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arXiv:1110.5944 · Physical Review A

Communication cost of classically simulating a quantum channel with subsequent rank-1 projective measurement

A. Montina

Published Dec 29, 2011• 15 citations
arXivHTMLPDF

Abstract

A process of preparation, transmission, and subsequent projective measurement of a qubit can be simulated by a classical model with only two bits of communication and some amount of shared randomness. However, to the best of our knowledge, no model for $n$ qubits with a finite amount of classical communication is known at present. A lower bound for the communication cost can provide useful hints for a generalization. It is known, for example, that the amount of communication must be greater than $c{2}^{n}$, where $c\ensuremath{\simeq}0.01$. The proof uses a quite elaborate theorem of communication complexity. Using a mathematical conjecture known as the ``double cap conjecture,'' we strengthen this result by presenting a geometrical and extremely simple derivation of the lower bound ${2}^{n}\ensuremath{-}1$. Only rank-1 projective measurements are involved in the derivation.

DOI
10.1103/physreva.84.060303

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A process of preparation, transmission, and subsequent projective measurement of a qubit can be simulated by a classical model with only two bits of communication and some amount of shared randomness. However, to the best of our knowledge, no model for nnn qubits with a finite amount of classical communication is known at present. A lower bound for the communication cost can provide useful hints for a generalization. It is known, for example, that the amount of communication must be greater than c2nc{2}^{n}c2n, where c\ensuremath≃0.01c\ensuremath{\simeq}0.01c\ensuremath≃0.01. The proof uses a quite elaborate theorem of communication complexity. Using a mathematical conjecture known as the ``double cap conjecture,'' we strengthen this result by presenting a geometrical and extremely simple derivation of the lower bound 2n\ensuremath−1{2}^{n}\ensuremath{-}12n\ensuremath−1. Only rank-1 projective measurements are involved in the derivation.
Quantum theory is, in some sense, “non-classical.” For instance, the behavior of entangled systems (i.e., shared quantum information) under measurements cannot, in general, be explained by shared classical information. With classical communication, on the other hand, both the correlations entanglement leads to as well as quantum channels can be reproduced in principle. Here, crucial questions are whether the required communications is finite; if so, then its exact amount is related to the “degree of non-classicality” of the quantum primitive. We apply information-theoretic results such as the reverse Shannon theorem for determining the required communication in the asymptotic limit. The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating the process of preparation, transmission through the channel, and subsequent measurement of a quantum state. At present, only little is known about this quantity. Our generic procedure allows for systematically evaluating the communication complexity of channels in any general probabilistic theory, in particular quantum theory. The procedure is constructive and provides the most efficient classical protocols. We illustrate it by evaluating the communication complexity of sending single qubits over a noiseless quantum channel with some finite sets of quantum states and measurements. As a second application, we determine the classical-communication rate required for the simulation of the behavior under measurements of entangled states. Here, the communication cost can be directly interpreted as the “non-classicality” of the correlation. A particular example is the simulation of non-maximally entangled pure qubit pairs, where we find the required communication rate to behave monotonically with the strength of the entanglement. For different measures of non-locality, such as the number of required non-local (PR) boxes, another behavior had been reported for the single-shot scenario.
We study the prepare-and-measure scenario in which Alice transmits a quantum system to Bob, who then performs a quantum measurement. The quantum state of the system is unknown to Bob, and the measurement is unknown to Alice. It has recently been shown that shared randomness and two bits of classical communication are necessary and sufficient to simulate the transmission of a qubit. We show that the communication cost can be reduced to an average of 1.891.891.89 bits. We then study restricted sets of state preparations: First, for a restriction to real-valued qubit states, if the communication of a classical trit is sufficient, we show that the corresponding protocol must have a convoluted form. We then reduce the smallest qubit scenario requiring two bits of classical communication to only 666 state preparations and 555 measurements. For a qutrit, it is not known whether the communication cost is finite; we identify a scenario that requires at least 555 classical messages, already for the simulation of the real qutrit. Finally, we develop a method for restricted sets of states, that allows us to lower bound the classical communication cost based solely on the set of quantum states.
A long-standing open question is to characterize the classical communication cost of simulating the correlations arising from measurements on finite-dimensional quantum systems. We consider this question in the prepare-and-measure scenario. First we construct a classical simulation model involving a finite average number of bits of one-way communication that simulates exactly the correlations arising from arbitrary ddd-dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for d≥4d\geq 4d≥4, hence strengthening a recent result of de Gois, Santos and Vieira. Our first result relies on a more general simulation theorem which can be applied to certain continuous variable (i.e. infinite-dimensional) quantum systems, and even beyond quantum theory, e.g. to generalized probabilistic theories. In particular, we show that multimode Gaussian states of bounded average energy and general Gaussian measurements admit an exact classical simulation with a finite expected number of bits. Overall, our results show that the classical simulation cost of quantum communication is essentially as high as it could be.
The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating a process of state preparation, transmission through the channel and subsequent measurement. It establishes a limit on the power of quantum communication in terms of classical resources. We show that classical simulations employing a finite amount of communication can be derived from a special class of hidden variable theories where quantum states represent statistical knowledge about the classical state and not an element of reality. This special class has attracted strong interest very recently. The communication cost of each derived simulation is given by the mutual information between the quantum state and the classical state of the parent hidden variable theory. Finally, we find that the communication complexity for single qubits is smaller than 1.28 bits. The previous known upper bound was 1.85 bits.
We consider the process consisting of preparation, transmission through a quantum channel, and subsequent measurement of quantum states. The communication complexity of the channel is the minimal amount of classical communication required for classically simulating it. Recently, we reduced the computation of this quantity to a convex minimization problem with linear constraints. Every solution of the constraints provides an upper bound on the communication complexity. In this paper, we derive the dual maximization problem of the original one. The feasible points of the dual constraints, which are inequalities, give lower bounds on the communication complexity, as illustrated with an example. The optimal values of the two problems turn out to be equal (zero duality gap). By this property, we provide necessary and sufficient conditions for optimality in terms of a set of equalities and inequalities. We use these conditions and two reasonable but unproven hypotheses to derive the lower bound $n\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$ for a noiseless quantum channel with capacity equal to nnn qubits. This lower bound can have interesting consequences in the context of the recent debate on the reality of the quantum state.
In [Toner and Bacon, Phys. Rev. Lett. 91, 187904 (2003)], 111 bit of communication was proven to be enough to simulate the statistics of local projective measurements over the maximally entangled state. Ever since then, the question of whether 111 bit is also enough for the case of generalized measurements has been open. In this thesis, we retort to inefficiency-resistant Bell functionals, a powerful technique to prove lower bounds communication complexity, to numerically study this question. The results obtained suggest that, indeed, as is the case with projective measurements, 111 bit of communication suffices to simulate POVMs over maximally entangled qubits.