Quantum Entanglement and Communication Complexity

Distributed computing, involving multiple servers collaborating on designated computations, faces a critical challenge in optimizing interserver communication---an issue central to the study of communication complexity. Quantum resources offer advantages over classical methods in addressing this challenge. In this work, we investigate a distributed computing scenario with multiple senders and a single receiver, establishing a scalable advantage of multipartite quantum entanglement in mitigating communication complexity. Specifically, we demonstrate that when the receiver and the senders share a multiqubit Greenberger-Horne-Zeilinger (GHZ) state---a quintessential form of genuine multipartite entanglement---certain global functions of the distributed inputs can be computed with only one bit of classical communication from each sender. In contrast, without entanglement, two bits of communication are required from all but one sender. Consequently, quantum entanglement reduces communication overhead by $n\ensuremath{-}1$ bits for $n$ senders, allowing for arbitrary scaling with an increasing number of senders. We also show that the entanglement-based protocol exhibits significant robustness under white noise, thereby establishing the potential for experimental realization of this quantum advantage.

Several questions related to the complexity of communication over channels with noise are addressed. We compare some of our results to wellknown results in information theory. In particular we compare the following two problems. Assuming that the communication channel between two processors P1 and P2 makes an error with probability ε≫0, the identification problem is to determine whether P1 and P2 have the same n-bit integer. The decoding problem is for P2 to determine the n-bit integer of P1. For the latter problem we show that given any arbitrarily large constant λ≫0, there exists an ε, 0≪ε≪1/2, for which no scheme requiring less than λn bits of communication can guarantee (for large n) any bound q≪1 on the error probability. On the other hand, given any arbitrarily small constant γ≫0 and any ε, 0≪ε≪1/2, the identification problem can be solved with (1+γ)n bits of (one-way) communication with an error probability bounded by c2-αn, where c and α are positive constants. These techniques are extended to other problems, and a one-bit output Boolean function is shown to exhibit a similar behavior to that of the decoding problem regardless of how the input bits are partitioned among the two processors.

We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ (0, 1)n and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple [i,j,b] such that the edge (i,j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω(√n) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins.For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(√[3] n log n) bits of communication.

In this paper, we present an efficient deterministic algorithm for consensus in presence of Byzantine failures. Our algorithm achieves consensus on an $L$-bit value with communication complexity $O(nL + n^4 L^{0.5} + n^6)$ bits, in a network consisting of $n$ processors with up to $t$ Byzantine failures, such that $t<n/3$. For large enough $L$, communication complexity of the proposed algorithm approaches $O(nL)$ bits. In other words, for large $L$, the communication complexity is linear in the number of processors in the network. This is an improvement over the work of Fitzi and Hirt (from PODC 2006), who proposed a probabilistically correct multi-valued Byzantine consensus algorithm with a similar complexity for large $L$. In contrast to the algorithm by Fitzi and Hirt, our algorithm is guaranteed to be always error-free. Our algorithm require no cryptographic technique, such as authentication, nor any secret sharing mechanism. To the best of our knowledge, we are the first to show that, for large $L$, error-free multi-valued Byzantine consensus on an $L$-bit value is achievable with $O(nL)$ bits of communication.

We theoretically study the reversible process of quantum entanglement state by means of weak measurement and corresponding reversible operation. We present a protocol of the reversion operation in two bodies based on the theory of reversion of single photon and then expend it in quantum communication channels. The theoretical results demonstrate that the protocol does not break the information transmission after a weak measurement and a reversible measurement with the subsequent process in the transmission path. It can reverse the perturbed entanglement intensity evolution to its original state. Under the condition of different weak measurement intensity the protocol can reverse the perturbed quantum entanglement system perfectly. In the process we can get the classical information described by information gain from the quantum system through weak measurement operation. On the other hand, in order to realize complete reversibility, the classical information of the quantum entanglement system must obey a limited range we present in this paper in the reverse process.

In this work, we study the impact of quantum entanglement on the two-point correlation function and the associated primordial power spectrum of mean square vacuum fluctuation in a bipartite quantum field theoretic system. The field theory that we consider is the effective theorry of axion field arising from Type IIB string theory compactified to four dimensions. We compute the expression for the power spectrum of vacuum fluctuation in three different approaches, namely (1) field operator expansion (FOE) technique with the quantum entangled state, (2) reduced density matrix (RDM) formalism with mixed quantum state and (3) the method of non-entangled state (NES). For massless axion field, in all these three formalism, we reproduce, at the leading order, the exact scale invariant power spectrum which is well known in the literature. We observe that due to quantum entanglement, the sub-leading terms for these thee formalisms are different. Thus, such correction terms break the degeneracy among the analysis of the FOE, RDM and NES formalisms in the super-horizon limit. On the other hand, for massive axion field we get a slight deviation from scale invariance and exactly quantify the spectral tilt of the power spectrum in small scales. Apart from that, for massless and massive axion field, we find distinguishable features of the power spectrum for the FOE, RDM, and NES on the large scales, which is the result of quantum entanglement. We also find that such large-scale effects are comparable to or greater than the curvature radius of the de Sitter space. Most importantly, in near future if experiments probe for early universe phenomena, one can detect such small quantum effects. In such a scenario, it is possible to test the implications of quantum entanglement in primordial cosmology.

In the $k$-party set disjointness problem, the goal is to determine whether given subsets $S_1,S_2,\dots,S_k\subseteq\{1,2,\dots,n\}$ have empty intersection. We study this problem in the number-on-the-forehead model of communication, where the $i$th party knows all the sets except for $S_i,$ and prove a lower bound of $\Omega(n/4^k)^{1/4}$ on the randomized and nondeterministic communication complexity. This lower bound is close to tight. Previous lower bounds for set disjointness with $k\geq3$ parties were weaker than $\Omega(n/2^{k^3})^{1/(k+1)}.$ We also prove that solving $\ell$ instances of set disjointness requires $\ell\cdot\Omega(n/4^k)^{1/4}$ bits of communication, even to achieve correctness probability exponentially close to $1/2.$ This gives the first direct product result for multiparty set disjointness, solving an open problem due to Beame et al. (2005). Finally, we construct a read-once $\{\wedge,\vee\}$-circuit of depth $3$ with exponentially small discrepancy for up to $k\approx\frac12\l...

In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f (x, y) = D(|x ∧ y|) for some D: {0, 1,..., n} → {0, 1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov’s result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as disjointness. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f: {0, 1} n/4 → {0, 1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1, x2,..., xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Ω(d), where d is the approximate degree of f. From this result, Razborov’s lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over Z n 2. The method of this paper has recently inspired important progress in multiparty communication complexity. 1

We introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games when Alice and Bob are allowed to use a limited amount (c bits) of one-way classical communication compared to their value when they are allowed to use the same amount of one-way quantum communication (c qubits). The key quantity here is the ratio between the quantum and classical value. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the previous quotient is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (PNAS 113(12):3191–3196, 2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical versus quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

Time fractional evolution of a single quantum state and entangled state

Multiparticle entanglement enables quantum simulations, quantum computing, and quantum-enhanced metrology. Yet, there are few methods to produce and measure such entanglement while maintaining single-qubit resolution as the number of qubits is scaled up. Using atom chips and fiber-optical cavities, we have developed a method based on nondestructive collective measurement and conditional evolution to create symmetric entangled states and perform their tomography. We demonstrate creation and analysis of entangled states with mean atom numbers up to 41 and experimentally prove multiparticle entanglement. Our method is independent of atom number and should allow generalization to other entangled states and other physical implementations, including circuit quantum electrodynamics.

We study deterministic one-way communication complexity of functions with Hankel communication matrices. In this paper some structural properties of such matrices are established and applied to the one-way two-party communication complexity of symmetric Boolean functions. It is shown that the number of required communication bits does not depend on the communication direction, provided that neither direction needs maximum complexity. Moreover, in order to obtain an optimal protocol, it is in any case sufficient to consider only the communication direction from the party with the shorter input to the other party. These facts do not hold for arbitrary Boolean functions in general. Next, gaps between one-way and two-way communication complexity for symmetric Boolean functions are discussed. Finally, we give some generalizations to the case of multiple parties.

We propose a probabilistic two-party communication complexity scenario with a prior nonmaximally entangled state, which results in less communication than that is required with only classical random correlations. A simple all-optical implementation of this protocol is presented and demonstrates our conclusion.

In this adventure, two quantum characters interact, because Henry has constructed a quantum suit for his cat, Schred. Henry and Schred end up in a quantum-entangled state. Remarkably, by measuring one of two entangled systems, the state of the other system is immediately collapsed, even if they are far apart. This bizarre feature of entanglement implies non-locality—synchronization or “collusion” between quantum objects, regardless of their distance. Cosmology provides an explanation: the universe emerged from a unified state describable as a quantum-entangled “hologram”. This notion resonates with the ancient Hindu view that the common essence of all things, the Brahman, can be revealed at every level of the natural hierarchy. The appendix to this chapter discusses operators that create entanglement.

We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on $k$ vertices. For $k\geq4$, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication ($k=3$), which is approximately 0.79. We raise the question whether for some $k$ the exponent per edge can be below $2/3$, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalise to higher order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.