The concepts of quantum correlation and communication complexity have been proposed to quantify the worst-case computational resources needed in generating bipartite classical distributions or bipartite quantum states in the single-shot setting by Zhang (Proc. 3rd Innov Theor Comput., pp. 39-59, 2012). The former characterizes the minimal size of the initial shared state needed if no communication is allowed, and the latter characterizes the minimal amount of communication needed if sharing nothing at the beginning. In this paper, we generalize these two concepts to approximate bipartite cases and various multipartite cases. 1. To generate a bipartite classical distribution P (x, y) approximately, we show that the minimal amount of quantum communication needed can be characterized completely by the approximate PSD-rank of P . This result is obtained based on the fact that the cost to approximate a bipartite quantum state equals that to approximate its exact purifications, which also implies a result for a general bipartite quantum state. 2. In the bipartite case, correlation complexity is always equivalent to communication complexity, but in multipartite cases, this is not true any more. For multipartite pure states, we show that both of them could be characterized by local ranks of subsystems. 3. For any bipartite quantum state, its correlation complexity is always equivalent to that of its optimal purification, while in multipartite cases the former could be smaller strictly than the latter. We characterize the relationship between them by giving upper and lower bounds. 4. For multipartite classical distributions, we show that the approximate correlation complexity could also be characterized by the generalized PSD-rank.
The concepts of quantum correlation and communication complexity have been proposed to quantify the worst-case computational resources needed in generating bipartite classical distributions or bipartite quantum states in the single-shot setting by Zhang (Proc. 3rd Innov Theor Comput., pp. 39-59, 2012). The former characterizes the minimal size of the initial shared state needed if no communication is allowed, and the latter characterizes the minimal amount of communication needed if sharing nothing at the beginning. In this paper, we generalize these two concepts to approximate bipartite cases and various multipartite cases. 1. To generate a bipartite classical distribution P (x, y) approximately, we show that the minimal amount of quantum communication needed can be characterized completely by the approximate PSD-rank of P . This result is obtained based on the fact that the cost to approximate a bipartite quantum state equals that to approximate its exact purifications, which also implies a result for a general bipartite quantum state. 2. In the bipartite case, correlation complexity is always equivalent to communication complexity, but in multipartite cases, this is not true any more. For multipartite pure states, we show that both of them could be characterized by local ranks of subsystems. 3. For any bipartite quantum state, its correlation complexity is always equivalent to that of its optimal purification, while in multipartite cases the former could be smaller strictly than the latter. We characterize the relationship between them by giving upper and lower bounds. 4. For multipartite classical distributions, we show that the approximate correlation complexity could also be characterized by the generalized PSD-rank.
We investigate the fundamental problem of generating bipartite classical distributions or quantum states. By designing efficient communication protocols and proving their optimality, we establish a number of intriguing connections to fundamental measures in optimization, convex geometry, and information theory. 1) To generate a classical distribution P(x,y), we tightly characterize the minimum amount of quantum communication needed by the psd-rank of P (as a matrix), a measure recently proposed by Fiorini et al. (Proc. 44th ACM Symp. Theory Comput., pp. 95-106, 2012) in studies of the minimum size of extended formulations of optimization problems such as TSP. This echos the previous characterization for the optimal classical communication cost by the nonnegative rank of P. The result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it. 2) When an approximation ϵ is allowed to generate a distribution (X,Y)~P, we present a classical protocol of the communication cost O((C(X,Y)+1)/ϵ, where C(X,Y) is common information, a well-studied measure in information theory introduced by Wyner (IEEE Trans. Inf. Theory, 21 (2):163-179, 1975). This also links nonnegative rank and common information, two seemingly unrelated quantities in different fields. 3) For approximately generating a quantum pure state |ψ〉, we completely characterize the minimum cost by a corresponding approximate rank, closing a possibly exponential gap left in Ambainis etal. (SIAM J. Comput., 32 (6):1570-1585, 2003).
We study the correlation complexity (or equivalently, the communication complexity) of generating a bipartite quantum state ρ. When ρ is a pure state, we completely characterize the complexity for approximately generating ρ by a corresponding approximate rank, closing a gap left in Ambainis, Schulman, Ta-Shma, Vazirani and Wigderson (SIAM Journal on Computing, 32(6):1570-1585, 2003). When ρ is a classical distribution P(x,y), we tightly characterize the complexity of generating P by the psd-rank, a measure recently proposed by Fiorini, Massar, Pokutta, Tiwary and de Wolf (STOC 2012). We also present a characterization of the complexity of generating a general quantum state ρ.
We investigate the fundamental problem of generating bipartite classical distributions or quantum states. By designing efficient communication protocols and proving their optimality, we establish a number of intriguing connections to fundamental measures in optimization, convex geometry, and information theory. 1. To generate a classical distribution P(x, y), we tightly characterize the minimum amount of quantum communication needed by the psd-rank of P (as a matrix), a measure recently proposed by Fiorini, Massar, Pokutta, Tiwary and de Wolf (Proceedings of the 44th A CM Symposium on Theory of Computing, pages 95–106, 2012) in studies of the minimum size of extended formulations of optimization problems such as TSP. This echos the previous characterization for the optimal classical communication cost by the nonnegative rank of P. The result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it. 2. When an approximation of ε is allowed to generate a distribution (X, Y) ∼ P, we present a classical protocol of the communication cost O((C(X, Y) + 1)/ε), where C(X, Y) is common information, a well-studied measure in information theory introduced by Wyner (IEEE Transactions on Information Theory, 21(2):163–179, 1975). This also links nonnegative rank and common information, two seemingly unrelated quantities in different fields. 3. For approximately generating a quantum pure state |ψ〉, we completely characterize the minimum cost by a corresponding approximate rank, closing a possibly exponential gap left in Ambainis, Schulman, Ta-Shma, Vazirani and Wigderson (SIAM Journal on Computing, 32(6):1570–1585, 2003).
We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information k, showing that it is Θ(n(k+1)) for all 0≤k≤n. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case (k=0), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs Ω(n) bits of mutual information in the corresponding distribution. We study the same question in the distributional quantum setting, and show a lower bound of Ω((n(k+1))1/4), and an upper bound, matching up to a logarithmic factor. We show that there are total Boolean functions fd on 2n inputs that have distributional communication complexity O(logn) under all distributions of information up to o(n), while the (interactive) distributional complexity maximised over all distributions is Θ(logd) for 6n≤d≤2n/100. We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error ε is multiplicative in log(1/ε) -- the previous upper bounds had the dependence of more than 1/ε.
We consider a realistic setting of quantum tasks that generate shared bipartite quantum states. Suppose \alice and \bob are located at different places and need to produce a target shared quantum state ρ. In order to save quantum communication, they can choose to share a proper smaller quantum state σ first, and then turn σ to ρ by performing only local quantum operations and classical communications (LOCC). We hope σ is the optimal such that the quantum communication needed is as little as possible, which is called the generation cost of ρ. In this paper, for an arbitrary bipartite ρ, we characterize its generation cost completely by proving that it is exactly equivalent to the logarithm of the Schmidt number of ρ. Similar quantum schemes where classical communication is not allowed have actually been considered. By comparing the two settings, we are able to look into the role that classical communication plays in these fundamental tasks, where we exhibit some instances in which classical communication is not helpful completely.
This dissertation is motivated by the following fundamental questions: (a) are there any exponential gaps between quantum and classical communication complexities? (b) what is the role of entanglement in assisting quantum communications? (c) how to characterize the nonlocality of quantum operations? We study four specific problems below. 1. The communication complexity of the Hamming Distance problem. The Hamming Distance problem is for two parties to determine whether or not the Hamming distance between two n-bit strings is more than a given threshold. We prove tighter quantum lower bounds in the general two-party, interactive communication model. We also construct an efficient classical protocol in the more restricted Simultaneous Message Passing model, improving previous results. 2. The Log-Equivalence Conjecture. A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the two-party, interactive model. The answer is believed to be No. Razborov proved this conjecture for the most general class of functions so far. We prove this conjecture for a broader class of functions that we called block-composed functions. Our proof appears to be the first demonstration of the dual approach of the polynomial method in proving new results. 3. Classical simulations of bipartite quantum measurement. We define a new ix concept that measures the nonlocality of bipartite quantum operations. From this measure, we derive an upper bound that shows the limitation of entanglement in reducing communication costs. 4. The maximum tensor norm of bipartite superoperators. We define a maximum tensor norm to quantify the nonlocality of bipartite superoperators. We show that a bipartite physically realizable superoperator is bi-local if and only if its maximum tensor norm is 1. Furthermore, the estimation of the maximum tensor norm can also be used to prove quantum lower bounds on communication complexities.