Direct Sum Theorem for Bounded Round Quantum Communication Complexity
نویسنده
چکیده
We prove a direct sum theorem for bounded round entanglement-assisted quantum communication complexity. To do so, we use the fully quantum definition for information cost and complexity that we recently introduced, and use both the fact that information is a lower bound on communication, and the fact that a direct sum property holds for quantum information complexity. We then give a protocol for compressing a single copy of a protocol down to its quantum information cost, up to terms depending on the number of rounds and the allowed increase in error. Two important tools to derive this protocol are a smooth conditional min-entropy bound for a oneshot quantum state redistribution protocol, and the quantum substate theorem of Jain, Radhakrishnan and Sen (FOCS’02) to transform this bound into a von Neumann conditional entropy bound. This result further establishes the newly introduced notions of quantum information cost and complexity as the correct quantum generalisations of the classical ones in the standard communication complexity setting. Finding such a quantum generalisation of information complexity was one of the open problem recently raised by Braverman (STOC’12).
منابع مشابه
A Direct Sum Theorem in Communication Complexity via Message Compression
We prove lower bounds for the direct sum problem for two-party bounded error randomised multipleround communication protocols. Our proofs use the notion of information cost of a protocol, as defined by Chakrabarti et al. [CSWY01] and refined further by Bar-Yossef et al. [BJKS02]. Our main technical result is a ‘compression’ theorem saying that, for any probability distribution μ over the inputs...
متن کاملA New, Fully Quantum Notion of Information Complexity, and an Application to Direct Sum for Bounded Round Quantum Communication Complexity
Direct Sum for Bounded Round Quantum Communication Complexity Dave Touchette 1 We present the first general direct sum theorem for quantum communication complexity that holds for more than a single round of communication. A direct sum theorem states that to compute n tasks simultaneously requires as much resources as the amount of the given resource required for computing them separately. By a ...
متن کاملA direct product theorem for bounded-round public-coin randomized communication complexity
A strong direct product theorem for a problem in a given model of computation states that, in order to compute k instances of the problem, if we provide resource which is less than k times the resource required for computing one instance of the problem with constant success probability, then the probability of correctly computing all the k instances together, is exponentially small in k. In thi...
متن کاملImproved direct sum theorem in classical communication complexity
For a function f : X ×Y → Z , the m-fold direct sum is the function f : X × Y → Z, defined by f(〈x1, . . . , xm〉, 〈y1, . . . , ym〉) ∆ = 〈f(x1, y1), . . . , f(xm, ym)〉. We show the following direct sum theorem for classical communication protocols, R(f) = Ω( m k R(f)) where R(f) is the the k-round private coins communication complexity of f and R(f) is the k-round public coin complexity of f . I...
متن کاملEfficient Communication Using Partial Information
We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver. We use our simulation method to obtain several results in communication complexity. • We prove a new direct sum the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1409.4391 شماره
صفحات -
تاریخ انتشار 2014