We find, in an analysis involving four prior probabilities (p's), that the information-theoretic-based comparative noninformativity test devised by Clarke, and applied by Slater in a quantum setting, yields a ranking (p_{F_{q=1}} > p_{B} > p_{B_{q=1}trunc} >p_{F}) fully consistent with Srednicki's recently-stated criterion for priors of ``biasedness to pure states''. Two of the priors are formed by extending certain metrics of quantum-theoretic interest from three- to four-dimensions -- by incorporating the q-parameter (nonextensivity/Tsallis index/escort parameter). The three-dimensional metrics are the Bures (minimal monotone) metric over the two-level quantum systems and the Fisher information metric over the corresponding family of Husimi distributions. The priors p_{B} and p_{F} are the (independent-of-q) normalized volume elements of these metrics, and p_{F_{q=1}} is the normalized volume element of the q-extended Fisher information metric, with q set to 1. While originally intended to similarly be the q-extension of the Bures metric, with q then set to 1, the prior p_{B_{q=1}trunc}, actually entails the truncation of the only off-diagonal entry of the extended Bures metric tensor. Without this truncation, the q-extended Bures volume element is null, as is also the case in two other quantum scenarios we examine.
We consider a family of prior probability distributions of particular interest, all being defined on the three-dimensional convex set of two-level quantum systems. Each distribution is, following recent work of Petz and Sudar, taken to be proportional to the volume element of a monotone metric on that Riemannian manifold. We apply an entropy-based test (a variant of one recently developed by Clarke) to determine which of two priors is more noninformative in nature. This involves converting them to posterior probability distributions based on some set of hypothesized outcomes of measurements of…
We study the interrelationships between the Fisher information metric recently introduced, on the basis of maximum entropy considerations, by Brody and Hughston (quant-ph/9906085) and the monotone metrics, as explicated by Petz and Sudar. This new metric turns out to be not strictly monotone in nature, and to yield (via its normalized volume element) a prior probability distribution over the Bloch ball of two-level quantum systems that is less noninformative than those obtained from any of the monotone metrics, even the minimal monotone (Bures) metric. We best approximate the additional inform…
The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (σ_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_α). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},σ_{1}^2,α) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < α< \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featu…
When prior partial information about a state to be cloned is available, it can be cloned with a fidelity higher than that of universal quantum cloning. We experimentally verify this intriguing relationship between the cloning fidelity and the prior information by reporting the first experimental optimal quantum state-dependent cloning, using nuclear magnetic resonance techniques. Our experiments may further have important implications into many quantum information processing protocols.
We find a protocol transmitting two quantum states crossly in the butterfly network only with prior entanglement between two senders. This protocol requires only one qubit transmission or two classical bits transmission in each channel in the butterfly network. It is also proved that it is impossible without prior entanglement. More precisely, an upper bound of average fidelity is given in the butterfly network when prior entanglement is not allowed.
In quantum Bayesian inference problems, any conclusions drawn from a finite number of measurements depend not only on the outcomes of the measurements but also on a prior. Here we show that, in general, the prior remains important even in the limit of an infinite number of measurements. We illustrate this point with several examples where two priors lead to very different conclusions given the same measurement data.
The principle of superposition is an intriguing feature of Quantum Mechanics, which is regularly exploited at various instances. A recent work [PRL \textbf{116}, 110403 (2016)] shows that the fundamentals of Quantum Mechanics restrict the superposition of two arbitrary pure states of a quantum system, even though it is possible to superpose two quantum states with partial prior knowledge. The prior knowledge imposes geometrical constraints on the choice of input pure states. We discuss an experimentally feasible protocol to superpose multiple pure states of a $d$ dimensional quantum system and…
We investigate a scheme for implementing quantum dense coding via atomic ensembles, where prior distribution of the quantum entangled state is not needed. Our scheme also combines another two distinct advantages: atomic ensembles qubits with long coherent time serve as memory qubits and phtonic qubit as flying qubit. Thus our suggestion may offer a simple and easy way of demonstrating quantum dense coding experimentally via atomic ensembles.
Nonlocality without entanglement(NLWE) is a nonlocal phenomenon that occurs in quantum state discrimination of multipartite separable states. In the discrimination of orthogonal separable states, the term NLWE is used when the quantum states cannot be discriminated perfectly by local operations and classical communication. In this case, the occurrence of NLWE is independent of nonzero prior probabilities of quantum states being prepared. Recently, it has been found that the occurrence of NLWE can depend on nonzero prior probabilities in minimum-error discrimination of nonorthogonal separable s…
We demonstrate that two spatially separated parties (Alice and Bob) can utilize shared prior quantum entanglement, and classical communications, to establish a synchronized pair of atomic clocks. In contrast to classical synchronization schemes, the accuracy of our protocol is independent of Alice or Bob's knowledge of their relative locations or of the properties of the intervening medium.
It is known that the dual of the general adversary bound can be used to build quantum query algorithms with optimal complexity. Despite this result, not many quantum algorithms have been designed this way. This paper shows another example of such algorithm. We use the learning graph technique from arXiv:1105.4024 to give a quantum algorithm for $k$-distinctness problem that runs in $o(n^{3/4})$ queries, for a fixed $k$, given some prior knowledge on the structure of the input. The best known quantum algorithm for the unconditional problem uses $O(n^{k/(k+1)})$ queries.
This paper explores the role of a priori knowledge in the optimization of quantum information processing by investigating optimum unambiguous discrimination problems for both the qubit and qutrit states. In general, a priori knowledge in optimum unambiguous discrimination problems can be classed into two types: a priori knowledge of discriminated states themselves and a priori probabilities of preparing the states. It is clarified that whether a priori probabilities of preparing discriminated states are available or not, what type of discriminators one should design just depends on what kind o…
This paper offers examples of concrete numerical applications of Bayesian quantum-state assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in the average of outcome values of N identical von Neumann projective measurements performed on N identically prepared three-level systems. In particular the large-N limit will be considered. Three kinds of prior kno…
It is shown that, despite strong nonlinearity, entanglement of formation of two-qubit state can be measured without prior state reconstruction. Collective measurements on small number of copies are provided that allow to determine quantum concurrence {\it via} estimation of only {\it four} parameters. It is also pointed out that another entanglement measure based on so called negativity can also be measured in similar way. The result is related to general problem: what kind of information can be extracted efficiently from unknown quantum state ?
The optimal phase estimation strategy is derived when partial a priori knowledge on the estimated phase is available. The structure of the optimal measurements, estimators and the optimal probe states is analyzed. The results fill the gap in the literature on the subject which until now dealt almost exclusively with two extreme cases: almost perfect knowledge (local approach based on Fisher information) and no a priori knowledge (global approach based on covariant measurements). Special attention is paid to a natural a priori probability distribution arising from a diffusion process.
Machine learning has achieved success in many areas because of its powerful fitting ability, so we hope it can help us to solve some significant physical quantitative problems, such as quantum correlation. In this research we will use neural networks to predict the value of quantum discord. Quantum discord is a measure of quantum correlation which is defined as the difference between quantum mutual information and classical correlation for a bipartite system. Since the definition contains an optimization term, it makes analytically solving hard. For some special cases and small systems, such a…
The aim of this work is to develop a framework for realising quantum network algorithms with the use of prior knowledge about the structure of the network. We seek to obtain computational methods that allows us to locally determine network properties in a quantum superposition and drive the walk behaviour accordingly. In particular, we consider a network that consists of different types of edges, such that the transitions between nodes result in extra edge-dependent phase shift. We combine amplitude amplification and phase estimation to develop an algorithm for exploring such networks. In the …
We describe a universal information compression scheme that compresses any pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalisation procedure that enables any classical compression algorithm to be utilised in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel-Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining hig…
This paper offers examples of concrete numerical applications of Bayesian quantum-state-assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in absolute frequencies of the outcomes of N identical von Neumann projective measurements performed on N identically prepared three-level systems. Various small values of N as well as the large-N limit are considered…
For certain infinite and finite-dimensional thermal systems, we obtain --- incorporating quantum-theoretic considerations into Bayesian thermostatistical investigations of Lavenda --- high-temperature expansions of priors over inverse temperature beta induced by volume elements ("quantum Jeffreys' priors) of Bures metrics. Similarly to Lavenda's results based on volume elements (Jeffreys' priors) of (classical) Fisher information metrics, we find that in the limit beta -> 0, the quantum-theoretic priors either conform to Jeffreys' rule for variables over [0,infinity], by being proportional to …
If an unstable particle used in a two-path interference experiment decays before reaching a detector, which-way information becomes available which reduces the detected interference fringe visibility ${\cal V}$. Here we argue that even when an unstable particle does {\em not} decay while in the interferometer, {\em a priori} which-way information is still available in the form of path predictability ${\cal P}$ which depends on the particle's decay rate $Γ$. We further demonstrate that in a matter-wave Mach-Zehnder interferometer using an excited atom with an appropriately tuned cavity, ${\cal …
Proper and improper mixed states serve as different prior beliefs for quantum state retrodiction
Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
We present two new schemes for quantum teleportation between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference frame alignment during execution, suiting them to situations of rapid reference frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially --- this includes the case of qubit teleportation under arbitrary SU(2) reference frame uncertainty--- while communicating no informatio…
We investigate the dependence of the fidelity of a Mach-Zehnder quantum interferometer on the prior information about the phase, for Fock state input and for maximally entangled (N00N) state input. For no prior information, the fidelity for Fock state input is greater than for N00N state input. In the limit of a narrow distribution describing the prior information, we find that both Fock and N00N state inputs lead to nearly equal fidelity.
Duan, Giedke, Cirac and Zoller (quant-ph/9908056) and, independently, Simon (quant-ph/9909044) have recently found necessary and sufficient conditions for the separability (classical correlation) of the Gaussian two-party (continuous variable) states. Duan et al remark that their criterion is based on a "much stronger bound" on the total variance of a pair of Einstein-Podolsky-Rosen-type operators than is required simply by the uncertainty relation. Here, we seek to formalize and test this particular assertion in both classical and quantum-theoretic frameworks. We first attach to these states …
Quantum state tomography is both a crucial component in the field of quantum information and computation, and a formidable task that requires an incogitably large number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, e…
We numerically study the performance of the displacement based quantum receiver for the discrimination of weak 3- and 4-phase-shift keyed (PSK) coherent state signals. We show that due to the nontrivial asymmetry of the receiver structure, optimization of the prior probability increases the mutual information and achieves sub-shot-noise limit discrimination. Moreover, we estimate the cutoff rate for a 4-PSK signal and confirm that the prior probability optimization shortens the code length for a given decoding error criterion. Such consideration for the asymmetric channel matrix is essential i…
We consider the problem of discriminating quantum states, where the task is to distinguish two different quantum states with a complete classical knowledge about them, and the problem of classifying quantum states, where the task is to distinguish two classes of quantum states where no prior classical information is available but a finite number of physical copies of each classes are given. In the case the quantum states are represented by coherent states of light, we identify intermediate scenarios where partial prior information is available. We evaluate an analytical expression for the mini…
Continuing our earlier work (quant-ph/0401060), we give two alternative proofs of the result that a noiseless qubit channel has identification capacity 2: the first is direct by a "maximal code with random extension" argument, the second is by showing that 1 bit of entanglement (which can be generated by transmitting 1 qubit) and negligible (quantum) communication has identification capacity 2. This generalises a random hashing construction of Ahlswede and Dueck: that 1 shared random bit together with negligible communication has identification capacity 1. We then apply these results to prove …
Hybrid Cramér-Rao bound for Quantum Bayes-Point Estimation with Nuisance Parameters
Graduate School of Informatics and Engineering, The University of Electro-Communications (UEC)（电气通信大学信息与工程研究生院）；Institute for Advanced Science, The University of Electro-Communications (UEC)（电气通信大学高等科学研究所）
In quantum process tomography, it is possible to express the experimenter's prior information as a sequence of quantum operations, i.e., trace-preserving completely positive maps. In analogy to de Finetti's concept of exchangeability for probability distributions, we give a definition of exchangeability for sequences of quantum operations. We then state and prove a representation theorem for such exchangeable sequences. The theorem leads to a simple characterization of admissible priors for quantum process tomography and solves to a Bayesian's satisfaction the problem of an unknown quantum ope…
The generalized notion of noncontextuality provides an avenue to explore the fundamental departure of quantum theory from a classical explanation. Recently, extracting a different form of quantum advantage in various information processing tasks has received an upsurge of interest. In a recent work [D. Schmid and R. W. Spekkens, Phys. Rev. X 8, 011015 (2018)] it has been demonstrated that discrimination of two nonorthogonal pure quantum states entails contextual advantage when the states are supplied with equal prior probabilities. We generalized the work to arbitrary prior probabilities as we…
The sequential unambiguous state discrimination (SSD) of two states prepared in arbitrary prior probabilities is studied, and compared with three strategies that allow classical communication. The deviation from equal probabilities contributes to the success in all the tasks considered. When one considers at least one of the parties succeeds, the protocol with probabilistic cloning is superior to others, which is not observed in the special case with equal prior probabilities. We also investigate the roles of quantum correlations in SSD, and show that the procedure requires discords but reject…
Robust multi-hypothesis quantum-state discrimination under unknown common unitary perturbations via least favorable priors
Graduate School of Engineering Science, The University of Osaka（大阪大学工程科学研究科）；Center for Education in Liberal Arts and Sciences, The University of Osaka（大阪大学通识教育中心）；Center for Quantum Information and Quantum Biology, The University of Osaka（大阪大学量子信息与量子生物学中心）
The fidelity of a quantum transformation is strongly linked with the prior partial information of the state to be transformed. We illustrate this interesting point by proposing and demonstrating the superior cloning of coherent states with prior partial information. More specifically, we propose two simple transformations that under the Gaussian assumption optimally clone symmetric Gaussian distributions of coherent states as well as coherent states with known phases. Furthermore, we implement for the first time near-optimal state-dependent cloning schemes relying on simple linear optics and f…
Superposition, arguably the most fundamental property of quantum mechanics, lies at the heart of quantum information science. However, how to create the superposition of any two unknown pure states remains as a daunting challenge. Recently, it is proved that such a quantum protocol does not exist if the two input states are completely unknown, whereas a probabilistic protocol is still available with some prior knowledge about the input states [M. Oszmaniec \emph{et al.}, Phys. Rev. Lett. 116, 110403 (2016)]. The knowledge is that both of the two input states have nonzero overlaps with some giv…
Two questions are suggested as having priority when trying to bring together Quantum Mechanics and General Relativity. Both questions have a scope which goes well beyond Physics, and in particular Quantum Mechanics and General Relativity.
It is believed that the more we have {\it a priori} information on input states, the better we can make the quality of clones in quantum cloning machines. This common sense idea was confirmed several years ago by analyzing a situation, where the input state is either one of two non-orthogonal states. If the {\it a priori} information is measured by the Shannon entropy, common sense predicts that the quality of the clone becomes poorer with increasing $N$, where $N$ is the number of possible input states. We show, however, that the {\it a priori} information measured by the Shannon entropy does…
We describe protocols for passive atomic clocks based on quantum interrogation of the atoms. Unlike previous techniques, our protocols are adaptive and take advantage of prior information about the clock's state. To reduce deviations from an ideal clock, each interrogation is optimized by means of a semidefinite program for atomic state preparation and measurement whose objective function depends on the prior information. Our knowledge of the clock's state is maintained according to a Bayesian model that accounts for noise and measurement results. We implement a full simulation of a running cl…
