Marcin Marciniak | University of Gdansk (original) (raw)

Papers by Marcin Marciniak

Physical review research, Dec 19, 2019

Banach Center Publications, 2007

Physical review, Mar 6, 2023

We analyze recent approaches to quantum Markovianity and how they relate to the proper definition... more We analyze recent approaches to quantum Markovianity and how they relate to the proper definition of quantum memory. We point out that the well-known criterion of information backflow may not correctly report character of the memory falsely signaling its quantumness. Therefore, as a complement to the well-known criteria, we propose several concepts of elementary dynamical maps. Maps of this type do not increase distinguishability of states which are indistinguishable by von Neumann measurements in a given basis. Those notions and convexity allows us to define general classes of processes without quantum memory in a weak and strong sense. Finally, we provide a practical characterization of the most intuitive class in terms of the new concept of witness of quantum information backflow.

Journal of Geometry and Physics, Sep 1, 2018

We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutati... more We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative n-torus T n θ in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups G θ = (A n θ , ∆) such that for each θ, G θ is the final object in the category of all compact quantum groups acting on T n θ in a filtration preserving way. We describe in details the structure of the C*-algebra A n θ and provide a concrete example of its representation in bounded operators. Moreover, we compute the Haar measure of G θ. For θ = 0, the quantum group G 0 is nothing but the classical group T n ⋊ Sn, where Sn is the symmetric group. For general θ, G θ is still an extension of the classical group T n by the classical group Sn. It turns out that for n = 2, the algebra A 2 θ coincides with the algebra of the quantum double-torus described by Hajac and Masuda. Using a variation of the little subgroup method we show that irreducible representations of G θ are in one-to-one correspondence with irreducible representations of T n ⋊ Sn.

We extend the theory of decomposable maps by giving a detailed description of k-positive maps. A ... more We extend the theory of decomposable maps by giving a detailed description of k-positive maps. A relation between transposition and modular theory is established. The structure of positive maps in terms of modular theory (the generalized Tomita-Takesaki scheme) is examined.

Linear & Multilinear Algebra, Jul 1, 2013

Let K and H be finite dimensional Hilbert spaces and let P denote the cone of all positive linear... more Let K and H be finite dimensional Hilbert spaces and let P denote the cone of all positive linear maps acting from B(K) into B(H). We show that each map of the form φ(X) = AXA * or φ(X) = AX T A * is an exposed point of P.

Journal of physics, Jul 19, 2006

It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyap... more It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyapunov exponents in the Heisenberg picture. Differences among various quantizations of Lyapunov exponents are clarified.

Journal of physics, Jul 16, 2001

arXiv (Cornell University), Sep 30, 2005

A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map ... more A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map while ϕ 2 is a co-CP map. It is known that if m = n = 2 then every positive map is decomposable. Given an extremal unital positive map ϕ : M 2 (C) → M 2 (C) we construct concrete maps (not necessarily unital) ϕ 1 and ϕ 2 which give a decomposition of ϕ. We also show that in most cases this decomposition is unique.

arXiv (Cornell University), Sep 30, 2005

A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map ... more A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map while ϕ 2 is a co-CP map. It is known that if m = n = 2 then every positive map is decomposable. Given an extremal unital positive map ϕ : M 2 (C) → M 2 (C) we construct concrete maps (not necessarily unital) ϕ 1 and ϕ 2 which give a decomposition of ϕ. We also show that in most cases this decomposition is unique.

arXiv (Cornell University), Oct 7, 2005

A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map ... more A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map while ϕ 2 is a co-CP map. A partial characterization of decomposability for maps ϕ : M 2 (C) → M 3 (C) is given.

Reports on Mathematical Physics, Dec 1, 1997

... W adys aw A. Majewski E-mail The Corresponding Author. Marcin Marciniak E-mail The Correspond... more ... W adys aw A. Majewski E-mail The Corresponding Author. Marcin Marciniak E-mail The Corresponding Author. ... 100 (1976), 62. [8] WA Majewski: Remarks on Quantum Characteristic Exponents, preprint, Gdansk Uni versity, 1996. [9] WA Majewski and M. Kuna: J. Math. Phys. ...

Proceedings of the American Mathematical Society, 1998

In this paper we investigate a structure of the fixed point algebra under an action of compact ma... more In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a C *-algebra B. We also show that the categories of C-comodules in B and inner endomorphisms restricted to the fixed point algebra coincide when the relative commutant of the fixed point algebra is trivial. Next we show a version of the Tannaka duality theorem for twisted unitary groups.

Physical Review A, 2001

The necessary and sufficient condition of separability of a mixed state of any systems is present... more The necessary and sufficient condition of separability of a mixed state of any systems is presented, which is practical in judging the separability of a mixed state. This paper also presents a method of finding the disentangled decomposition of a separable mixed state.

Quantum Information Processing

We determine filtering and master equations for a quantum system interacting with wave packet of ... more We determine filtering and master equations for a quantum system interacting with wave packet of light in a continuous-mode squeezed number state. We formulate the problem of conditional evolution of a quantum system making use of model of repeated interactions and measurements. In this approach, the quantum system undergoes a sequence of interactions with an environment defined by a chain of harmonic oscillators. We assume that the environment is prepared in an entangled state being a discrete analogue of a continuous-mode squeezed number state. We present a derivation of a discrete stochastic dynamics that depends on the results of measurement performed on the field after its interaction with the system. In this paper, we consider a photon counting measurement scheme. By taking a continuous time limit, we finally obtain differential stochastic equations for the system. Analytical formulae for quantum trajectories and exclusive probability densities that allow to fully characterize...

arXiv (Cornell University), May 8, 2014

In [11, 12] the operator space theory was applied to study bipartite Bell inequalities. The aim o... more In [11, 12] the operator space theory was applied to study bipartite Bell inequalities. The aim of the paper is to follow this line of research and use the operator space technique to analyze the steering scenario. We obtain a bipartite steering functional with unbounded largest violation of steering inequality, as well as we can construct all ingredients explicitly. It turns out that the unbounded largest violation is obtained by non maximally entangled state. Moreover, we focus on the bipartite dichotomic case where we construct a steering functional with unbounded largest violation of steering inequality. This phenomenon is different to the Bell scenario where only bounded largest violation can be obtained by any bipartite dichotomic Bell functional.

Physical Review Letters, Oct 20, 2015

We construct steering inequalities which exhibit unbounded violation. The concept was to exploit ... more We construct steering inequalities which exhibit unbounded violation. The concept was to exploit the relationship between steering violation and uncertainty relation. To this end we apply mutually unbiased bases and anti-commuting observables, known to exibit the strongest uncertainty. In both cases, we are able to procure unbounded violations. Our approach is much more constructive and transparent than the operator space theory approach employed to obtain large violation of Bell inequalities. Importantly, using anti-commuting observables we are able to obtain a dichotomic steering inequality with unbounded violation. So far there is no analogous result for Bell inequalities. Interestingly, both the dichotomic inequality and one of our inequalities can not be directly obtained from existing uncertainty relations, which strongly suggest the existence of an unknown kind of uncertainty relation.

arXiv (Cornell University), Dec 1, 2014

The aim of this paper is to indicate possible applications of operator systems in qualitative des... more The aim of this paper is to indicate possible applications of operator systems in qualitative description of varoius scenarios while studying non-locality. To this end we study in details the notion of generalized noncommuting cube. Following ideas of [5] and [3] we show in systematic way that various classes of Tsirelson's correlation boxes as well as NPA hierarchies can be described by using various operator system tensor products of generalized non-commuting cubes. Moreover, we show also that noncommuting cubes can be applied for the description of steering assemblages. Next we study some aproximation properties of noncommuting cubes by finite dimensional models. Finaly, we indicate possibility to use the framework operator systems for studying Bell and steering inequalities.

Physical review research, Dec 19, 2019

Banach Center Publications, 2007

Physical review, Mar 6, 2023

We analyze recent approaches to quantum Markovianity and how they relate to the proper definition... more We analyze recent approaches to quantum Markovianity and how they relate to the proper definition of quantum memory. We point out that the well-known criterion of information backflow may not correctly report character of the memory falsely signaling its quantumness. Therefore, as a complement to the well-known criteria, we propose several concepts of elementary dynamical maps. Maps of this type do not increase distinguishability of states which are indistinguishable by von Neumann measurements in a given basis. Those notions and convexity allows us to define general classes of processes without quantum memory in a weak and strong sense. Finally, we provide a practical characterization of the most intuitive class in terms of the new concept of witness of quantum information backflow.

Journal of Geometry and Physics, Sep 1, 2018

We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutati... more We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative n-torus T n θ in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups G θ = (A n θ , ∆) such that for each θ, G θ is the final object in the category of all compact quantum groups acting on T n θ in a filtration preserving way. We describe in details the structure of the C*-algebra A n θ and provide a concrete example of its representation in bounded operators. Moreover, we compute the Haar measure of G θ. For θ = 0, the quantum group G 0 is nothing but the classical group T n ⋊ Sn, where Sn is the symmetric group. For general θ, G θ is still an extension of the classical group T n by the classical group Sn. It turns out that for n = 2, the algebra A 2 θ coincides with the algebra of the quantum double-torus described by Hajac and Masuda. Using a variation of the little subgroup method we show that irreducible representations of G θ are in one-to-one correspondence with irreducible representations of T n ⋊ Sn.

We extend the theory of decomposable maps by giving a detailed description of k-positive maps. A ... more We extend the theory of decomposable maps by giving a detailed description of k-positive maps. A relation between transposition and modular theory is established. The structure of positive maps in terms of modular theory (the generalized Tomita-Takesaki scheme) is examined.

Linear & Multilinear Algebra, Jul 1, 2013

Let K and H be finite dimensional Hilbert spaces and let P denote the cone of all positive linear... more Let K and H be finite dimensional Hilbert spaces and let P denote the cone of all positive linear maps acting from B(K) into B(H). We show that each map of the form φ(X) = AXA * or φ(X) = AX T A * is an exposed point of P.

Journal of physics, Jul 19, 2006

It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyap... more It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyapunov exponents in the Heisenberg picture. Differences among various quantizations of Lyapunov exponents are clarified.

Journal of physics, Jul 16, 2001

arXiv (Cornell University), Sep 30, 2005

A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map ... more A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map while ϕ 2 is a co-CP map. It is known that if m = n = 2 then every positive map is decomposable. Given an extremal unital positive map ϕ : M 2 (C) → M 2 (C) we construct concrete maps (not necessarily unital) ϕ 1 and ϕ 2 which give a decomposition of ϕ. We also show that in most cases this decomposition is unique.

arXiv (Cornell University), Sep 30, 2005

A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map ... more A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map while ϕ 2 is a co-CP map. It is known that if m = n = 2 then every positive map is decomposable. Given an extremal unital positive map ϕ : M 2 (C) → M 2 (C) we construct concrete maps (not necessarily unital) ϕ 1 and ϕ 2 which give a decomposition of ϕ. We also show that in most cases this decomposition is unique.

arXiv (Cornell University), Oct 7, 2005

A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map ... more A map ϕ : Mm(C) → Mn(C) is decomposable if it is of the form ϕ = ϕ 1 + ϕ 2 where ϕ 1 is a CP map while ϕ 2 is a co-CP map. A partial characterization of decomposability for maps ϕ : M 2 (C) → M 3 (C) is given.

Reports on Mathematical Physics, Dec 1, 1997

... W adys aw A. Majewski E-mail The Corresponding Author. Marcin Marciniak E-mail The Correspond... more ... W adys aw A. Majewski E-mail The Corresponding Author. Marcin Marciniak E-mail The Corresponding Author. ... 100 (1976), 62. [8] WA Majewski: Remarks on Quantum Characteristic Exponents, preprint, Gdansk Uni versity, 1996. [9] WA Majewski and M. Kuna: J. Math. Phys. ...

Proceedings of the American Mathematical Society, 1998

In this paper we investigate a structure of the fixed point algebra under an action of compact ma... more In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a C *-algebra B. We also show that the categories of C-comodules in B and inner endomorphisms restricted to the fixed point algebra coincide when the relative commutant of the fixed point algebra is trivial. Next we show a version of the Tannaka duality theorem for twisted unitary groups.

Physical Review A, 2001

The necessary and sufficient condition of separability of a mixed state of any systems is present... more The necessary and sufficient condition of separability of a mixed state of any systems is presented, which is practical in judging the separability of a mixed state. This paper also presents a method of finding the disentangled decomposition of a separable mixed state.

Quantum Information Processing

We determine filtering and master equations for a quantum system interacting with wave packet of ... more We determine filtering and master equations for a quantum system interacting with wave packet of light in a continuous-mode squeezed number state. We formulate the problem of conditional evolution of a quantum system making use of model of repeated interactions and measurements. In this approach, the quantum system undergoes a sequence of interactions with an environment defined by a chain of harmonic oscillators. We assume that the environment is prepared in an entangled state being a discrete analogue of a continuous-mode squeezed number state. We present a derivation of a discrete stochastic dynamics that depends on the results of measurement performed on the field after its interaction with the system. In this paper, we consider a photon counting measurement scheme. By taking a continuous time limit, we finally obtain differential stochastic equations for the system. Analytical formulae for quantum trajectories and exclusive probability densities that allow to fully characterize...

arXiv (Cornell University), May 8, 2014

In [11, 12] the operator space theory was applied to study bipartite Bell inequalities. The aim o... more In [11, 12] the operator space theory was applied to study bipartite Bell inequalities. The aim of the paper is to follow this line of research and use the operator space technique to analyze the steering scenario. We obtain a bipartite steering functional with unbounded largest violation of steering inequality, as well as we can construct all ingredients explicitly. It turns out that the unbounded largest violation is obtained by non maximally entangled state. Moreover, we focus on the bipartite dichotomic case where we construct a steering functional with unbounded largest violation of steering inequality. This phenomenon is different to the Bell scenario where only bounded largest violation can be obtained by any bipartite dichotomic Bell functional.

Physical Review Letters, Oct 20, 2015

We construct steering inequalities which exhibit unbounded violation. The concept was to exploit ... more We construct steering inequalities which exhibit unbounded violation. The concept was to exploit the relationship between steering violation and uncertainty relation. To this end we apply mutually unbiased bases and anti-commuting observables, known to exibit the strongest uncertainty. In both cases, we are able to procure unbounded violations. Our approach is much more constructive and transparent than the operator space theory approach employed to obtain large violation of Bell inequalities. Importantly, using anti-commuting observables we are able to obtain a dichotomic steering inequality with unbounded violation. So far there is no analogous result for Bell inequalities. Interestingly, both the dichotomic inequality and one of our inequalities can not be directly obtained from existing uncertainty relations, which strongly suggest the existence of an unknown kind of uncertainty relation.

arXiv (Cornell University), Dec 1, 2014

The aim of this paper is to indicate possible applications of operator systems in qualitative des... more The aim of this paper is to indicate possible applications of operator systems in qualitative description of varoius scenarios while studying non-locality. To this end we study in details the notion of generalized noncommuting cube. Following ideas of [5] and [3] we show in systematic way that various classes of Tsirelson's correlation boxes as well as NPA hierarchies can be described by using various operator system tensor products of generalized non-commuting cubes. Moreover, we show also that noncommuting cubes can be applied for the description of steering assemblages. Next we study some aproximation properties of noncommuting cubes by finite dimensional models. Finaly, we indicate possibility to use the framework operator systems for studying Bell and steering inequalities.