Coherence and contextuality as quantum resources

Abstract

In this thesis, we explore the intersection of two fundamental subfields of quantum information theory: quantum coherence and contextuality. Despite their apparent differences, both areas address key issues relevant to the foundations and applications of quantum theory. By developing a novel graph-theoretic approach, extending a framework recently introduced by Galvão and Brod (Phys. Rev. A 101, 062110, 2020), we establish a formal connection between inequality-based witnesses of quantum coherence and noncontextuality inequalities. Our key contributions include: the development of a graph-theoretic framework for generating coherence and contextuality witnesses; a formal mapping between the inequalities that follows from the work by Galvão and Brod to existing noncontextuality inequalities; the conceptualization of a relational form of quantum coherence; a proof of contextual advantage for the task of quantum interrogation; and the discovery of an infinite family of coherence witnesses that also require quantum states in Hilbert spaces of specific dimensions.

Figures (36)

• Figure 1: Joint measurements, and the notion of a compatibility scenario. A device prepares boxes sent to two workers, lined up in sequence one after the other. (left) Each worker is allowed to make one single dichotomic measurement, and for each box, they choose pairs of measurements $\{x_i,x_j\}$ for each of them to make. It does not matter the order in which the measurements are made. For each box, they mark their joint outcomes and at the end of the day, they end up with statistics $\{p(s_is_j|x_ix_j)\}$ related to their joint operations. The pairs $\{x_i,x_j\}$ are called contexts, and the operation is said to be a joint measurement having joint outcomes $s_i$ and $s_j$, respectively. (right) Alternatively, a single worker can perform both measurements and report their joint statistics. In our \ref{['example: boxes']}, it is simple to see that this is possible because, say, estimating $x_1$ (the number of shoes inside the box) and $x_3$ (whether the shoes are for running) can both be learned jointly when the worker opens the box and be used to gather the final statistics $\{p(s_is_j|x_ix_j)\}$.
• Figure 2: Testing the assumption of ideal measurements. (left) If the measurements performed are ideal, they always yield the same outcome when repeated over the same system. For instance, if the first worker performs the measurement $x_1$ and finds out that instead of a pair of shoes, there is some other number of shoes inside the box, say, one, when the second worker makes the measurement $x_1$ the same outcome must be observed. (right) If the measurements are ideal, they cannot disturb the outcomes of measurements in the same context. Hence if the first worker makes a measurement $x_1$ and obtains $\neg 2$, the second worker makes a measurement $x_2$, testing if the shoes are black or not, and obtains $\mathrm{black}$, because $\{x_1,x_2\}$ is a context if a third measurement of $x_1$ is made afterward, the outcome of it must always be $\neg 2$.
• Figure 3: Kochen--Specker noncontextual ontological model. If there is a ks noncontextual model for a given behavior $B$ in a scenario, the model provides a 'story' for it similarly as above. Some initial source of randomness $\mu(\lambda)$ prepares the system in some unknown state $\lambda$ that completely characterizes the state of affairs of the physical system. Once this is done, measurements in the same context can be interpreted as merely revealing pre-determined properties that are deterministically fixed by $\lambda$ and by the chosen measurement. The probability of observing these outcomes is entirely independent of which context they are being performed in. In the figure, as soon as the shoes are put inside the box, measurements merely reveal the properties of the shoes, which are fixed. The case shows that there are two shoes and their color is black.
• Figure 4: Example of an exclusivity graph. The nodes of the graph represent events that can be related to some compatibility scenario. Not all events are necessarily present. There exists an edge between nodes if and only if the events are exclusive. For example $00|11$ and $11|10$ are exclusive because of the measurement labeled as $1$, as well as $11|10$ and $00|00$ because of the measurement labeled as $0$.
• Figure 5: Exclusivity graph for the kcbs inequality. Nodes of the graph represent events present in the kcbs inequality \ref{['eq: exclusivity KCBS inequality']}. There exists an edge between nodes if, and only if (iff) the events are exclusive, i.e., the same measurement in both events yields different measurement outcomes. For example, $01|01$ and $01|40$ are exclusive because of the measurement labeled as $0$. The events $01|01$ and $01|12$ are exclusive because of the measurement labeled as $1$.

Theorems & Definitions (279)

• Definition 2.1: Basis-dependent coherence
• Definition 2.2: Coherence rank
• Definition 2.3: Coherence quantifiers
• Example 2.4: Dimension gap between coherence quantifiers
• Definition 2.5: Witness
• Definition 2.6: Operator witnesses
• Definition 2.7: Basis-dependent imaginarity
• Definition 2.8: Set coherence
• Definition 2.9: Set imaginarity
• Example 3.1: Joint measurements and (compatibility) scenarios