The aim of the manuscript is to characterize monotone metric in the space of Markov map. Here, metric may not be Riemanian, or equivalently, may not be induced from an inner product. So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov and Petz characterized all the monotone metrics in the classical and quantum state space. As for channels, however, only a little is known about its geometrical structures. In that author's previous manuscript, the upper and the lower bound of monotone channel metric was derived using resource conversion theory, and it is proved that any monotone metric cannot be Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to build a geometric theory consistent across probability distributions and channels. To dispense with the assumption that a metric is Riemanian, we introduce some assumptions on asymptotic behavior;weak asymptotic additivity and lower asymptotic continuity. The proof utilizes resource conversion technique. In the end of the paper, an implication on quantum state metrics is discussed.
The aim of the manuscript is to characterize monotone metric in the space of Markov map. Here, metric may not be Riemanian, or equivalently, may not be induced from an inner product. So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov and Petz characterized all the monotone metrics in the classical and quantum state space. As for channels, however, only a little is known about its geometrical structures. In that author's previous manuscript, the upper and the lower bound of monotone channel metric was derived using resource conversion theory, and it is proved that any monotone metric cannot be Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to build a geometric theory consistent across probability distributions and channels. To dispense with the assumption that a metric is Riemanian, we introduce some assumptions on asymptotic behavior;weak asymptotic additivity and lower asymptotic continuity. The proof utilizes resource conversion technique. In the end of the paper, an implication on quantum state metrics is discussed.
The aim of the manuscript is to characterize monotone metric in the space of Markov map. Here, metric may not be Riemanian, or equivalently, may not be induced from an inner product. So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov and Petz characterized all the monotone metrics in the classical and quantum state space. As for channels, however, only a little is known about its geometrical structures. In that author's previous manuscript, the upper and the lower bound of monotone channel metric was derived using resource conversion theory, and it is proved that any monotone metric cannot be Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to build a geometric theory consistent across probability distributions and channels. To dispense with the assumption that a metric is Riemanian, we introduce some assumptions on asymptotic behavior;weak asymptotic additivity and lower asymptotic continuity. The proof utilizes resource conversion technique. In the end of the paper, an implication on quantum state metrics is discussed.
The aim of the manuscript is to characterize monotone `metric' in the space of Markov map. Here, `metric' means the square of the norm defined on the tangent space, and not necessarily induced from an inner product (this property hereafter will be called inner-product-assumption), different from usual metric used in differential geometry. As for metrics in So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov proved the monotone metric in probability distribution space is unique up to constant multiple, and identical to Fisher information metric. Petz characterized all the monotone metrics in the quantum state space using operator mean. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. (Notably, we do not use the inner-product-assumption.) To obtain this result, `resource conversion' technique, which is widely used in quantum information, is used. We consider distillation from and formation to a family of channels. Under these axioms, we identify the largest and the smallest `metrics'. Interestingly, they are not induced from any inner product, i.e., not a metric. Indeed, one can prove that any `metric' satisfying our axioms can not be a metric. This result has some impact on the axiomatic study of the monotone metric in the space of classical and quantum states, since both conventional theory relies on the inner-product-assumption. Also, we compute the lower and the upper bound for some concrete examples.
So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. Under these axioms, we identify the largest and the smallest metrics. Also, we studied asymptotic theory of metric in parallel and adaptive repetition settings, and applied them to the study of channel estimation. First we express the achievable lower bound of the mean square error (MSE) of an estimate by a monotone channel metric, and show this equals O(1/n) for noisy channels, where n is the number of times of channel use. This result shows Heisenberg rate, or O(1/n^2)-rate of the MSE observed in case of estimation of unitary, collapses with very small arbitrary noise.
Probability theory is a measure theory – with a soul. Mark Kac Section 2.6 was devoted to classical ensembles , that is to say ensembles defined by probability measures on the set of classical probability distributions over N events. In this chapter quantum ensembles are defined by choosing probability measures on the set of density matrices of size N . A warning should be issued first: there is no single, naturally distinguished measure in M ( N ) , so we have to analyse several measures, each of them with different physical motivations, advantages and drawbacks. This is in contrast to the set of pure quantum states, where the Fubini–Study measure is the only natural choice for a measure that defines ‘random states’. A simple way to define a probability measure goes through a metric. Hence we will start this chapter with a review of the metrics defined on the set of mixed quantum states. Monotone metrics In Section 2.5 we explained how the Fisher metric holds its distinguished position due to the theorem of Čencov, which states that the Fisher metric is the unique monotone metric on the probability simplex Δ N –1 . Now that the latter has been replaced by the space of quantum states M ( N ) we must look again at the question of metrics. Since the uniqueness in the classical case came from the behaviour under stochastic maps, we turn our attention to stochastic quantum maps – the completely positive, trace preserving maps discussed in Chapter 10. A distance D in the space of quantum states M ( N ) is called monotone if it does not grow under the action of a stochastic map Φ, If a monotone distance is geodesic the corresponding metric on M ( N ) is called monotone.
Quantum monotone metric was introduced by Petz [Linear Algebra Appl. 244, 81–96 (1996)], and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always the one based on completely positive, trace preserving maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics that have monotonicity under completely positive, trace non-increasing maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity, and monotonicity with respect to positive operators.
Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures. They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels. In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information. Petz and Sudár (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices. We analyse the SM quantum information in light of their work. We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics. We define a new metric C_L as a lower bound among the SM quantum informations. We look at properties of C_L and show that it is invariant but not monotone.