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European Journal of Physics

Information gain when measuring an unknown qubit

Gunnar G. Bjork

Published Oct 20, 2017• 5 citations
Publisher

Abstract

In quantum information the fundamental information-containing system is the qubit. A measurement of a single qubit can at most yield one classical bit. However, a dichotomous measurement of an unknown qubit will yield much less information about the qubit state. We use Bayesian inference to compute how much information one progressively gets by making sucessive, individual measurements on an ensemble of identically prepared qubits. Perhaps surprisingly, even if the measurements are arranged so that each measurement yields one classical bit, that is, the two possible measurement outcomes are a priori equiprobable, it takes almost a handful of measurements before one has gained one bit of information about the gradually concentrated qubit probability density. We also show that by following a strategy that reaps the maximum information per measurement, we are led to a mutually unbiased basis as our measurement bases. This is a pleasing, although not entirely surprising, result.

DOI
10.1088/1361-6404/aa94e4

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In quantum information the fundamental information-containing system is the qubit. A measurement of a single qubit can at most yield one classical bit. However, a dichotomous measurement of an unknown qubit will yield much less information about the qubit state. We use Bayesian inference to compute how much information one progressively gets by making sucessive, individual measurements on an ensemble of identically prepared qubits. Perhaps surprisingly, even if the measurements are arranged so that each measurement yields one classical bit, that is, the two possible measurement outcomes are a priori equiprobable, it takes almost a handful of measurements before one has gained one bit of information about the gradually concentrated qubit probability density. We also show that by following a strategy that reaps the maximum information per measurement, we are led to a mutually unbiased basis as our measurement bases. This is a pleasing, although not entirely surprising, result.
Quantum state estimation is one of the most extensively studied topics in the field of quantum theory and quantum information. This process is essential in quantum protocols to ensure that the estimated state matches the true state. Typically, the quantum state of identical systems can be estimated from the statistics of measurement outcomes across different measurement bases. In this research, we study adaptive-measurement strategies for quantum state estimation used to identify a pure state of two-level systems and compare the obtained estimated state to the state obtained when the measurement bases are chosen randomly. The adaptive measurements in this study include the maximum-information-gain strategy, and the confirmation strategy. In this work, we simulate the outcome of the measurement on a system and use the Bayesian estimation method to estimate the obtained state. We then adaptively change the basis of the next measurement in accordance with the estimated state and the used adaptive strategy. We find that the maximum-information-gain strategy provides the fastest convergence and the most accurate state estimation compared to the confirmation strategy and the randomly-chosen-measurement-basis strategy, especially in the presence of measurement noise. Keywords: Quantum state estimation, State tomography, Adaptive measurements
Incompatibility of certain measurements—impossibility of obtaining deterministic outcomes simultaneously—is a well known property of quantum mechanics. This feature can be utilized in many contexts, ranging from Bell inequalities to device dependent QKD protocols. Typically, in these applications the measurements are chosen from a predetermined set based on a classical random variable. One can naturally ask, whether the non-determinism of the outcomes is due to intrinsic hiding property of quantum mechanics, or rather by the fact that classical, incoherent information entered the system via the choice of the measurement. Authors Rozpedek et al (2017 New J. Phys. 19 023038) examined this question for a specific case of two mutually unbiased measurements on systems of different dimensions. They have somewhat surprisingly shown that in case of qubits, if the measurements are chosen coherently with the use of a controlled unitary, outcomes of both measurements can be guessed deterministically. Here we extend their analysis and show that specifically for qubits, measurement result for any set of measurements with any a priori probability distribution can be faithfully guessed by a suitable state preparation and measurement. We also show that up to a small set of specific cases, this is not possible for higher dimensions. This result manifests a deep difference in properties of qubits and higher dimensional systems and suggests that these systems might offer higher security in specific cryptographic protocols. More fundamentally, the results show that the impossibility of predicting a result of a measurement is not caused solely by a loss of coherence between the choice of the measurement and the guessing procedure.
Performing a quantum measurement yields two different results: a classical outcome drawn from a probability distribution, according to Born's rule, and a quantum outcome corresponding to the post-measurement state. Quantum devices that provide both outcomes can be described through quantum instruments. In a realistic scenario, one can expect that the observer's obtained classical and quantum outcomes are non-ideal: this can be due to experimental limitations, but could also be explained by adversarial interference, that is, a second party that disturbs the device through a concealed measurement to obtain information. The second scenario can be interpreted through quantum compatibility, as it implies that both the observer's instrument and the adversary's measurement can be performed simultaneously. In this work, we show how the noise of the observer's device relates to the amount of information that the adversary can obtain. We study scenarios in which the adversary aims to acquire information on the same basis as the observer's measurement, or on a mutually unbiased basis with respect to the observer's basis. In both cases, we derive necessary and sufficient conditions for the compatibility of a single qubit non-ideal quantum instrument and a noisy meter, from which we obtain the maximum amount of information that the adversary can extract in terms of the noise parameters of the observer's instrument. Finally, we provide the device implementation from the adversary's point of view for the same basis scenario.
We determine the optimal measurement that maximizes the average information gain about the state of a qubit system. The qubit is prepared in one of two known states with known prior probabilities. To treat the problem analytically we employ the formalism developed for the maximum confidence quantum state discrimination strategy and obtain the POVM which optimizes the information gain for the entire parameter space of the system. We show that the optimal measurement coincides exactly with the minimum-error quantum measurement only for two pure states, or when the two states have the same Bloch radius or they are on the same diagonal of the Bloch disk.
We calculate the single-outcome information gain of quantum measurements on a completely unknown qubit state randomly chosen from different state ensembles. It is shown that the information gain decreases when the ensemble size increases. We obtain analytic single-outcome information gain formulas for continuous equatorial and spherical state ensembles, which show that the states in the former ensemble are relatively easier to identify. The squared fidelity between post-measurement state and pre-measurement state is also calculated, and the qualitative tradeoff relations between information gain and fidelity are illustrated by our results.
In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives no information about the outcomes of measurements in another basis. Here we relax the no-information condition by allowing the ddd outcomes to be predicted according to a predefined probability distribution $q=({q}_{0},...,{q}_{d\ensuremath{-}1})$. The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of mutually equibiased bases (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For d=2,3d=2,3d=2,3, we derive a set of d+1d+1d+1 MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution qqq, leading, for d=3d=3d=3, to the restriction $1/3\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1/2$, where $\ensuremath{\mu}={\ensuremath{\sum}}_{k=0}^{2}{q}_{k}^{2}$. To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index $\ensuremath{\mu}$, employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a $3\ifmmode\times\else\texttimes\fi{}3$ system. Our results reveal that not all bases in a set of LLL MEBs can contribute to the entanglement detection. A constraint, dependent on the probability distribution qqq, limits the number of MEBs that can be employed.