In this thesis, we focus on two well-connected topics: quantum measurement and quan-
tum uncertainty relation. The first topic is the well-known “pre- and postselected mea-
surements”. Aharonov, Bergmann, and Lebowitz (ABL) [1, 2] coined the term “pre- and
postselections” to address the issue of temporal asymmetry in quantum mechanics. In the later years, Aharonov, Albert, and Vaidman (AAV) [3] introduced the notion of “weak
value” of an observable and it is considered to be one of the strangest findings in a pre-
and postselected (PPS) system for being a complex number and it can take values outside the max-min range of the eigenvalues of the observable when the overlap between the pre-and postselections is very small. Although the construction of the PPS system (includ-ing ABL and AAV) remained under controversies because of its unusual and contrasting behaviour compared to the standard quantum system, the discovery of weak values, the weak values of tensor product observables and higher moment weak values have provided immense practical applications, to resolve some paradoxes and some unthinkable stuffs.
One of the main studies in this thesis is to derive and show applications of product and
higher moment weak values.
The second topic of this thesis is uncertainty relation. The Robertson-Heisenberg uncer-
tainty relation (RHUR) [4] is one of the most important uncertainty relations in quantum
mechanics after the discovery of the uncertainty principle due to Heisenberg [5]. The
RHUR describes the difficulty of jointly sharp preparation of a quantum state for incom-
patible observables. It is needless to mention how important uncertainty relations are
fundamentally and practically in quantum systems. There are three types of uncertainty
relations in three different contexts which we derive and study here. The first uncertainty
relation is derived in PPS system. The second uncertainty relation (or more precisely
uncertainty equality), is based on standard deviation for mixed states and along with, we derive our third uncertainty relation based on skew information (which was introduced by Wigner and Yanase [6] to quantify the quantum uncertainty in measurements under conservation laws) and its extended versions by Dyson and others. Following are the themes
of this thesis. We demonstrate that pairwise orthogonal postselections can be used to obtain higher moment weak values. By measuring only local weak values (defined as single system weak values in a multipartite scenario), product weak values can be obtained. As applications, we use product and higher moment weak values to reconstruct quantum states showing advantages over previous works in terms of number of required measurement operators and experimental feasibility. Additionally, a necessary separability criteria is given using
product weak values to detect entanglement. For some classes of entangled states, posi-
tive partial transpose (PPT) criteria is achieved by cleverly choosing product observables
and postselections.
As PPS systems are useful practically as well as fundamentally, then an immediate ques-
tion can be asked whether there exists any uncertainty relation which can give the limita-
tions on joint sharp preparation of the given pre- and post-selected states when noncom-
muting observables are measured. We confirm the existance of Robertson-Heisenberg like
uncertainty relation for two incompatible observables in a PPS system. The newly defined
standard deviation and the uncertainty relation in the PPS system have physical meanings
which we present here. We demonstrate two unusual properties in the PPS system using
our uncertainty relation. First, commuting observables can disturb each other’s measure-
ment results in a PPS system which is in fully contrast with the RHUR. Second, unlike
the standard quantum system, the PPS system makes it feasible to prepare a quantum state
(preselection) sharply for noncommuting observables. Some applications of uncertainty
and uncertainty relation in the PPS system are provided.
We derive uncertainty equalities for skew informations of two arbitrary incompatible op-
erators which contains the commutator of the two incompatible operators. For the first
time, we derive state-independent uncertainty relations based on the skew information for a collection of arbitrary operators. We derive standard deviation based sum and product uncertainty equalities for mixed states, a generalization of the previous works where only pure states were considered. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase-Dyson skew information and the Fisher information as special cases hold. Finally, we provide a scheme to determine the Wigner-Yanase-Dyson skew information of an unknown observable which can be performed in experiment using the notion of weak values.
