) to denote the restriction to ensembles of n states. It is well-known [1] that for qubit channels the maximum can be achieved with an ensemble containing at most four states.In general, a qubit channel maps the Bloch sphere to an ellipsoid [6,9]. If the channel is unital, i.e, if Φ(I) = I, then the ellipsoid is centered at the origin, and the capacity is achieved with a pair of orthogonal inputs whose images, which are also states of minimal entropy [6], are the endpoints of the major axis of the ellipsoid. For nonunital channels the ellipsoid is displaced from the origin, and examples are known [2,11] for which the capacity is achieved with two non-orthogonal inputs which are not mapped onto states of minimal entropy. Here we present * king@neu.edu † ruskai@mediaone.net examples of non-unital channels which require three input states to achieve capacity. Furthermore these channels are non-extreme points in the set of all qubit channels, and this property is essential to our construction. To motivate our strategy, we consider the capacity of two well-known channels which have rotational symmetry about an axis of the Bloch sphere, and we maximize (1) over ensembles consisting of a pair of states on a line either parallel or orthogonal to this axis.First, let Φ D denote the shifted depolarizing channel which contracts the Bloch sphere to a sphere of radius µ and shifts it up until it touches the unit sphere, i.e, if ρ is written in the formWhen µ = 0.5, the image of the Bloch sphere satisfiesas shown in Fig. 1, and the poles ρ = 1 2 I ± σ 3 are mapped to the states 1 2 I + σ 3 and 1 2 I which have entropy 0 and 1 respectively (using base 2 for logarithms). If we restrict the input ensemble to a vertical line, the χ-quantity (1) is maximized by a convex combination of the poles, so that the average output state Φ D (ρ) = 1 2 I + zσ 3 lies on the z-axis. One finds that this vertical capacity, which we denote C V , is achieved when the output average is at z = 0.6 and its value is(This is also the capacity of a quantum-classical channel Φ QC , ...
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