Characterizing the projective simulability of quantum measurements and instruments

Abstract

In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength.

The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined.

For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.

Popular Abstract

Is it not beautiful to throw oneself into the abyss of unknowingness? In her poem, Utopia, Wisława Szymborska wrote: "As if all you can do here is leave and plunge, never to return, into the depths. Into unfathomable life." The conventional interpretation is of a political nature; however, I would like to stretch it to the extreme and read it through the eyes of a physicist. Through these new lenses, the isolated island called Utopia, where "The Tree of Understanding, dazzlingly straight and simple, sprouts by the spring called Now I Get It", is an allegorical description of classical physics. Seemingly disentangled and complete, it is nevertheless merely a small part of the story, part of something far bigger. The realm of quantum mechanics stretches across the island and far into the sea surrounding it, where physics becomes a conundrum.

In an attempt to tangent the tiny world we know very little about, quantum measurements and instruments play a key role. In a classical physics picture, the attributes of physical objects are well-defined and revealed by performing a measurement. In contrast, the characteristics of quantum physical systems can be viewed as emerging in the course of the measurement process itself. Depending on the measurement procedure used, different aspects of the information are obtained.

In quantum mechanics, the general measurement procedures that can be performed on a quantum system are described by mathematical objects called Positive Operator-Valued Measures (POVMs). A special class of such measurements is the Projective-Valued Measures (PVMs). The POVMs that do not belong to this set are complementarily called non-projective measurements. PVMs are generally easy to implement in an experiment and are considered standard measurements in quantum mechanics. However, for many purposes, they do not give a good picture of the characteristics. Instead, it is more advantageous to use non-projective POVMs to reveal more useful information. However, such measurements are generally difficult to implement experimentally as they require more resources.

It is known that, in the set of non-projective POVMs, there exist measurements that can be simulated by a mixture of PVMs. These POVMs offer no advantage over PVMs, as they reveal the same informational aspects and can therefore be implemented using the simpler projective schemes. On the other hand, for the POVMs that cannot be simulated projectively, we cannot resort to simpler experimental implementations, and the data retrieved might be more interesting. Such measurements are genuinely non-projective. It is, however, known that some of these genuinely non-projective POVMs can be easily made projectively simulable by perturbing them with noise. Some POVMs require more noise and some less. If a POVM requires a large amount of noise, it is more non-projective than one that needs less noise. Very non-projective POVMs are of greater interest as their statistics are very far from resembling the statistics of PVMs. Therefore, it might be worth the trouble of implementing them.

In the spirit of this, this thesis is about drawing the border between projectively simulable POVMs and genuinely non-projective POVMs. To this end, we characterize the degree of non-projectiveness among genuinely non-projective POVMs and ask which quantum measurements are the most non-projective.

To extend the picture of measurements, the thesis also treats quantum instruments. Imagine an electron moving in space. When we measure its properties, e.g., position, we register a classical value. Hence, a quantum measurement device takes in a quantum state and outputs a readable classical outcome. However, the act of measuring perturbs the state of the electron, leaving it in a changed state after the measurement has been performed. A quantum instrument is a mathematical model that describes both the classical outcome and the post-measurement state of the electron. Thus, a quantum instrument has an associated measurement part and can therefore be seen as an extension of the measurement. Consequently, the same questions can be asked in the context of instruments. For this purpose, a search algorithm is developed for finding the most non-projective instruments.

• author: Strömberg, Julia
• supervisor: Armin Tavakoli; Gabriele Cobucci; Shishir Khandelwal
• organization: Mathematical Physics; Department of Physics
• course: PHYM01 20261
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• language: English
• id: 9229631
• date added to LUP: 2026-06-03 08:22:49
• date last changed: 2026-06-03 08:22:49