How much of quantum mechanics is really needed to defy Extended Church-Turing Thesis?
This paper has been withdrawn by the author as one of the coauthors needs institutional permission.
quant-ph↗
arXiv subjects
Sudhir Kumar Singh
Publications and source records attributed to Sudhir Kumar Singh.
Selfish Distributed Compression over Networks: Correlation Induces Anarchy
We consider the min-cost multicast problem (under network coding) with multiple correlated sources where each terminal wants to losslessly reconstruct all the sources. We study the inefficiency brought forth by the selfish behavior of the terminals in this scenario by modeling it as a noncooperative game among the terminals. The degradation in performance due to the lack of regulation is measured by the {\it Price of Anarchy} (POA), which is defined as the ratio between the cost of the worst possible \textit{Wardrop equilibrium} and the socially optimum cost. Our main result is that in contrast with the case of independent sources, the presence of source correlations can significantly increase the price of anarchy. Towards establishing this result, we first characterize the socially optimal flow and rate allocation in terms of four intuitive conditions. Next, we show that the Wardrop equilibrium is a socially optimal solution for a different set of (related) cost functions. Using this, we construct explicit examples that demonstrate that the POA $> 1$ and determine near-tight upper bounds on the POA as well. The main techniques in our analysis are Lagrangian duality theory and the usage of the supermodularity of conditional entropy.
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To Broad-Match or Not to Broad-Match : An Auctioneer's Dilemma ?
We initiate the study of an interesting aspect of sponsored search advertising, namely the consequences of broad match-a feature where an ad of an advertiser can be mapped to a broader range of relevant queries, and not necessarily to the particular keyword(s) that ad is associated with. Starting with a very natural setting for strategies available to the advertisers, and via a careful look through the algorithmic lens, we first propose solution concepts for the game originating from the strategic behavior of advertisers as they try to optimize their budget allocation across various keywords. Next, we consider two broad match scenarios based on factors such as information asymmetry between advertisers and the auctioneer, and the extent of auctioneer's control on the budget splitting. In the first scenario, the advertisers have the full information about broad match and relevant parameters, and can reapportion their own budgets to utilize the extra information; in particular, the auctioneer has no direct control over budget splitting. We show that, the same broad match may lead to different equilibria, one leading to a revenue improvement, whereas another to a revenue loss. This leaves the auctioneer in a dilemma - whether to broad-match or not. This motivates us to consider another broad match scenario, where the advertisers have information only about the current scenario, and the allocation of the budgets unspent in the current scenario is in the control of the auctioneer. We observe that the auctioneer can always improve his revenue by judiciously using broad match. Thus, information seems to be a double-edged sword for the auctioneer.
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Some Directions beyond Traditional Quantum Secret Sharing
We investigate two directions beyond the traditional quantum secret sharing (QSS). First, a restriction on QSS that comes from the no-cloning theorem is that any pair of authorized sets in an access structure should overlap. From the viewpoint of application, this places an unnatural constraint on secret sharing. We present a generalization, called assisted QSS (AQSS), where access structures without pairwise overlap of authorized sets is permissible, provided some shares are withheld by the share dealer. We show that no more than $λ-1$ withheld shares are required, where $λ$ is the minimum number of {\em partially linked classes} among the authorized sets for the QSS. Our result means that such applications of QSS need not be thwarted by the no-cloning theorem. Secondly, we point out a way of combining the features of QSS and quantum key distribution (QKD) for applications where a classical information is shared by quantum means. We observe that in such case, it is often possible to reduce the security proof of QSS to that of QKD.
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Capacity as a Fundamental Metric for Mechanism Design in the Information Economy
The auction theory literature has so far focused mostly on the design of mechanisms that takes the revenue or the efficiency as a yardstick. However, scenarios where the {\it capacity}, which we define as \textit{``the number of bidders the auctioneer wants to have a positive probability of getting the item''}, is a fundamental concern are ubiquitous in the information economy. For instance, in sponsored search auctions (SSA's) or in online ad-exchanges, the true value of an ad-slot for an advertiser is inherently derived from the conversion-rate, which in turn depends on whether the advertiser actually obtained the ad-slot or not; thus, unless the capacity of the underlying auction is large, key parameters, such as true valuations and advertiser-specific conversion rates, will remain unknown or uncertain leading to inherent inefficiencies in the system. In general, the same holds true for all information goods/digital goods. We initiate a study of mechanisms, which take capacity as a yardstick, in addition to revenue/efficiency. We show that in the case of a single indivisible item one simple way to incorporate capacity constraints is via designing mechanisms to sell probability distributions, and that under certain conditions, such optimal probability distributions could be identified using a Linear programming approach. We define a quantity called {\it price of capacity} to capture the tradeoff between capacity and revenue/efficiency. We also study the case of sponsored search auctions. Finally, we discuss how general such an approach via probability spikes can be made, and potential directions for future investigations.
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Exploration via design and the cost of uncertainty in keyword auctions
We present a deterministic exploration mechanism for sponsored search auctions, which enables the auctioneer to learn the relevance scores of advertisers, and allows advertisers to estimate the true value of clicks generated at the auction site. This exploratory mechanism deviates only minimally from the mechanism being currently used by Google and Yahoo! in the sense that it retains the same pricing rule, similar ranking scheme, as well as, similar mathematical structure of payoffs. In particular, the estimations of the relevance scores and true-values are achieved by providing a chance to lower ranked advertisers to obtain better slots. This allows the search engine to potentially test a new pool of advertisers, and correspondingly, enables new advertisers to estimate the value of clicks/leads generated via the auction. Both these quantities are unknown a priori, and their knowledge is necessary for the auction to operate efficiently. We show that such an exploration policy can be incorporated without any significant loss in revenue for the auctioneer. We compare the revenue of the new mechanism to that of the standard mechanism at their corresponding symmetric Nash equilibria and compute the cost of uncertainty, which is defined as the relative loss in expected revenue per impression. We also bound the loss in efficiency, as well as, in user experience due to exploration, under the same solution concept (i.e. SNE). Thus the proposed exploration mechanism learns the relevance scores while incorporating the incentive constraints from the advertisers who are selfish and are trying to maximize their own profits, and therefore, the exploration is essentially achieved via mechanism design. We also discuss variations of the new mechanism such as truthful implementations.
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Diversification in the Internet Economy:The Role of For-Profit Mediators
We investigate market forces that would lead to the emergence of new classes of players in the sponsored search market. We report a 3-fold diversification triggered by two inherent features of the sponsored search market, namely, capacity constraints and collusion-vulnerability of current mechanisms. In the first scenario, we present a comparative study of two models motivated by capacity constraints - one where the additional capacity is provided by for-profit agents, who compete for slots in the original auction, draw traffic, and run their own sub-auctions, and the other, where the additional capacity is provided by the auctioneer herself, by essentially acting as a mediator and running a single combined auction. This study was initiated by us in \cite{SRGR07}, where the mediator-based model was studied. In the present work, we study the auctioneer-based model and show that this model seems inferior to the mediator-based model in terms of revenue or efficiency guarantee due to added capacity. In the second scenario, we initiate a game theoretic study of current sponsored search auctions, involving incentive driven mediators who exploit the fact that these mechanisms are not collusion-resistant. In particular, we show that advertisers can improve their payoffs by using the services of the mediator compared to directly participating in the auction, and that the mediator can also obtain monetary benefit, without violating incentive constraints from the advertisers who do not use its services. We also point out that the auctioneer can not do very much via mechanism design to avoid such for-profit mediation without losing badly in terms of revenue, and therefore, the mediators are likely to prevail.
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For-profit mediators in sponsored search advertising
A mediator is a well-known construct in game theory, and is an entity that plays on behalf of some of the agents who choose to use its services, while the rest of the agents participate in the game directly. We initiate a game theoretic study of sponsored search auctions, such as those used by Google and Yahoo!, involving {\em incentive driven} mediators. We refer to such mediators as {\em for-profit} mediators, so as to distinguish them from mediators introduced in prior work, who have no monetary incentives, and are driven by the altruistic goal of implementing certain desired outcomes. We show that in our model, (i) players/advertisers can improve their payoffs by choosing to use the services of the mediator, compared to directly participating in the auction; (ii) the mediator can obtain monetary benefit by managing the advertising burden of its group of advertisers; and (iii) the payoffs of the mediator and the advertisers it plays for are compatible with the incentive constraints from the advertisers who do dot use its services. A simple intuition behind the above result comes from the observation that the mediator has more information about and more control over the bid profile than any individual advertiser, allowing her to reduce the payments made to the auctioneer, while still maintaining incentive constraints. Further, our results indicate that there are significant opportunities for diversification in the internet economy and we should expect it to continue to develop richer structure, with room for different types of agents to coexist.
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Capacity constraints and the inevitability of mediators in adword auctions
One natural constraint in the sponsored search advertising framework arises from the fact that there is a limit on the number of available slots, especially for the popular keywords, and as a result, a significant pool of advertisers are left out. We study the emergence of diversification in the adword market triggered by such capacity constraints in the sense that new market mechanisms, as well as, new for-profit agents are likely to emerge to combat or to make profit from the opportunities created by shortages in ad-space inventory. We propose a model where the additional capacity is provided by for-profit agents (or, mediators), who compete for slots in the original auction, draw traffic, and run their own sub-auctions. The quality of the additional capacity provided by a mediator is measured by its {\it fitness} factor. We compute revenues and payoffs for all the different parties at a {\it symmetric Nash equilibrium} (SNE) when the mediator-based model is operated by a mechanism currently being used by Google and Yahoo!, and then compare these numbers with those obtained at a corresponding SNE for the same mechanism, but without any mediators involved in the auctions. Such calculations allow us to determine the value of the additional capacity. Our results show that the revenue of the auctioneer, as well as the social value (i.e. efficiency), always increase when mediators are involved; moreover even the payoffs of {\em all} the bidders will increase if the mediator has a high enough fitness. Thus, our analysis indicates that there are significant opportunities for diversification in the internet economy and we should expect it to continue to develop richer structure, with room for different types of agents and mechanisms to coexist.
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A combinatorial approach for studying LOCC transformations of multipartite states
We develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among $n$ agents. Further, we discuss a new graph theoretic metric on a class of multi-partite states, and its implications.
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Generalized Quantum Secret Sharing
We explore a generalization of quantum secret sharing (QSS) in which classical shares play a complementary role to quantum shares, exploring further consequences of an idea first studied by Nascimento, Mueller-Quade and Imai (Phys. Rev. {\bf A64} 042311 (2001)). We examine three ways, termed inflation, compression and twin-thresholding, by which the proportion of classical shares can be augmented. This has the important application that it reduces quantum (information processing) players by replacing them with their classical counterparts, thereby making quantum secret sharing considerably easier and less expensive to implement in a practical setting. In compression, a QSS scheme is turned into an equivalent scheme with fewer quantum players, compensated for by suitable classical shares. In inflation, a QSS scheme is enlarged by adding only classical shares and players. In a twin-threshold scheme, we invoke two separate thresholds for classical and quantum shares based on the idea of information dilution.
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Quantum seals
A quantum seal is a way of encoding a message into quantum states, so that anybody may read the message with little error, while authorized verifiers can detect that the seal has been broken. We present a simple extension to the Bechmann-Pasquinucci majority-voting scheme that is impervious to coherent attacks, and further, encompasses sealing quantum messages by means of quantum encryption. The scheme is relatively easy to implement, requiring neither entanglement nor controlled operations during the state preparation, reading or verification stages.
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Assisted Quantum Secret Sharing
A restriction on quantum secret sharing (QSS) that comes from the no-cloning theorem is that any pair of authorized sets in an access structure should overlap. From the viewpoint of application, this places an unnatural constraint on secret sharing. We present a generalization, called assisted QSS (AQSS), where access structures without pairwise overlap of authorized sets is permissible, provided some shares are withheld by the share dealer. We show that no more than $λ-1$ withheld shares are required, where $λ$ is the minimum number of {\em partially linked classes} among the authorized sets for the QSS. This is useful in QSS schemes where the share dealer is honest by definition and is equivalent to a secret reconstructor. Our result means that such applications of QSS need not be thwarted by the no-cloning theorem.
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Combinatorial Approaches in Quantum Information Theory
We investigate the exploitation of various combinatorial properties of graphs and set systems to study several issues in quantum information theory. We characterize the combinatorics of distributed EPR pairs for preparing multi-partite entanglement in a real communication network. This combinatorics helps in the study of various problems in multi-party case by just reducing to the two-party case. Particularly, we use this combinatorics to (1) study various possible and impossible transformations of multi-partite states under LOCC, thus presenting an entirely new approach, not based on entropic criterion, to study such state transformations. (2) present a protocol and proof of its unconditional security for quantum key distribution amongst several trusted parties. (3) propose an idea to combine the features of quantum key distribution and quantum secret sharing. We investigate all the above issues in great detail and finally conclude briefly with some open research directions based on our research.
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Unconditionally Secure Multipartite Quantum Key Distribution
We consider the problem of secure key distribution among $n$ trustful agents: the goal is to distribute an identical random bit-string among the $n$ agents over a noisy channel such that eavesdroppers learn little about it. We study the general situation where the only resources required are secure bipartite key distribution and authenticated classical communication. Accordingly, multipartite quantum key distribution can be proven unconditionally secure by reducing the problem to the biparitite case and invoking the proof of security of bipartite quantum key distribution.
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Characterizing the combinatorics of distributed EPR pairs for multi-partite entanglement
We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set $N$ of $n$ agents in a network using apriori multi-partite entanglement states amongst subsets of $N$. We propose protocols for establishing multi-partite entanglement in the above cases.
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Multi-partite Quantum Entanglement versus Randomization: Fair and Unbiased Leader Election in Networks
In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of $O(\log n)$ distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, $n$ is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number $O(n/\log n)$ of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number $O(\log n)$ of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions.
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Computing sharp and scalable bounds on errors in approximate zeros of univariate polynomials
There are several numerical methods for computing approximate zeros of a given univariate polynomial. In this paper, we develop a simple and novel method for determining sharp upper bounds on errors in approximate zeros of a given polynomial using Rouche's theorem from complex analysis. We compute the error bounds using non-linear optimization. Our bounds are scalable in the sense that we compute sharper error bounds for better approximations of zeros. We use high precision computations using the LEDA/real floating-point filter for computing our bounds robustly.
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