Super-additivity in communication of classical information through quantum channels from a quantum parameter estimation perspective

Abstract

We point out a contrasting role the entanglement plays in communication and estimation scenarios. In the first case it brings noticeable benefits at the measurement stage (output super-additivity), whereas in the latter it is the entanglement of the input probes that enables significant performance enhancement (input super-additivity). We identify a weak estimation regime where a strong connection between concepts crucial to the two fields is demonstrated; the accessible information and the Holevo quantity on one side and the quantum Fisher information related quantities on the other. This allows us to shed new light on the problem of super-additivity in communication using the concepts of quantum estimation theory. of merit used in estimation and communication approaches are usually different [14]. The central problem of quantum communication and estimation theories is to identify the potential benefits coming from exploiting entanglement in the input states as well as at the measurement stage of the protocols. Interestingly, unlike in the communication problem, there is a great number of examples demonstrating significant gains coming from the use of entangled input probes in quantum estimation protocols, with applications ranging from optical and atomic interferometry, via magnetometry to spectroscopy and atomic clocks stabilization [17][18][19][20][21][22][23]. At the same time, in a typical estimation protocol utilizing unentangled input probes collective measurements are typically irrelevant. Thus a contrasting picture emerges: when thinking of capacities of quantum channels gains in information processing arising from utilizing entanglement are at the measurement stage whereas it is the input stage where entanglement makes a difference in the estimation scenarios.The goal of this paper is to better understand the connections between the two fields from the point of view of the super-additivity issue, understood here as a general question of utility of entanglement in estimation/ communication protocols. We show that in the weak estimation regime, where the amount of information extractable on the parameter is very small compared to the prior information, i.e. the error of estimation is large compared with the variance of prior distribution, the accessible information as well as the Holevo quantity can be expressed using Fisher information (FI)-like concepts, which allows us to discuss utility of entanglement in communication using the properties of quantities well understood within the field of quantum estimation. We also point out that in a communication problem encoding a large number of independent parameters is favored even at the cost of their limited estimation precision, which makes the estimation strategy of learning a given parameter with highest possible accuracy not likely to be useful for the communication purposes. In order to make the paper self-contained we also recall some of the recent results on applications of rate-distortion theory in quantum estimation, which is another way of connecti...

Classical capacity per unit cost for quantum channels

“…The second inequality is a consequence of the fact that F is the smallest type of quantum Fisher information [33,43] so it has to be lower than the respective REQFI. The above inequality is in agreement with the results of [20], in which it was shown that for narrow continuous prior distributions, that is for distributions with small variance β = x 2 ≪ 1, in the protocol utilizing states with rank r of the Hilbert space with dimension d, the capacity can be expressed as…”

“…The quantity J θ=0 [ρ θ ] appearing in the above formula is a relative entropy quantum Fisher information (REQFI) [20,33,34] at θ = 0 for a continuously parametrized family of quantum states…”

“…In that picture many fundamental results have been obtained concerning both specific transmission protocols [2][3][4][5][6][7] and general optimal bounds [8][9][10][11][12][13]. The application of laws of quantum mechanics has allowed for a deeper analysis of communication [14][15][16][17] and showing such effects as output and input superadditivity [18][19][20] or finite optimal rates even for noiseless channels, emerging from nonclassical phenomena such as entanglement or the Heisenberg uncertainty principle.…”

Low-cost limit of classical communication with restricted quantum measurements

“…Inserting equations (19) and (22) into (10), the leading-order expansion of mutual information in the cost parameter ζ is upper bounded by the following expression:…”

“…Suppose first that the measurement is composed only of type-Z operators. Inserting equation (27) into equation (19) and recalling thatˆ|…”