Abstract We propose an extension of the quantum entropy power inequality for finite dimensional quantum systems, and prove a conditional quantum entropy power inequality by using the majorization relation as well as the concavity of entropic functions also given by Audenaert et al (2016 J. Math. Phys . 57 052202). Here, we make particular use of the fact that a specific local measurement after a partial swap operation (or partial swap quantum channel) acting only on finite dimensional bipartite subsystems does not affect the majorization relation for the conditional output states when a separable ancillary subsystem is involved. We expect our conditional quantum entropy power inequality to be useful, and applicable in bounding and analyzing several capacity problems for quantum channels.
Abstract We propose an extension of the quantum entropy power inequality for finite dimensional quantum systems, and prove a conditional quantum entropy power inequality by using the majorization relation as well as the concavity of entropic functions also given by Audenaert et al (2016 J. Math. Phys . 57 052202). Here, we make particular use of the fact that a specific local measurement after a partial swap operation (or partial swap quantum channel) acting only on finite dimensional bipartite subsystems does not affect the majorization relation for the conditional output states when a separable ancillary subsystem is involved. We expect our conditional quantum entropy power inequality to be useful, and applicable in bounding and analyzing several capacity problems for quantum channels.
We propose a generalization of the quantum entropy power inequality involving condi-tional entropies. For the special case of Gaussian states, we give a proof based on perturba-tion theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels. 1 Classical and quantum entropy-power inequalities The entropy power inequality, proposed by Shannon [27] and later established with increasing rigor by Stam [29] and Blachman [5], has become a fundamental tool in classical informa-tion theory. Shannon’s original application of the entropy power inequality is a lower bound on the capacity of an additive (but potentially non-Gaussian) noise channel [27, Theorem 18]. However, the usefulness of the entropy power inequality is especially evident in multi-terminal information theory. Among the most well-known applications are the characterization of the Gaussian broadcast channel by Bergman [4], the Gaussian two-description problem by Ozarow [25] and the quadratic Gaussian CEO problem by Oohama [24]. In these multi-user settings, Fano’s inequality by itself is insufficient to characterize different tradeoffs. A more recent application proposed by Liu and Viswanath [23] uses entropy power inequalities to solve certain optimization problems. The entropy power inequality lower bounds the differential entropy of the convolution of two independent random variables X, Y taking values in Rn. Its covariance-preserving version states that H( λX + 1 − λY) ≥ λH(X) + (1 − λ)H(Y) for all 0 ≤ λ ≤ 1. (1) Eq. (1) can be shown [22, 32] to be equivalent to the more commonly used statement e2H(X+Y)/n ≥ e2H(X)/n + e2H(Y)/n. The latter explains the terminology as e2H(X)/n is the power, i.e., variance of a Gaussian ran-dom variable with identical entropy as H(X). Inequalities such as (1) are closely related to Log-Sobolev inequalities (see e.g., [31]) as well as Brunn-Minkowski-type inequalities [6]. Gen-eralizations to free probability [30] and quantum states have been considered. 1 ar
We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of “quantum majorization” from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.
Shannon's entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule: f (√a X + √1 - aY) ≥ af(X)+(1 - a)f(Y) ∀a ∈ [0,1]. Here, X and Y are continuous random variables and the function f is either the differential entropy or the entropy power. König and Smith [IEEE Trans. Inf. Theory 60(3), 1536-1548 (2014)] and De Palma, Mari, and Giovannetti [Nat. Photonics 8(12), 958-964 (2014)] obtained quantum analogues of these inequalities for continuous-variable quantum systems, where X and Y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields. In this paper, we similarly establish a class of EPI analogues for d-level quantum systems (i.e., qudits). The underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a ∈ [0,1] and acts like a finite-dimensional analogue of a beam splitter with transmissivity a, converting a two-qudit product state into a single qudit state. We refer to this channel as a partial swap channel because of the particular way its output interpolates between the states of the two qudits in the input as a is changed from zero to one. We obtain analogues of Shannon's EPI, not only for the von Neumann entropy and the entropy power for the output of such channels, but also for a much larger class of functions. This class includes the Rényi entropies and the subentropy. We also prove a qudit analogue of the entropy photon number inequality (EPnI). Finally, for the subclass of partial swap channels for which one of the qudit states in the input is fixed, our EPIs and EPnI yield lower bounds on the minimum output entropy and upper bounds on the Holevo capacity.
We prove the multimode conditional quantum Entropy Power Inequality for bosonic quantum systems. This inequality determines the minimum conditional von Neumann entropy of the output of the most general linear mixing of bosonic quantum modes among all the input states of the modes with given conditional entropies. Bosonic quantum systems constitute the mathematical model for the electromagnetic radiation in the quantum regime, which provides the most promising platform for quantum communication and quantum key distribution. We apply our multimode conditional quantum Entropy Power Inequality to determine new lower bounds to the squashed entanglement of a large family of bosonic quantum Gaussian states. The squashed entanglement is one of the main entanglement measures in quantum communication theory, providing the best known upper bound to the distillable key. Exploiting this result, we determine a new lower bound to the squashed entanglement of the multimode bosonic Gaussian channels that are extreme,i.e., that cannot be decomposed as a non-trivial convex combination of quantum channels. The squashed entanglement of a quantum channel provides an upper bound to its secret-key capacity,i.e., the capacity to generate a secret key shared between the sender and the receiver. Lower bounds to the squashed entanglement are notoriously hard to prove. Our results contribute to break this barrier and will stimulate further research in the field of quantum communication with bosonic quantum systems.
Shannon’s entropy power inequality (EPI) can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule: f(a X+1−a Y)≥af(X)+(1−a)f(Y)∀ a∈[0,1]. Here, X and Y are continuous random variables and the function f is either the differential entropy or the entropy power. König and Smith [IEEE Trans. Inf. Theory 60(3), 1536–1548 (2014)] and De Palma, Mari, and Giovannetti [Nat. Photonics 8(12), 958–964 (2014)] obtained quantum analogues of these inequalities for continuous-variable quantum systems, where X and Y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields. In this paper, we similarly establish a class of EPI analogues for d-level quantum systems (i.e., qudits). The underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a ∈ [0, 1] and acts like a finite-dimensional analogue of a beam splitter with transmissivity a, converting a two-qudit product state into a single qudit state. We refer to this channel as a partial swap channel because of the particular way its output interpolates between the states of the two qudits in the input as a is changed from zero to one. We obtain analogues of Shannon’s EPI, not only for the von Neumann entropy and the entropy power for the output of such channels, but also for a much larger class of functions. This class includes the Rényi entropies and the subentropy. We also prove a qudit analogue of the entropy photon number inequality (EPnI). Finally, for the subclass of partial swap channels for which one of the qudit states in the input is fixed, our EPIs and EPnI yield lower bounds on the minimum output entropy and upper bounds on the Holevo capacity.
Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L2 L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.