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.
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.
Nonlocality is at the heart of quantum information processing. In this paper we investigate the minimum amount of classical communication required to simulate a nonlocal quantum measurement. We derive general upper bounds, which in turn translate to systematic classical simulations of quantum communication protocols.As a concrete application, we prove that any quantum communication protocol with shared entanglement for computing a Boolean function can be simulated by a classical protocol whose cost does not depend on the amount of the shared entanglement. This implies that if the cost of communication is a constant, quantum and classical protocols, with shared entanglement and shared coins, respectively, compute the same class of functions.Furthermore, we describe a new class of efficient quantum communication protocols based on fast quantum algorithms. While some of them have efficient classical simulations by our method, others appear to be good candidates for separating quantum v.s. classical protocols, and quantum protocols with v.s. without shared entanglement.Yet another application is in the context of simulating quantum correlations using local hidden variable models augmented with classical communications. We give a constant cost, approximate simulation of quantum correlations when the number of correlated variables is a constant, while the dimension of the entanglement and the number of possible measurements can be arbitrary.Our upper bounds are expressed in terms of some tensor norms on the measurement operator. Those norms capture the nonlocality of bipartite operators in their own way and may be of independent interest and further applications.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form F(x,y)=f(x1∧y1,…,xn∧yn), when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors.
In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘AND2 are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of f.
Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.
A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions F of the following form: F(x,y)=fn(x1⋅y1,x2⋅y2,...,xn⋅yn), where fn is a {\em symmetric} Boolean function on n Boolean inputs, and xi, yi are the i'th bit of x and y, respectively. His proof critically depends on the symmetry of fn. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F is the ``block-composition'' of a ``building block'' gk:{0,1}k×{0,1}k→{0,1}, and an fn:{0,1}n→{0,1}, such that F(x,y)=fn(gk(x1,y1),gk(x2,y2),...,gk(xn,yn)), where xi and yi are the i'th k-bit block of x,y∈{0,1}nk, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.
We initiate the study of quantifying nonlocality of a bipartite measurement by the minimum amount of classical communication required to simulate the measurement. We derive general upper bounds in terms of some tensor norms of the measurement operator. As applications, we show that (a) if the amount of communication is a constant, then quantum and classical communication protocols with an unlimited amount of shared entanglement or shared randomness compute the same class of functions; and (b) it requires only a constant amount of communication to classically generate an approximation of the output distribution resulting from local measurements on an entangled quantum state, as long as the number of measurement outcomes is a constant.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for And-functions of the form F(x,y) = f(x₁ ∧ y₁, …, x_n ∧ y_n), when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all And-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘And₂ are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized - up to polynomial loss - by the logarithm of the De Morgan sparsity of f. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett [Arkadev Chattopadhyay et al., 2026] on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general And-functions.
We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a,b distributed according to some pre-specified joint distribution p(a,b∣x,y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa.~~~By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. ~~~ Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.