The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, called the combined-probability space, is presented here for a $d$ -level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, called the combined-probability space, is presented here for a $d$-level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.

arXiv:1702.01680Physical Review AAug 7, 20171 citation

Quantum probability theory

Miklós Rédei, Stephen J. Summers

arXiv:quant-ph/0601158Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsJun 1, 2007110 citations

Secrecy Performance of User and Antenna Selection With Channel Estimation Inaccuracy

Donghun Lee

This paper presents the secrecy performance of a user and antenna selection at transmitter (i.e., combined scheme) over channel estimation inaccuracy in wiretap channels with multiple antennas. This paper develops the precise formulations of the combined system for the secrecy outage probability and the probability of the non-zero secrecy capacity. This work develops the diversity order for the secrecy outage probability in the presence and absence of channel inaccuracy at the combined system and eavesdropper. This work provides various numerical evaluations, showing that, the diversity order of the combined system is changed only by combined system`s channel inaccuracy not by eavesdropper's channel inaccuracy. Also, the channel inaccuracy of the eavesdropper acts on the signal to noise ratio (SNR) gain of the combined system.This paper presents the secrecy performance of a user and antenna selection at transmitter (i.e., combined scheme) over channel estimation inaccuracy in wiretap channels with multiple antennas. This paper develops the precise formulations of the combined system for the secrecy outage probability and the probability of the non-zero secrecy capacity. This work develops the diversity order for the secrecy outage probability in the presence and absence of channel inaccuracy at the combined system and eavesdropper. This work provides various numerical evaluations, showing that, the diversity order of the combined system is changed only by combined system`s channel inaccuracy not by eavesdropper's channel inaccuracy. Also, the channel inaccuracy of the eavesdropper acts on the signal to noise ratio (SNR) gain of the combined system.

IEEE Transactions on Vehicular TechnologyFeb 18, 202012 citations

Quantum probability theory

Schleissinger, Sebastian

Mathematical surveys and monographsSep 22, 20250 citations

Negative probability in the framework of combined probability

Mark Burgin

Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of conventional probability. So, it is natural that negative probability also has different axiomatic frameworks. In the previous publications (Burgin, 2009; 2010), negative probability was mathematically formalized and rigorously interpreted in the context of extended probability. In this work, the axiomatic system that synthesizes conventional probability and negative probability is constructed in the form of combined probability. In a mathematically rigorous way, both theoretical concepts - combined probability and extended probability - stretch conventional probability so that it can takes negative values. After introducing axioms for combined probability, we study its properties, as well as relations to extended probability and conventional probability.Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of conventional probability. So, it is natural that negative probability also has different axiomatic frameworks. In the previous publications (Burgin, 2009; 2010), negative probability was mathematically formalized and rigorously interpreted in the context of extended probability. In this work, the axiomatic system that synthesizes conventional probability and negative probability is constructed in the form of combined probability. In a mathematically rigorous way, both theoretical concepts - combined probability and extended probability - stretch conventional probability so that it can takes negative values. After introducing axioms for combined probability, we study its properties, as well as relations to extended probability and conventional probability.

arXiv:1306.1166May 31, 20132 citations

d

-level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.