Volume 57, Issue 1, January 2016
 SPECIAL ISSUE: OPERATOR ALGEBRAS AND QUANTUM INFORMATION THEORY


Inequalities for operator space numerical radius of 2 × 2 block matrices
View Description Hide DescriptionIn this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space , when X is a numerical radius operator space. Moreover, we establish several inequalities for operator space numerical radius and the maximal numerical radius norm of 2 × 2 operator matrices and their offdiagonal parts. One of our main results states that if (X, (O n )) is an operator space, then for all .

Positivity of linear maps under tensor powers
View Description Hide DescriptionWe investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely copositive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist nontrivial maps with this property and that for twodimensional Hilbert spaces there is no nontrivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such nontrivial “tensorstable positive maps” to a oneparameter family of maps and show that an affirmative answer would imply the existence of nonpositive partial transpose bound entanglement. As an application, we show that any tensorstable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the wellknown cbnorm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communicationsassisted quantum capacity, and that moreover it is a strong converse rate for this task.

On the constrained classical capacity of infinitedimensional covariant quantum channels
View Description Hide DescriptionThe additivity of the minimal output entropy and that of the χcapacity are known to be equivalent for finitedimensional irreducibly covariant quantum channels. In this paper, we formulate a list of conditions allowing to establish similar equivalence for infinitedimensional covariant channels with constrained input. This is then applied to bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper [Giovannetti et al., Commun. Math. Phys. 334(3), 15531571 (2015)] to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particular, this implies a multimode generalization of the “threshold condition,” obtained for single mode in Schäfer et al. [Phys. Rev. Lett. 111, 030503 (2013)], and the proof of the fact that under this condition the classical “Gaussian capacity” resulting from optimization over only Gaussian inputs is equal to the full classical capacity. Complex structures correspond to different squeezings, each with its own normal modes, vacuum and coherent states, and the gauge. Thus our results apply, e.g., to multimode channels with a squeezed Gaussian noise under the standard input energy constraint, provided the squeezing is not too large as to violate the generalized threshold condition. We also investigate the restrictiveness of the gaugecovariance condition for single and multimode bosonic Gaussian channels.

Continuity of the maximumentropy inference: Convex geometry and numerical ranges approach
View Description Hide DescriptionWe study the continuity of an abstract generalization of the maximumentropy inference—a maximizer. It is defined as a rightinverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3 × 3 matrices.

Various notions of positivity for bilinear maps and applications to tripartite entanglement
View Description Hide DescriptionWe consider bilinear analogues of spositivity for linear maps. The dual objects of these notions can be described in terms of Schmidt ranks for tritensor products and Schmidt numbers for tripartite quantum states. These tripartite versions of Schmidt numbers cover various kinds of biseparability, and so we may interpret witnesses for those in terms of bilinear maps. We give concrete examples of witnesses for various kinds of three qubit entanglement.

Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm
View Description Hide DescriptionWe develop the notions of hypercontractivity (HC) and the logSobolev (LS) inequality for completely bounded norms of oneparameter semigroups of superoperators acting on matrix algebras. We prove the equivalence of the completely bounded versions of HC and LS under suitable hypotheses. We also prove a version of the Gross lemma which allows LS at general q to be deduced from LS at q = 2.

On the convex structure of process positive operator valued measures
View Description Hide DescriptionMeasurements on quantum channels are described by socalled process positive operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger than the usual extremality given by the convex structure. This property motivates the introduction and investigation of the convex structure of POVMs, which generalizes both the usual convex and C*convex structures. We show that extremal points and faces of the set of process POVMs are closely related to extremal points and faces of POVMs, for a certain subalgebra . We also give a characterization of extremal and pure POVMs.

Private algebras in quantum information and infinitedimensional complementarity
View Description Hide DescriptionWe introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinitedimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

Measurement theory in local quantum physics
View Description Hide DescriptionIn this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a onetoone correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the wellknown result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the DoplicherHaagRoberts and DoplicherRoberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.
