Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. In this work we characterize the quantum version of this task in the one-shot setting using the smooth entropy framework and one-shot operational quantities. We further establish asymptotic equipartition properties for our correlation measures. We also introduce entanglement versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling.
Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. In this work we characterize the quantum version of this task in the one-shot setting using the smooth entropy framework and one-shot operational quantities. We further establish asymptotic equipartition properties for our correlation measures. We also introduce entanglement versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling.
Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.
The novel ways in which correlations may be encoded non-locally in quantum mechanics have led to the promise of new technologies. This, in turn, has led to the first attempts at the development of quantum networks. Nonetheless, our understanding of quantum correlations and the limits of how we may process them over a quantum network remains nascent. This thesis studies the limits of processing quantum correlations between spatially separated parties in various settings and the mathematical tools necessary to determine such limits. To do so, this thesis studies three general problems in quantum network theory. The first problem is the minimum resources to prepare a quantum correlation between non-communicative receivers, which may be viewed as a star network. This study is separated into three pieces. The first piece is a near-tight one-shot quantum generalization of a problem introduced by Wyner, distributed source simulation. Establishing our one-shot bounds builds on recent work on soft-covering without smoothing and relies on establishing a variety of properties for one-shot entropic quantities, as well as improving our understanding of the smooth entropy framework in quantum network information theory. As distributed source simulation cannot be achieved to arbitrary error for sufficiently entangled states, the second piece characterizes the minimal error of converting one entangled state to another using zero communication. This relies on establishing new properties of the fidelity measure. The third piece establishes an entanglement-assisted version of the distributed source simulation task that can be achieved to arbitrary error for arbitrary quantum systems. This result builds on recent work on multipartite state splitting and establishing new properties of one-shot multipartite max mutual information quantities. The second problem is the extraction of perfect classical correlation from a multipartite quantum source where the receivers do not communicate. This may be seen as the reverse process of distributed source simulation. We establish that this is asymptotically possible in quantum mechanics for the first time and establish the asymptotic rate for classical-quantum states. The technical barrier in establishing the asymptotic rate is establishing a limit on how much noiseless correlation can be extracted without communication. Our major technical contribution is establishing such a limit by generalizing a technical lemma of Gács and Körner to classical-quantum states. The final problem is distributed state discrimination of quantum product states. This is motivated by its relevance to both understanding non-locality in quantum mechanics and developing quantum cryptographic schemes. Our study again breaks into three key pieces. First, we study the framework of local operations and simultaneous classical or quantum communication (LOSCC and LOSQC respectively) for product states, which is relevant to quantum cryptography. We establish the first separation between quantum and classical communication in the simultaneous communication model using only product state inputs. We also establish a new uncertainty relation that implies error bounds for discriminating product states using LOSQC. Second, we study state discrimination under local operations and classical communication (LOCC). We develop a new framework for zero-error state discrimination under LOCC. Our framework provides a simple proof of orthogonal product states that cannot be distinguished using LOCC--- a property introduced by Bennett et al. Finally, we study the thermodynamics of zero-error distributed state discrimination in the framework of Faist et al. This allows us to resolve an open problem from the aforementioned paper of Bennett et al.
Recently, Cheng et al generalized quantum state splitting (QSS) to the multipartite setting for applications in quantum network information theory. Furthermore, Cheng and Gao recently noted that unipartite convex splitting reduces to soft covering for classical-quantum states. Soft covering is a common primitive in (classical, network) information theory that does not seem to need an extension to the multipartite setting, e.g. when establishing rates for many party distributed source simulation (DSS). This suggests a gap in our understanding of how these tasks are related. In this work we clarify this relation. We first identify “incoherent” QSS without entanglement assistance with DSS in the bipartite setting. Motivated by this, we identify “coherent” DSS as a special case of multipartite QSS. We then establish a one-shot rate region for multipartite QSS in terms of smooth multipartite mutual information quantities, which in turn implies a one-shot rate region for coherent DSS. Lastly, we establish a strong asymptotic equipartition property for mutual information quantities to establish a strong converse rate region for multipartite QSS and coherent DSS.
We study a generalized version of Wyner's common information problem (also coined the distributed source simulation problem). The original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and target distributions is the normalized relative entropy. Our generalization involves changing the distance measure to the unnormalized and normalized Rényi divergences of order α=1+s∈[0,2]. We show that the minimum rate needed to ensure the Rényi divergences between the distribution induced by a code and the target distribution vanishes remains the same as the one in Wyner's setting, except when the order α=1+s=0. This implies that Wyner's common information is rather robust to the choice of distance measure employed. As a byproduct of the proofs used to the establish the above results, the exponential strong converse for the common information problem under the total variation distance measure is established.
We consider the task of faithfully simulating a distributed quantum measurement, wherein we provide a protocol for the three parties, Alice, Bob and Charlie, to simulate a repeated action of a distributed quantum measurement using a pair of non-product approximating measurements by Alice and Bob, followed by a stochastic mapping at Charlie. The objective of the protocol is to utilize minimum resources, in terms of classical bits needed by Alice and Bob to communicate their measurement outcomes to Charlie, and the common randomness shared among the three parties, while faithfully simulating independent repeated instances of the original measurement. To achieve this, we develop a mutual covering lemma and a technique for random binning of distributed quantum measurements, and, in turn, characterize a set of sufficient communication and common randomness rates required for asymptotic simulatability in terms of single-letter quantum information quantities. In the special case, where the Charlie’s action is restricted to a deterministic mapping, we develop a one-shot performance characterization of the distributed faithful simulation problem. Furthermore, using these results we address a distributed quantum rate-distortion problem, where we characterize the achievable rate distortion region through a single-letter inner bound. Finally, via a technique of single-letterization of multi-letter quantum information quantities, we provide an outer bound for the rate-distortion region.
We consider state redistribution of a "hybrid" information source that has both classical and quantum components. The sender transmits classical and quantum information at the same time to the receiver, in the presence of classical and quantum side information both at the sender and at the decoder. The available resources are shared entanglement, and noiseless classical and quantum communication channels. We derive one-shot direct and converse bounds for these three resources, represented in terms of the smooth conditional entropies of the source state. The two bounds coincide in the asymptotic limit of infinitely many copies and vanishingly small error. Various coding theorems for two-party communication tasks are obtained by reduction from our results.