Uncertainty principle, one of the most iconic features of quantum mechanics, was originally viewed as a fundamental limitation. Since the inception of quantum information science, researchers began to use it to achieve quantum advantages. To better understand the origin of these advantages, an essential question is: To what extent can the uncertainty principle's signatures be simulated by a single measurement? As a single measurement clearly cannot demonstrate the uncertainty principle, such a simulation, if exists, implies the claimed advantages may either stem from other quantum features, or just be reproducible in a less resourceful way. In this work, we report a series of noise-robust no-go theorems, showing that strong enough signatures of uncertainty principle cannot be simulated by a single measurement, even when assisted by quantum pre- or post-processing. This signature is modelled by complementary instruments. We completely characterise complementary instruments by a numerically feasible measure and show that they are necessary and sufficient resources for the advantage in an operational task that aims to unambiguously send classical information.
Uncertainty principle, one of the most iconic features of quantum mechanics, was originally viewed as a fundamental limitation. Since the inception of quantum information science, researchers began to use it to achieve quantum advantages. To better understand the origin of these advantages, an essential question is: To what extent can the uncertainty principle's signatures be simulated by a single measurement? As a single measurement clearly cannot demonstrate the uncertainty principle, such a simulation, if exists, implies the claimed advantages may either stem from other quantum features, or just be reproducible in a less resourceful way. In this work, we report a series of noise-robust no-go theorems, showing that strong enough signatures of uncertainty principle cannot be simulated by a single measurement, even when assisted by quantum pre- or post-processing. This signature is modelled by complementary instruments. We completely characterise complementary instruments by a numerically feasible measure and show that they are necessary and sufficient resources for the advantage in an operational task that aims to unambiguously send classical information.
Abstract Can quantum processes be simulated using only classical resources? This question delineates the boundary between classical and quantum models and clarifies the origin of quantum advantage in information processing. We address this question through the task of quantum channel simulation, where a sender (Alice) holds the classical description of a quantum state and wishes to transmit it to a receiver (Bob) for measurement. Prior work has shown that, for qubit channels, 2 bits of forward communication with shared randomness suffice to reproduce the statistics of any single-qubit measurement. We argue, however, that true channel simulation requires reproducing statistics of joint measurements—including entangled effects—on Alice’s state and an auxiliary system held by Bob. Such scenarios naturally arise in network communication, where some nodes know the state, while others do not. We prove that a perfect qubit channel cannot be simulated with any finite amount of classical communication, even using the most general multi-round, bidirectional protocols. We further show that this no-go result is rooted in the necessity of reproducing statistics associated with entangled effects. On the other hand, we show that noisy qubit channels, such as depolarizing channels, admit classical simulation, though the required communication diverges as noise decreases.
Measurement incompatibility is a fundamental quantum resource that enables advantages in many quantum information tasks, including cryptography and communication. However, unavoidable interactions between a system and its environment can degrade or even completely destroy measurement incompatibility; such a process is referred to as a measurement-incompatibility-annihilating channel. Recently, the capability of noisy quantum dynamics to preserve measurement incompatibility has been characterized within the resource theory of measurement incompatibility preservability. This motivates studying how to purify the preservation of measurement incompatibility and how to activate it from a measurement-incompatibility-annihilating channel. To this end, we first introduce our figures of merit as robustness-based resource monotones within this resource theory. We demonstrate that while pre-filtering operations can strengthen measurement incompatibility preservability, they cannot activate it from an incompatibility-annihilating channel, establishing a no-go theorem. Furthermore, we explicitly demonstrate that an incompatibility-annihilating channel can be stochastically activated via post-filtering operations. Our results provide a practical framework for exploiting measurement incompatibility in quantum information processing.
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.
Complementarity is a phenomenon explaining several core features of quantum theory, such as the well-known uncertainty principle. Roughly speaking, two objects are said to be complementary if being certain about one of them necessarily forbids useful knowledge about the other. Two quantum measurements that do not commute form an example of complementary measurements, and this phenomenon can also be defined for ensembles of states. Although a key quantum feature, it is unclear whether complementarity can be understood more operationally, as a necessary resource in some quantum information task. Here we show this is the case, and relates to a novel task which we term η-unambiguous exclusion. As well as giving complementarity a clear operational definition, this also uncovers the foundational underpinning of unambiguous exclusion tasks for the first time. We further show that a special type of measurement complementarity is equivalent to advantages in certain encryption tasks. Finally, our analysis suggest that complementarity of measurement and state ensemble can be interpreted as strong forms of measurement incompatibility and quantum steering, respectively.
One of quantum theory's salient features is its apparent indeterminism, i.e. measurement outcomes are typically probabilistic. We formally define and address whether this uncertainty is unavoidable or whether post-quantum theories can offer a predictive advantage while conforming to the Born rule on average. We present a no-go claim combining three aspects: predictive advantage, no-signalling, and reliable intersubjectivity between quantum observers. The results of the analysis lead to the conclusion that there exists a fundamental limitation on genuine predictive advantage. However, we uncover a fascinating possibility: When the assumption of reliable intersubjectivity between different observers is violated, subjective predictive advantage can, in principle, exist. This, in turn, entails an epistemic boundary between different observers of the same theory. The findings reconcile us to quantum uncertainty as an aspect of limits on Nature's predictability.
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.