Bounded-error quantum state identification and exponential separations in communication complexity

An open problem in communication complexity proposed by several authors is to prove that for every Boolean function $f,$ the task of computing $f(x\wedge y)$ has polynomially related classical and quantum bounded-error complexities. We solve a variant of this question. For every $f,$ we prove that the task of computing, on input \mbox{$x$ and $y,$} \emph{both} of the quantities $f(x\wedge y)$ and $f(x\vee y)$ has polynomially related classical and quantum bounded-error complexities. We further show that the quantum bounded-error complexity is polynomially related to the classical deterministic complexity and the block sensitivity of $f.$ This result holds regardless of prior entanglement.

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Quantum Entanglement and Communication Complexity

BRICS Report Series ◽

10.7146/brics.v4i40.18966 ◽

1997 ◽

Vol 4 (40) ◽

Cited By ~ 3

Author(s):

Harry Buhrman ◽

Richard Cleve ◽

Wim Van Dam

Keyword(s):

Quantum Entanglement ◽

Quantum State ◽

Communication Channel ◽

Communication Complexity ◽

Complexity Problem

We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement results in less communication than is required with only classical random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among remote parties.

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On the role of shared entanglement

Quantum Information and Computation ◽

10.26421/qic8.1-2-6 ◽

2008 ◽

Vol 8 (1&2) ◽

pp. 82-95

Author(s):

D. Gavinsky

Keyword(s):

Lower Bounds ◽

Message Passing ◽

Communication Complexity ◽

Optimal Solution ◽

Famous Theorem ◽

Trade Offs ◽

Wide Range ◽

Bounded Error ◽

Strong Direct Product Theorem ◽

Apparent Similarity

Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known ``entanglement analogue'' for the famous theorem by Newman, saying that the number of shared random bits required for solving any communication problem can be at most logarithmic in the input length (i.e., using more than $\asO[]{\log n}$ shared random bits would not reduce the complexity of an optimal solution). In this paper we prove that the same is not true for entanglement. We establish a wide range of tight (up to a polylogarithmic factor) entanglement vs.\ communication trade-offs for relational problems. The low end is:\ for any $t>2$, reducing shared entanglement from $log^tn$ to $\aso[]{log^{t-2}n}$ qubits can increase the communication required for solving a problem almost exponentially, from $\asO[]{log^tn}$ to $\asOm[]{\sqrt n}$. The high end is:\ for any $\eps>0$, reducing shared entanglement from $n^{1-\eps}\log n$ to $\aso[]{n^{1-\eps}/\log n}$ can increase the required communication from $\asO[]{n^{1-\eps}\log n}$ to $\asOm[]{n^{1-\eps/2}/\log n}$. The upper bounds are demonstrated via protocols which are \e{exact} and work in the \e{simultaneous message passing model}, while the lower bounds hold for \e{bounded-error protocols}, even in the more powerful \e{model of 1-way communication}. Our protocols use shared EPR pairs while the lower bounds apply to any sort of prior entanglement. We base the lower bounds on a strong direct product theorem for communication complexity of a certain class of relational problems. We believe that the theorem might have applications outside the scope of this work.

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Communication complexity and the reality of the wave function

Modern Physics Letters A ◽

10.1142/s0217732315300013 ◽

2015 ◽

Vol 30 (01) ◽

pp. 1530001 ◽

Cited By ~ 7

Author(s):

Alberto Montina

Keyword(s):

Quantum State ◽

Communication Complexity ◽

Quantum Communication ◽

Communication Process ◽

Minimal Amount ◽

Quantum Processes ◽

Independence Property ◽

Classical Communication ◽

Asymptotic Equipartition Property ◽

Quantum Communication Complexity

In this review, we discuss a relation between quantum communication complexity and a long-standing debate in quantum foundation concerning the interpretation of the quantum state. Is the quantum state a physical element of reality as originally interpreted by Schrödinger? Or is it an abstract mathematical object containing statistical information about the outcome of measurements as interpreted by Born? Although these questions sound philosophical and pointless, they can be made precise in the framework of what we call classical theories of quantum processes, which are a reword of quantum phenomena in the language of classical probability theory. In 2012, Pusey, Barrett and Rudolph (PBR) proved, under an assumption of preparation independence, a theorem supporting the original interpretation of Schrödinger in the classical framework. The PBR theorem has attracted considerable interest revitalizing the debate and motivating other proofs with alternative hypotheses. Recently, we showed that these questions are related to a practical problem in quantum communication complexity, namely, quantifying the minimal amount of classical communication required in the classical simulation of a two-party quantum communication process. In particular, we argued that the statement of the PBR theorem can be proved if the classical communication cost of simulating the communication of n qubits grows more than exponentially in n. Our argument is based on an assumption that we call probability equipartition property. This property is somehow weaker than the preparation independence property used in the PBR theorem, as the former can be justified by the latter and the asymptotic equipartition property of independent stochastic sources. The probability equipartition property is a general and natural hypothesis that can be assumed even if the preparation independence hypothesis is dropped. In this review, we further develop our argument into the form of a theorem.

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QUANTUM KOLMOGOROV COMPLEXITY AND ITS APPLICATIONS

International Journal of Quantum Information ◽

10.1142/s0219749907003171 ◽

2007 ◽

Vol 05 (05) ◽

pp. 729-750 ◽

Cited By ~ 19

Author(s):

CATERINA E. MORA ◽

HANS J. BRIEGEL ◽

BARBARA KRAUS

Keyword(s):

Quantum Computation ◽

Quantum State ◽

Kolmogorov Complexity ◽

Communication Complexity ◽

Quantum States ◽

Qubit State ◽

Binary String ◽

Classical Communication ◽

Pure Quantum State ◽

Definition Of

Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore, we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.

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