Quantum addition imparts less disorder than mixing and commutes with incoherent channels

We prove a generalized version of a previously conjectured inequality by Zhang et al (Zhang et al 2018 Phys. Lett. A 382, 1516–23) for the quantum addition operation defined by Datta et al (Audenaert et al 2016 J. Math. Phy. 57, 052202) in the context of proving an entropy power inequality for qubits. We also show that the quantum addition operation commutes with an incoherent channel, which may have possible implications for resource theories of coherence in optical settings.

Shannon proposed [1] an entropic power inequality (EPI) for classical continuous random variables, where the addition of two random variables was taken in the sense of convolution. Shannon's conjectured entropy power inequality for continuous random variables X, Y with probability densities p X and p Y respectively, in terms of the differential entropy H was expressed in the following form where denotes the convolution operation on the probability densities p X , p Y . Various proofs of this inequality, beginning with the original proof by Stam [2] have come up over the years [3][4][5]. A quantum generalization of the above inequality was obtained for the continuous variable (CV) systems considering the beamsplitter merging operation as the analogue of convolution [6]. Datta, Ozols and Audenart [7] have recently found the EPI for finite dimensional quantum systems through the qudit addition channel [8]. A conditional EPI is also discussed in recent literature [9]. Essentially, this channel can be realized as a unitary evolution in Ä   , where the parameter α ä [0,1] is the weight of the addition, and S is the two qudit swap gate å ñá Ä ñá The Kraus operators corresponding to this map are expressible in the form It has been proven in [7] that for any concave function f which depends solely on the spectrum of a state ρ, the following relation holds true for α ä [0,1] If f is chosen as the exponential of the von Neumann entropy, we obtain the qudit analog to the classic EPI. For another suitable choice of f, the resulting inequality is a qudit analog of the entropic photon number inequality [7,10]. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

A reverse entropy power inequality
The entropic power inequality puts a lower bound on the entropy of the output of the qudit addition channel. In this section, we attempt to find an upper bound to the entropy of the output of the qudit addition channel.
Theorem (Reverse EP equality). If r and s are two qudit states and a  denotes quantum addition with weight a Î 0, 1 ( ), then the following equality holds ar a s r s ar a s r s Proof. Let us first analyze the difference in von Neumann entropies between the results of classical mixture and quantum addition of states ρ and σ.
Now, let us concentrate on the second term. Since α ρ+(1−α) σ is a positive semidefinite matrix, its logarithm, say K, is Hermitian and commutes with α ρ+(1−α) σ. Thus, Armed with this result, we now simplify the second term in the following way.  [ ]

( )
Here we have used the cyclicity of trace as well as the commutation relation in (7). Thus, the second term in the remainder vanishes and the proof is complete. , An immediate corollary to the theorem above is the following inequality for arbitrary α ä [0,1], which was conjectured [11] to hold for arbitrary qudit systems when a = Corollary. If r and s are two qudit states, then r s ar a s + - This result follows from the relation (5) and the non-negativity of quantum relative entropy. It is easy to see that for α ä (0,1), this inequality is strict if ρ and σ do not commute. This we term as the reverse entropy power inequality. Thus quantum addition of two density operators results in a density operator whose entropy is always lower compared to the corresponding classical mixture. One may thus wonder, does this difference in entropy capture some quantumness ?

Quantum addition commutes with incoherent channels
Now we go on to show the second result in the paper, which proves that the quantum addition operation commutes with the free operations in the resource theory of quantum coherence [12,13], viz. the incoherent operations (IOs).
Lemma. If s is an incoherent state and L denotes an incoherent channel, then the following equality holds r s r s L L =L