In this paper, we derive key-rate expressions for different quantum key distribution protocols. Our key-rate equations utilize multiple channel statistics, including those gathered from mismatched measurement bases - i.e., when Alice and Bob choose incompatible bases. In particular, we will consider an Extended B92 and a two-way semi-quantum protocol. For both these protocols, we demonstrate that their tolerance to noise is higher than previously thought - in fact, we will show the semi-quantum protocol can actually tolerate the same noise level as the fully quantum BB84 protocol. Along the way, we will also consider an optimal QKD protocol for various quantum channels. Finally, all the key-rate expressions which we derive in this paper are applicable to any arbitrary, not necessarily symmetric, quantum channel.
In @cite_11 , a modified two-basis BB84 was developed where the first basis was the standard computational @math basis ( @math ), while the second consisted of states @math and @math where @math . The authors of that work showed that for small @math , mismatched measurement bases can still be used to gain good channel estimates while also allowing @math and @math to use mismatched measurement bases to distill their key (since, for @math small, even with differing bases, their measurement results will be nearly correlated).
@cite_11
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We consider a modified version of the BB84 quantum key distribution protocol in which the angle between two different bases is less than π 4. We show that the channel parameter estimate becomes the same as the original protocol with sufficiently transmitted qubits. On the other hand, the statistical correlation between bits transmitted in one basis and those received in the other basis becomes stronger as the angle between two bases becomes narrower. If the angle is very small, the statistical correlation between bits transmitted in one basis and those received in the other basis is as strong as those received in the same basis as the transmitting basis, which means that the modified protocol can generate almost twice as long secret key as in the original protocol, provided that Alice and Bob choose two different bases with almost the same probability. We also point out that the reverse reconciliation often gives a different amount of secret key to the direct reconciliation over Pauli channels with our modified protocol. We consider the mismatched measurements in the BB84 quantum key distribution protocol, in which measuring bases are different from transmitting bases. We give a lower bound on the amount of a secret key that can be extracted from the mismatched measurements. Our lower bound shows that we can extract a secret key from the mismatched measurements with certain quantum channels, such as the channel over which the Hadamard matrix is applied to each qubit with high probability. Moreover, the entropic uncertainty principle implies that one cannot extract the secret key from both matched measurements and mismatched ones simultaneously, when we use the standard information reconciliation and privacy amplification procedure. Compressive Sensing theory details how a sparsely represented signal in a known basis can be reconstructed with an underdetermined linear measurement model. However, in reality there is a mismatch between the assumed and the actual bases due to factors such as discretization of the parameter space defining basis components, sampling jitter in A D conversion, and model errors. Due to this mismatch, a signal may not be sparse in the assumed basis, which causes significant performance degradation in sparse reconstruction algorithms. To eliminate the mismatch problem, this paper presents a novel perturbed orthogonal matching pursuit (POMP) algorithm that performs controlled perturbation of selected support vectors to decrease the orthogonal residual at each iteration. Based on detailed mathematical analysis, conditions for successful reconstruction are derived. Simulations show that robust results with much smaller reconstruction errors in the case of perturbed bases can be obtained as compared to standard sparse reconstruction techniques. The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.