Product numerical range in a space with tensor product structure

Puchała, Zbigniew; Gawron, Piotr; Miszczak, Jarosław Adam; Skowronek, Łukasz; Choi, Man-Duen; Życzkowski, Karol

doi:10.1016/j.laa.2010.08.026

Cited by 28 publications

(40 citation statements)

References 21 publications

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“…The differential topology and projection aspects of numerical range were investigated in [17,18] The standard notion of numerical range, often used in the theory of quantum information [19,20,21], can be generalized in several ways [22,23,24]. For instance, for an operator acting on a composite Hilbert space one defines the product numerical range [25] (also called local numerical range [24]) and a more general class of numerical ranges restricted to a specific class of states [21], W R (A) = {z : z = ψ|A|ψ , |ψ ∈ R ⊂ Ω D , ψ|ψ = 1}.…”

Section: Introductionmentioning

confidence: 99%

Restricted numerical shadow and the geometry of quantum entanglement

Puchała¹,

Miszczak²,

Gawron³

et al. 2012

J. Phys. A: Math. Theor.

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The restricted numerical range W R (A) of an operator A acting on a Ddimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the of set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A -a normalized probability distribution on the complex plane supported in W R (A). Its value at point z ∈ C is equal to the probability that the expectation value ψ|A|ψ is equal to z, where |ψ represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. Studying restricted shadows of operators of a fixed size D = N A N B we analyse the geometry of sets of separable and maximally entangled states of the N A × N B composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on D = 2 3 dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three-qubit system.

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Section: Introductionmentioning

confidence: 99%

Restricted numerical shadow and the geometry of quantum entanglement

Puchała¹,

Miszczak²,

Gawron³

et al. 2012

J. Phys. A: Math. Theor.

Self Cite

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“…It is known that the product of two circular disks is star-shaped [3,4,7,8]. In this section, we will prove some unexpected results that if K 1 is a circular disk, then for many closed sets K 2 , the product set is star-shaped.…”

Section: A Circular Disk and A Closed Setmentioning

confidence: 84%

“…The study will be more challenging. As pointed out in [8], the set K 1 · · · K s may not be simply connected in general. Nevertheless, our results in Section 5 and Proposition 6.2 imply the following.…”

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confidence: 99%

Minkowski product of convex sets and product numerical range

Li¹,

Pelejo²,

Poon³

et al. 2016

Operators and Matrices

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Abstract. Let K 1 ,K 2 be two compact convex sets in C . Their Minkowski product is the setWe show that the set K 1 K 2 is star-shaped if K 1 is a line segment or a circular disk. Examples for K 1 and K 2 are given so that K 1 and K 2 are triangles (including interior) and K 1 K 2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science.Additional results and open problems are presented.Mathematics subject classification (2010): 51M15, 15A60.

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“…This simplification to only separable states then allows us to study the geometry of 2-RDMs with a mathematical concept, called joint product numerical range [22][23][24][25][26][27], denoted by Π, of the Hamiltonian interaction terms. Π includes all the extreme points of the three-dimensional projections of 2-RDMs, and the projection itself, denoted by Θ, is a convex hull of Π.…”

Section: Introductionmentioning

confidence: 99%

Joint product numerical range and geometry of reduced density matrices

Chen

Guo

et al. 2016

Sci. China Phys. Mech. Astron.

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The reduced density matrices of a many-body quantum system form a convex set, whose threedimensional projection Θ is convex in R 3 . The boundary ∂Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of Π. We show that, a ruled surface on ∂Θ sitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

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Product numerical range in a space with tensor product structure

Cited by 28 publications

References 21 publications

Restricted numerical shadow and the geometry of quantum entanglement

Restricted numerical shadow and the geometry of quantum entanglement

Minkowski product of convex sets and product numerical range

Joint product numerical range and geometry of reduced density matrices

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