Reverse test and quantum analogue of classical fidelity and generalized fidelity

The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic. In the former setting, we prove that the upperbound and the lowerbund of D^{Q}({\rho}||{\sigma}) is D^{R}({\rho}||{\sigma}):=tr{\rho}ln{\sigma}^{1/2}{\rho}^{-1}{\sigma}^{1/2} and D({\rho}||{\sigma}):= tr{\rho}(ln{\rho}-ln{\sigma}), respectively. In the latter setting, we prove uniqueness of quantum relative entropy, that is, D^{Q}({\rho}||{\sigma}) should equal a constant multiple of D({\rho}||{\sigma}). In the analysis, we define and use reverse test and asymptotic reverse test, which are natural inverse of hypothesis test.Comment: A new proof of joint convexity of $D^R$ is added. Also, some technical correction, Title change

Reverse estimation theory, Complementality between RLD and SLD, and monotone distances

Many problems in quantum information theory can be vied as interconversion between resources. In this talk, we apply this view point to state estimation theory, motivated by the following observations. First, a monotone metric takes value between SLD and RLD Fisher metric. This is quite analogous to the fact that entanglement measures are sandwiched by distillable entanglement and entanglement cost. Second, SLD add RLD are mutually complement via purification of density matrices, but its operational meaning was not clear. To find a link between these observations, we define reverse estimation problem, or simulation of quantum state family by probability distribution family, proving that RLD Fisher metric is a solution to local reverse estimation problem of quantum state family with 1-dim parameter. This result gives new proofs of some known facts and proves one new fact about monotone distances. We also investigate information geometry of RLD, and reverse estimation theory of a multi-dimensional parameter family.Comment: Submitted to QIT. Full is in prepareatio

On interaction-free measurement

This manuscript is inspired by the paper [2]. In the paper, they investigate a method to detect existence of an object with arbitrarily small interaction. Below, we sketch their protocol to motivate the present manuscript. The object of their protocol is to detect whether the given blackbox interact with input states or not, with negligible distortion of the blackbox, and high detection probability. In this paper, we do two things. First, we prove the above mentioned protocol is optimal in a certain setting. The main tool here is adversary method, a classical method in query complexity. Second, we present a protocol to detect unitary operations with negligible error and no distortion of the input at all

Convertibility of Observables

Some problems of quantum information, cloning, estimation and testing of states, universal coding etc., are special example of the following `state convertibility' problem. In this paper, we consider the dual of this problem, 'observable conversion problem'. Given families of operators \{L_\theta}\}_{\theta\in\Theta} and \{M_\theta}\}_{\theta\in\Theta} , we ask whether there is a completely positive (sub) unital map which sends \{L_\theta}\} to \{M_\theta}\} for each {\theta}. We give necessary and sufficient conditions for the convertibility in some special cases

On maximization of measured ff-divergence between a given pair of quantum states

This paper deals with maximization of classical $f$-divergence between the distributions of a measurement outputs of a given pair of quantum states. $f$-divergence $D_{f}$ between the probability density functions $p_{1}$ and $p_{2}$ over a discrete set is defined as $D_{f}( p_{1}||p_{2}) :=\sum_{x}p_{2}(x) f\left(p_{1}(x)/p_{2}( x) \right)$. For example, Kullback-Leibler divergence and Renyi type relative entropy are well-known examples with good operational meanings. Thus, finding the maximal value $D_{f}^{\min}$ of measured measured $f$-divergence is also an interesting question. But so far the question is solved only for very restricted example of $f$. \ The purposes of the present paper is to advance the study further, by investigating its properties, rewriting the maximization problem to more tractable form, and giving closed formulas of the quantity in some special cases

On the First Order Asymptotic Theory of Quantum Estimation

We give a rigorous treatment on the foundation of the first order asymptotic theory of quantum estimation, with tractable and reasonable regularity conditions. Different from past works, we do not use Fisher information nor MLE, and an optimal estimator is constructed based on locally unbiased estimators. Also, we treat state estimation by local operations and classical communications (LOCC), and estimation of quantum operations.Comment: dedicated to Prof. Masafumi Akahir

The monodromy representations of local systems associated with Lauricella's FDF_D

We give the monodromy representations of local systems of twisted homology groups associated with Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations under mild conditions on parameters. Our representation is effective even in some cases when the system $F_D(a,b,c)$ is reducible. We characterize invariant subspaces under our monodromy representations by the kernel or image of a natural map from a finite twisted homology group to locally finite one.Comment: 18 pages, 1 figur

Pfaffian of Lauricella's hypergeometric system FAF_A

We give a Pfaffian system of differential equations annihilating Lauricella's hypergeometric series $F_A(a,b,c;x)$ of $m$-variables. This system is integrable of rank $2^m$. To express the connection form of this system, we make use of the intersection form of twisted cohomology groups with respect to integrals representing solutions of this system.Comment: 14 page

Reverse Test and Characterization of Quantum Relative Entropy

The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic. In the former setting, we prove that the upperbound and the lowerbund of $\mathrm{D}^{Q}(\rho||\sigma)$ is $\mathrm{D}^{R}(\rho||\sigma) :=\mathrm{tr}\,\rho\ln\sqrt{\rho}\sigma^{-1}\sqrt{\rho}$ and $\mathrm{D}(\rho||\sigma) :=$ $\mathrm{tr}\,\rho(\ln\rho-\ln\sigma)$, respectively. In the latter setting, we prove uniqueness of quantum relative entropy, that is, $\mathrm{D}^{Q}(\rho||\sigma)$ should equal a constant multiple of $\mathrm{D}(\rho||\sigma)$. In the analysis, we define and use reverse test and asymptotic reverse test, which are natural inverse of hypothesis test

Self-teleportation and its application on LOCC estimation and other tasks

A way to characterize quantum nonlocality is to see difference in the figure of merit between LOCC optimal protocol and globally optimal protocol in doing certain task, e.g., state estimation, state discrimination, cloning and broadcasting. Especially, we focus on the case where $n$ tensor of unknown states. Our conclusion is that separable pure states are more non-local than entangled pure states. More specifically, the difference in the figure of the merit is exponentially small if the state is entangled, and the exponent is log of the largest Schmidt coefficient. On the other hand, in many cases, estimation of separable states by LOCC is worse than the global optimal estimate by $O(\frac{1}{n})$. To show that the gap is exponentially small for entangled states, we propose self-teleportation protocol as the key component of construct of LOCC protocols. Objective of the protocol is to transfer Alice's part of quantum information by LOCC, using intrinsic entanglement of $| \phi_{\theta}> ^{\otimes n}$ without using any extra resources. This protocol itself is of interest in its own right