The entropy of the Gram matrix of a joint purification of an ensemble of K mixed states yields an upper bound for the Holevo information Chi of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain useful bounds for Chi. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For K=3 quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound Chi. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state fidelities in relation to the Holevo quantity.
The Holevo quantity and the $SU(2)$-invariant states have particular importance in quantum information processing. We calculate analytically the Holevo quantity for bipartite systems composed of spin-$j$ and spin-$\frac{1}{2}$ subsystems with $SU(2)$ symmetry, when the projective measurements are performed on the spin-$\frac{1}{2}$ subsystem. The relations among the Holevo quantity, the maximal values of the Holevo quantity and the states are analyzed in detail. In particular, we show that the Holevo quantity increases in the parameter region $F<F_d$ and decreases in region $F>F_d$ when $j$ in…
The Holevo bound is a keystone in many applications of quantum information theory. We propose "weak maximal Holevo quantity" with weak measurements as the generalization of the standard Holevo quantity which is defined as the optimal projective measurements. The scenarios that weak measurements is necessary are that only the weak measurements can be performed because for example the system is macroscopic or that one intentionally tries to do so such that the disturbance on the measured system can be controlled for example in quantum key distribution protocols. We evaluate systematically the we…
We present a family of easily computable upper bounds for the Holevo quantity of ensemble of quantum states depending on a reference state as a free parameter. These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve existing estimates in the literature. We also present upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on metric divergence of the ensemb…
We start with Fannes' type and Winter's type tight continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying Hilbert space is finite and obtain Fannes' type tight continuity bounds for the Holevo quantity in this case. In general case co…
Our everyday reality is characterized by objective information$\unicode{x2013}$information that is selected and amplified by the environment that interacts with quantum systems. Many observers can accurately infer that information indirectly by making measurements on fragments of the environment. The correlations between the system, $\mathcal{S}$, and a fragment, $\mathcal{F}$, of the environment, $\mathcal{E}$, is often quantified by the quantum mutual information or the Holevo quantity that bounds the classical information about $\mathcal{S}$ transmittable by a quantum channel $\mathcal{F}$.…
We study tree kinds of quantum fidelity. Usual Uhlmann's fidelity, minus of f-divergence when $f(x)=-\sqrt{x}$, and the one introduced by the author via reverse test. All of them are quantum extensions of classical fidelity, where the first one is the largest and the third one is the smallest. We characterize them in terms of convex optimization, and introduce their 'dual' quantity, or the polar of the minus of the fidelity. They turned out to be monotone increasing by unital completely positive maps, concave, and linked to its classical version via optimization about classical-to-quantum maps…
For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a state-independent constant. However, such a bound can be violated if the quantum system is entangled with another quantum system (called memory); the quantum correlation between the systems can reduce the measurement uncertainty. On the other hand, in a curved spacetime, the presence of the Hawking radiation can increase the uncertainty in quantum measurement. The interplay of quantum correlation in the curved spacetime has become an interesting arena for studying quantum uncertainty relations. Here we d…
The interpretation of non-Markovian effects as due to the information exchange between an open quantum system and its environment has been recently formulated in terms of properly regularized entropic quantities, as their revivals in time can be upper bounded by means of quantities describing the storage of information outside the open system [Phys. Rev. Lett. 127, 030401 (2021)]. Here, we elaborate on the wider mathematical framework of the theory, specifying the key properties that allow us to associate distinguishability quantifiers with the information flow from and towards the open system…
The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumness R, a measure that has been recently introduced in order to quantify how much incompatible are the parameters to be estimated. In particular, R is an upper bound for the renormalized d…
In this paper, we characterize the saturation of four universal inequalities in quantum information theory, including a variant version of strong subadditivity inequality for von Neumann entropy, the coherent information inequality, the Holevo quantity and average entropy inequalities. These results shed new light on quantum information inequalities.
Uncertainty principle is a striking and fundamental feature in quantum mechanics distinguishing from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and Holevo quantity. It shows that the bound derived by this method will be ti…
The uncertainty principle is a fundamental principle in quantum physics. It implies that the measurement outcomes of two incompatible observables can not be predicted simultaneously. In quantum information theory, this principle can be expressed in terms of entropic measures. Berta \emph{et al}. [\href{http://www.nature.com/doifinder/10.1038/nphys1734}{ Nature Phys. 6, 659 (2010) }] have indicated that uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this article, we obtain a lower bound for entropic uncertainty in the pre…
In the study of Markovian processes, one of the principal achievements is the equivalence between the $Φ$-Sobolev inequalities and an exponential decrease of the $Φ$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $Φ$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carré du champ operator, and prove a …
I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order $E Δt / \hbar$, where $E$ is the average energy of the signal, and $Δt$ is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitt…
The uncertainty principle determines the distinction between the classical and quantum worlds. This principle states that it is not possible to measure two incompatible observables with the desired accuracy simultaneously. In quantum information theory, Shannon entropy has been used as an appropriate measure to express the uncertainty relation. According to the applications of entropic uncertainty relation, studying and trying to improve the bound of this relation is of great importance. Uncertainty bound can be altered by considering an extra quantum system as the quantum memory $B$ which is …
Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative $\mathbb{L}_p$ spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among many other applications. In this paper, we consider yet another fundamental setting - matrix-valued functions - and prove new Sobolev-type inequalities for them. Our technical contributions are two-fold: (i) we establish a series of matrix Poincaré inequalities for separably convex functions and general functions with Gaussian unitary ensembles inputs; and …
We show how to use properties of the quantum conditional mutual information to obtain continuity bounds for information characteristics of quantum channels depending on their input dimension. First we prove tight estimates for variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. Then we obtain tight continuity bounds for output conditional mutual information for a single channel and for $n$ copies of a channel. As a result tight and close-to-tight continuity bounds for basic capacities of quantum channels depending on the input …
In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Renyi generalizations of the conditional quantum mutual information of a tripartite state on $ABC$ (with $C$ being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system $B$ (with it being left open to show that the Renyi quantity is monotone with respect to local operations on system $A$). Here we define a Renyi squashed entanglement and a Renyi quantum discord based on a R…
We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay a special attention to the case of a multi-mode quantum oscillator as an input system. First we prove continuity bounds for the output conditional mutual information for a single channel and for $n$ copies of a channel. Then we obtain estimates for variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result tight …
