Tomography of the two-qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the quantum key distribution scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so-called symmetric informationally complete positive-operator-valued measure (SIC POVM) qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimental data the deep connections that exist between SIC POVM tomography and discrete Wigner representations. We also emphasize the special role played by Bell states in this approach and propose a protocol for quantum key distribution during which a third party is able to concede or to deny a posteriori to the authorized users the ability to build a fresh cryptographic key.
Tomography of the two-qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the quantum key distribution scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so-called symmetric informationally complete positive-operator-valued measure (SIC POVM) qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimental data the deep connections that exist between SIC POVM tomography and discrete Wigner representations. We also emphasize the special role played by Bell states in this approach and propose a protocol for quantum key distribution during which a third party is able to concede or to deny a posteriori to the authorized users the ability to build a fresh cryptographic key.
Symmetric, informationally complete, positive operator-valued measurements (SIC-POVM) are fundamental in quantum information and foundations of quantum theory[1]. For example, SIC-POVMs are crucial for optimal quantum state tomography, quantum key distribution with optimal trade-off between security and key rate, device-independent certification of optimal randomness from one bit of entanglement, and information-based interpretations and operational reconstructions of quantum theory. Experimentally, photonic experiments of qubit and qutrit tomography, and quantum key distribution[2] have aimed for SIC-POVMs. However, these experiments rely on specific models for parts of the experimental setups and do not allow us to test whether the target measurements are truly SIC-POVMs in a "device independent" (DI) way that is, without relying on any modelling of the devices.
We study different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g. to perform full tomography of this state. We focus on two apparently simple cases, full tomography of one and two qubit systems. We analyze and compare those techniques according to two figures of merit. Our first criterion is the minimisation of the redundancy of the data acquired during the tomographic process. In the case of two-qubits tomography, we also analyze this process from the point of view of factorisability, so to say we analyze the possibility to realise the tomographic process through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions. The discrete Heisenberg-Weyl group is an essential ingredient of our approach. Possible extensions of our results to higher dimensions are discussed in the last section and in the conclusions.
An experimental demonstration of two-qutrit state tomography via one-qutrit symmetric informationally complete positive operator-valued measure (SIC-POVM) is presented. A two-qutrit state is encoded in the transversal paths of a spontaneous parametric down-converted photon pair. A spatial light modulator allows us to implement the necessary 81 POVM elements to reconstruct the state. The quality of the reconstruction is evaluated by comparing independent measurements with the predicted results calculated with the reconstructed state. Entanglement in the system is calculated via negativity and generalized robustness from the two-qutrit reconstructed density operator.
Traditionally, quantum information is formulated and implemented in terms of elementary quantum systems e.g. qubits. Now, it has been shown that certain appealing properties of entangled systems, like for instance nonlocality, are more robust against noise when they are realized at the level of higher dimensional systems, like qutrits (or more generally qudits with d strictly larger than 2). It has also been shown, in the case of the generalized Ekert protocol, (a so-called entanglement-based protocol) that confidentiality is enhanced when Quantum Key Distribution is realized with carriers of information of dimensionality strictly higher than 2 (this is because optimal cloning is less effective when the dimension d increases). This brought us to consider the possible generalization of the so-called Singapore protocol, a full tomographic qubit protocol, to dimensions strictly higher than 2. As we shall show, dimension 3 is, like dimension 2, privileged regarding the connections of the protocol with Wigner tomography. We also show that, although the qutrit version of the Singapore protocol is, just like Ekert's protocol, more robust against noise than its qubit counterpart, it is less appealing for what concerns the effective information rate (e.g. the number of qudits to be sent in average in order to exchange one classical bit of the key).
The characterization of quantum processes, e.g. communication channels, is an essential ingredient for establishing quantum information systems. For quantum key distribution protocols, the amount of overall noise in the channel determines the rate at which secret bits are distributed between authorized partners. In particular, tomographic protocols allow for the full reconstruction, and thus characterization, of the channel. Here, we perform quantum process tomography of high-dimensional quantum communication channels with dimensions ranging from 2 to 5. We can thus explicitly demonstrate the effect of an eavesdropper performing an optimal cloning attack or an intercept-resend attack during a quantum cryptographic protocol. Moreover, our study shows that quantum process tomography enables a more detailed understanding of the channel conditions compared to a coarse-grained measure, such as quantum bit error rates. This full characterization technique allows us to optimize the performance of quantum key distribution under asymmetric experimental conditions, which is particularly useful when considering high-dimensional encoding schemes.
Recently, Boyer et al. presented a novel semiquantum key distribution protocol [ Phys. Rev. Lett. 99 (2007) 140501 ] by using four quantum states, each of which is randomly prepared in the Z or X basis. Here we present a semi-quantum key distribution protocol by using maximally entangled states in which quantum Alice shares a secret key with classical Bob. Quantum Alice has the ability to prepare Bell states and perform Bell basis or computational basis measurement. Classical Bob is restricted to measuring, preparing a particle in the computational basis, reflecting or reordering the particles. The qubit efficiency of the protocol improves to 50% and the protocol can be modified to a measure-resend protocol or a protocol without quantum memory. We also show that the protocol is secure against eavesdropping.