Abstract We introduce a class of communication tests where the task is to communicate partial ignorance by means of a physical system. We present a full characterization of the implementations of these tests in the qubit case and partial results for qudits. A peculiar observation is that two physical systems with the same operational dimensions may differ with respect to implementations of these tasks, as is shown to be the case for the qubit and rebit. Finally, we consider the natural question whether some of the communication tests are more difficult than others. A new preordering that we call the ultraweak matrix majorization is presented to answer this question in a theory-independent way.
Abstract We introduce a class of communication tests where the task is to communicate partial ignorance by means of a physical system. We present a full characterization of the implementations of these tests in the qubit case and partial results for qudits. A peculiar observation is that two physical systems with the same operational dimensions may differ with respect to implementations of these tasks, as is shown to be the case for the qubit and rebit. Finally, we consider the natural question whether some of the communication tests are more difficult than others. A new preordering that we call the ultraweak matrix majorization is presented to answer this question in a theory-independent way.
Abstract We investigate the question which communication tasks can be accomplished within a given operational theory. The concrete task is to find out which communication matrices have a prepare-and-measure implementation with multiple states and a single measurement from a given theory, without using shared randomness. To set a general framework for this question we develop the ultraweak matrix majorization in the set of communication matrices. This preorder gives us means to determine when one communication task is more difficult than another. Furthermore, we introduce several monotones which can be used to compare and characterize the communication matrices. We observe that not only do the monotones allow us to compare communication matrices, but also their maximal values in a given theory are seen to relate to some physical properties of the theory. The maximal values can then be thought as ‘dimensions’, allowing us to compare different theories to each other. We analyse the introduced monotones one by one and demonstrate how the set of implementable communication matrices is different in several theories with the focus being mainly on the difference between classical and quantum theories of a given dimension.
We look into the task of differentiating between any two quantum channels and reconstructing them from the obtained measurement statistics with possibly limited information about the experimental set-up. We employ the communication matrix formalism where the measurement statistics of a prepare-and-measure scenario is represented as a stochastic communication matrix. In order to differentiate between any two quantum channels, the informational completeness of the set-up is both necessary and sufficient. On the other hand, if we want to uniquely characterize any quantum channel, in addition we also need to have a complete description of the set-up. We show that in many important cases we can deduce this information directly from the communication matrix of the set-up before applying the channel. Given that we trust the dimension of the system, we show that we can deduce the information completeness of the set-up directly from the rank of the communication matrix. Furthermore, we show that another quantity of the communication matrix, called the information storability, can be used to self-test the set-up (up to unitary or antiunitary freedom) for an important class of states and measurements. This provides us a semi-device-independent way to identify quantum channels from the prepare-and-measure statistics. Lastly, we consider scenarios where we might have some additional information about the channels or additional resources at our disposal which could help us relax some of the assumptions of the proposed scenario.
We introduce a generalization of the communication of partial ignorance where both parties of a prepare-and-measure setup receive inputs from a third party before a success metric is maximized over the measurements and preparations. We call these communication scenarios partial ignorance communication tasks of type Tn,m, where n indicates the number of possible correct answers in the communication task and m loosely represents the information about the correct answer that is revealed to the party preparing states. Various methods are used to obtain bounds on the success metrics, including semidefinite programs, ultraweak monotones for communication matrices, and frame theory for quantum states. The simplest scenarios in this alternative generalized prepare-and-measure setting, partial-ignorance communication tasks of types T3,1, T4,1, and T4,2, are analyzed exhaustively for bits and qudits. Finally, this generalized setting allows the introduction of operational equivalences to the preparations and measurements, allowing us to analyze and observe a contextual advantage for quantum theory in the partial ignorance communication task of type T4,1.
The notion of a qubit is ubiquitous in quantum information processing. In spite of the simple abstract definition of qubits as two-state quantum systems, identifying qubits in physical systems is often unexpectedly difficult. There is an astonishing variety of ways in which qubits can emerge from devices. What essential features are required for an implementation to properly instantiate a qubit? We give three typical examples and propose an operational characterization of qubits based on quantum observables and subsystems.
Quantum communication is often investigated in scenarios where only the dimension of Hilbert space is known. However, assigning a precise dimension is often an approximation of what is actually a higher-dimensional process. Here, we introduce and investigate quantum information encoded in carriers that nearly, but not entirely, correspond to standard qudits. We demonstrate the relevance of this concept for semi-device-independent quantum information by showing how small higher-dimensional components can significantly compromise the conclusions of established protocols. Then we provide a general method, based on semidefinite relaxations, for bounding the set of almost qudit correlations, and apply it to remedy the demonstrated issues. This method also offers a novel systematic approach to the well-known task of device-independent tests of classical and quantum dimensions with unentangled devices. Finally, we also consider viewing almost qubit systems as a physical resource available to the experimenter and determine the optimal quantum protocol for the well-known random access code.
Quantum information has the potential to disrupt the present computational landscape. Much of this potential rests on the existence of efficient quantum algorithms for classically intractable problems and of quantum cryptographic protocols for tasks that are provably impossible to realize classically. At the heart of many quantum advantages is one of the most counterintuitive features of quantum mechanics, known as entanglement. The central motivating question of this thesis is the following: if quantum devices will perform tasks that are beyond the reach of classical devices, can we hope to certify that they are performing these tasks correctly? Bell's theorem, a landmark result in physics, provides a partial answer to this question: it asserts that measurements on spatially isolated, but entangled, particles can result in outcomes that are correlated in a way that cannot be explained by any local hidden variable theory (such as Newtonian physics). A direct operational consequence of this theorem is that one can devise a statistical test to certify the presence of entanglement (and hence of genuine quantumness). Remarkably, nature allows us to take this certification one step further: in some cases, the correlation of measurement outcomes is sufficient to single out a unique quantum setup compatible with this correlation. This phenomenon is often referred to as self-testing, and is the central topic of this thesis. In the first part of this thesis, we review the basic terminology and results in the theory of self-testing. We then explore a concrete application to the problem of verifiably delegating a quantum computation. Our main technical contribution is a test that robustly certifies products of single-qubit Clifford measurements on many EPR pairs. We employ this test to obtain a protocol which allows a classical user to verifiably delegate her quantum computation to two spatially isolated quantum servers. The overall complexity of our protocol is near-optimal, requiring resources that scale almost linearly in the size of the circuit being delegated. In the second part of this thesis, the driving question is the following: what is the class of quantum states and measurements that can be certified through self-testing? Does self-testing only apply to a few special cases, like EPR pairs or copies of EPR pairs, or are these instances of a more general phenomenon? One of the main results of this thesis is that we settle this question for the case of bipartite states. We show the existence of a self-testing correlation for any pure bipartite entangled state of any finite local dimension. We then move on to explore the multipartite case, and we show that a significantly larger class of states can be self-tested than was previously known. This includes all multipartite partially entangled GHZ states, and more generally all multipartite qudit states which admit a Schmidt decomposition. In the final part of this thesis, we explore connections of the theory of self-testing to basic questions about entanglement and quantum correlation sets. In particular, we set out to understand the expressive power of infinite-dimensional quantum systems. We consider two questions: can spatially isolated quantum systems of infinite dimension produce correlations that are unattainable by finite-dimensional systems? Does there exist a correlation that cannot be attained exactly by spatially isolated quantum systems (not even infinite-dimensional ones), but can be approximated arbitrarily well by a sequence of finite or infinite-dimensional systems? The first question was posed by Tsirelson in 1993, and its answer has been elusive. One of the main results of this thesis is a resolution of this question. The second question is better known as the "non-closure of the set of quantum correlations", and was answered affirmatively in a breakthrough of Slofstra. We give a new elementary proof of this result which leverages one of our self-testing results and a phenomenon known as embezzlement.