arXiv subjects

Michał Oszmaniec

Publications and source records attributed to Michał Oszmaniec.

49 records · Page 3Linked to original sources

arXiv2016

Simulating positive-operator-valued measures with projective measurements

Standard projective measurements represent a subset of all possible measurements in quantum physics, defined by positive-operator-valued measures. We study what quantum measurements are projective simulable, that is, can be simulated by using projective measurements and classical randomness. We first prove that every measurement on a given quantum system can be realised by classical processing of projective measurements on the system plus an ancilla of the same dimension. Then, given a general measurement in dimension two or three, we show that deciding whether it is projective-simulable can be solved by means of semi-definite programming. We also establish conditions for the simulation of measurements using projective ones valid for any dimension. As an application of our formalism, we improve the range of visibilities for which two-qubit Werner states do not violate any Bell inequality for all measurements. From an implementation point of view, our work provides bounds on the amount of noise a measurement tolerates before losing any advantage over projective ones.

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arXiv2016

Creation of superposition of unknown quantum states

The superposition principle is one of the landmarks of quantum mechanics. The importance of quantum superpositions provokes questions about the limitations that quantum mechanics itself imposes on the possibility of their generation. In this work we systematically study the problem of creation of superpositions of unknown quantum states. First, we prove a no-go theorem that forbids the existence of a universal probabilistic quantum protocol producing a superposition of two unknown quantum states. Secondly, we provide an explicit probabilistic protocol generating a superposition of two unknown states, each having a fixed overlap with the known referential pure state. The protocol is proven to be unique and optimal. Moreover, it can be implemented on arbitrary Hilbert spaces. In the context of quantum optics it can be used to efficiently generate highly nonclassical or nongaussian states.

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arXiv2015

Communication strength of correlations violating monogamy relations

In any theory satisfying the no-signaling principle correlations generated among spatially separated parties in a Bell-type experiment are subject to certain constraints known as monogamy relations. Recently, in the context of the black hole information loss problem it was suggested that these monogamy relations might be violated. This in turn implies that correlations arising in such a scenario must violate the no-signaling principle and hence can be used to send classical information between parties. Here, we study the amount of information that can be sent using such correlations. To this aim, we first provide a framework associating them with classical channels whose capacities are then used to quantify the usefulness of these correlations in sending information. Finally, we determine the minimal amount of information that can be sent using signaling correlations violating the monogamy relation associated to the chained Bell inequalities.

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arXiv2014

Applications of differential geometry and representation theory to description of quantum correlations

One of the most important questions in quantum information theory is the so-called separability problem. It involves characterizing the set of separable (or, equivalently entangled) states among mixed states of a multipartite quantum system. In this thesis we study the generalization of this problem to types of quantum correlations that are defined in a manner analogous to entanglement. We start with the subset of set of pure states of a given quantum system and call states belonging to the convex hull of this subset "non-correlated" states. Consequently, the states laying outside the convex hull are referred to as "correlated". In this work we focus on cases when there exist a symmetry group that preserves the class of "non-correlated" pure states. The presence of symmetries allows to obtain a unified treatment of many types of seemingly unrelated types of correlations. We apply our general results to particular types of correlations: (i) entanglement of distinguishable particles, (ii) particle entanglement of bosons, (iii) "entanglement" of fermions, (iv) non-convex-Gaussian correlations in fermionic systems, (v) genuine multiparty entanglement, and finally (vi) refined notions of bipartite entanglement based on the concept of the Schmidt number. We investigate the natural problems and questions concerning the correlations defined above: (I) We provide explicit polynomial characterization of various types of correlations for pure states. (II) We examine cases in which it is possible to give a complete analytical characterization of correlated mixed states. (III) We derive a variety of polynomial criteria for detection of correlations in mixed states.(IV) We use the above criteria and the technique of measure concentration to study typical properties of correlations on sets of isospectral density matrices.

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arXiv2014

Geometry and topology of CC and CQ states

We show that multipartite mixed bipartite CC and CQ states are geometrically and topologically distinguished in the space of states. They are characterized by non-vanishing Euler-Poincaré characteristics on the topological side and by the existence of symplectic structure on the geometric side.

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arXiv2014

Fraction of isospectral states exhibiting quantum correlations

For several types of correlations: mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions and non-Gaussian correlations in fermionic systems we estimate the fraction of non-correlated states among the density matrices with the same spectra. We prove that for the purity exceeding some critical value (depending on the considered problem) fraction of non-correlated states tends to zero exponentially fast with the dimension of the relevant Hilbert space. As a consequence a state randomly chosen from the set of density matrices possessing the same spectra is asymptotically a correlated one. To prove this we developed a systematic framework for detection of correlations via nonlinear witnesses.

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arXiv2014

Classical simulation of fermionic linear optics augmented with noisy ancillas

Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure Fermionic Gaussian states, the corresponding computation scheme is classically simulable. We present an analytic characterisation of the set of convex-Gaussian states in the first non-trivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by De Melo et al. and to study in detail the geometrical properties of the set of convex-Gaussian states.

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arXiv2014

Which multiphoton states are related via linear optics?

We investigate which pure states of $n$ photons in $d$ modes can be transformed into each other via linear optics, without post-selection. In other words, we study the local unitary (LU) equivalence classes of symmetric many-qudit states. Writing our state as $f^\dagger|Ω\rangle$, with $f^\dagger$ a homogeneous polynomial in the mode creation operators, we propose two sets of LU-invariants: (a) spectral invariants, which are the eigenvalues of the operator $ff^\dagger$, and (b) moments, each given by the norm of the symmetric component of a tensor power of the initial state, which can be computed as vacuum expectation values of $f^k(f^\dagger)^k$. We provide scheme for experimental measurement of the later, as related to the post-selection probability of creating state $f^{\dagger k}|Ω\rangle$ from $k$ copies of $f^{\dagger}|Ω\rangle$.

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arXiv2013

A universal framework for entanglement detection

We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons and fermions. In each case properties of an entanglement measures are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entanglement measure reduces to the well-known many particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a non trivial lower bound of thus obtained generalized concurrence.

math-ph↗

arXiv2013

Critical sets of the total variance of state detect all SLOCC entanglement classes

We present a general algorithm for finding all classes of pure multiparticle states equivalent under Stochastic Local Operations and Classsical Communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also discuss the Morse indices of critical points which have the interpretation of the number of independent non-local perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.

math-ph↗

arXiv2013

Convexity of momentum map, Morse index, and quantum entanglement

We analyze form the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many `asymptotically' equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well defined measure of entanglement) of the state Var[v] attains maximum. We provide an algorithm for finding critical sets of Var[v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using the recent developments in the momentum map geometry known as Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

math-ph↗

arXiv2013

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e. by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.

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arXiv2012

On detection of quasiclassical states

We give a criterion of classicality for mixed states in terms of expectation values of a quantum observable. Using group representation theory we identify all cases when the criterion can be computed exactly in terms of the spectrum of a single operator.

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