Clean Positive Operator Valued Measures

IEEE Transactions on Information Theory, 2014

We study the distinguishability of bipartite quantum states by Positive Operator-Valued Measures with positive partial transpose (PPT POVMs). The contributions of this paper include: (1). We give a negative answer to an open problem of [M. Horodecki et.al, Phys. Rev. Lett. 90, 047902(2003)] showing a limitation of their method for detecting nondistinguishability. (2). We show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, can not be distinguished by PPT POVMs, even unambiguously. This result is much stronger than the previous known ones [1], [2]. (3). We study the entanglement cost of distinguishing quantum states. It is proved that 2/3|00 + 1/3|11 is sufficient and necessary for distinguishing three Bell states by PPT POVMs. An upper bound of entanglement cost of distinguishing a d ⊗ d pure state and its orthogonal complement is obtained for separable operations. Based on this bound, we are able to construct two orthogonal quantum states which cannot be distinguished unambiguously by separable POVMs, but finite copies would make them perfectly distinguishable by LOCC. We further observe that a two-qubit maximally entangled state is always enough for distinguishing a d ⊗ d pure state and its orthogonal complement by PPT POVMs, no matter the value of d. In sharp contrast, an entangled state with Schmidt number at least d is always needed for distinguishing such two states by separable POVMs. As an application, we show that the entanglement cost of distinguishing a d⊗d maximally entangled state and its orthogonal complement must be a maximally entangled state for d = 2, which implies that teleportation is optimal; and in general, it could be chosen as O(log d d).

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular , the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.

Journal of Mathematical Physics, 2005

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n 2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases.

Axioms, 2014

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple or more elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.

Journal of Physics A-mathematical and General, 2005

Similarly to quantum states, also quantum measurements can be 'mixed', corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are indecomposable, i.e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVMs form a convex set, and in this language the indecomposable apparatuses are represented by extremal points-the analogous of 'pure states' in the convex set of states. Differently from the case of states, however, indecomposable POVMs are not necessarily rank-one, e.g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVMs, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, 'informationally complete' measurements are analysed in this respect. The convex set of POVMs is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVMs.

Physical Review A, 2005

We propose a scheme that can realize a class of positive-operator-valued measures (POVMs) by performing a sequence of projective measurements on the original system, in the sense that for an arbitrary input state the probability distribution of the measurement outcomes is faithfully reproduced. A necessary and sufficient condition for a POVM to be realizable in this way is also derived. In contrast to the canonical approach provided by Neumark's theorem, our method has the advantage of requiring no auxiliary system. Moreover, an arbitrary POVM can be realized by utilizing our protocol on an extended space which is formed by adding only a single extra dimension.

Physical Review A, 2010

In the four-dimensional Hilbert space, there exist 16 Heisenberg-Weyl (HW) covariant symmetric informationally complete positive operator valued measures (SIC POVMs) consisting of 256 fiducial states on a single orbit of the Clifford group. We explore the structure of these SIC POVMs by studying the symmetry transformations within a given SIC POVM and among different SIC POVMs. Furthermore, we find 16 additional SIC POVMs by a regrouping of the 256 fiducial states, and show that they are unitarily equivalent to the original 16 SIC POVMs by establishing an explicit unitary transformation. We then reveal the additional structure of these SIC POVMs when the fourdimensional Hilbert space is taken as the tensor product of two qubit Hilbert spaces. In particular, when either the standard product basis or the Bell basis are chosen as the defining basis of the HW group, in eight of the 16 HW covariant SIC POVMs, all fiducial states have the same concurrence of 2/5. These SIC POVMs are particularly appealing for an experimental implementation, since all fiducial states can be connected to each other with just local unitary transformations. In addition, we introduce a concise representation of the fiducial states with the aid of a suitable tabular arrangement of their parameters.

Russian Journal of Mathematical Physics, 2014

Using Grothendieck approach to the tensor product of locally convex spaces we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.

Physical Review A, 2003

We study the local implementation of POVMs when we require only the faithful reproduction of the statistics of the measurement outcomes for all initial states. We first demonstrate that any POVM with separable elements can be implemented by a separable super-operator, and develop techniques for calculating the extreme points of POVMs under a certain class of constraint that includes separability and PPT-ness. As examples we consider measurements that are invariant under various symmetry groups (Werner, Isotropic, Bell-diagonal, Local Orthogonal), and demonstrate that in these cases separability of the POVM elements is equivalent to implementability via LOCC. We also calculate the extrema of these classes of measurement under the groups that we consider, and give explicit LOCC protocols for attaining them. These protocols are hence optimal methods for locally discriminating between states of these symmetries. One of many interesting consequences is that the best way to locally discriminate Bell diagonal mixed states is to perform a 2-outcome POVM using local von Neumann projections. This is true regardless of the cost function, the number of states being discriminated, or the prior probabilities. Our results give the first cases of local mixed state discrimination that can be analysed quantitatively in full, and may have application to other problems such as demonstrations of non-locality, experimental entanglement witnesses, and perhaps even entanglement distillation.

Journal of Physics A: Mathematical and Theoretical, 2018

We address the problem of quantum nonlocality with positive operator valued measures (POVM) in the context of Einstein-Podolsky-Rosen quantum steering. We show that, given a candidate for local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartite quantum state of finite dimension with POVMs can be formulated as a nesting problem of two convex objects. One consequence of this is the strengthening of the theorem that justifies choosing the LHS ensemble based on symmetry of the bipartite state. As a more practical application, we study the classic problem of the steerability of two-qubit Werner states with POVMs. We show strong numerical evidence that these states are unsteerable with POVMs up to a mixing probability of 1 2 within an accuracy of 10 −3 .