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Asymptotic security analysis of teleportation-based quantum cryptography

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Abstract

We prove that the teleportation-based quantum cryptography protocol presented in Gordon and Rigolin (Opt Commun 283:184, 2010), which is built using only orthogonal states encoding the classical bits that are teleported from Alice to Bob, is asymptotically secure against all types of individual and collective attacks. We then investigate modifications to that protocol leading to greater secret-key rates and to security against coherent attacks. In other words, we show an unconditional secure quantum key distribution protocol that does not need non-orthogonal quantum states to encode the bits of the secret key sent from Alice to Bob. We also revisit the security proof of the BB84 protocol by exploring the non-uniqueness of the Schmidt decomposition of its entanglement-based representation. This allows us to arrive at a secure transmission of the key for a slightly greater quantum bit error rate (quantum communication channel’s noise) when compared to its standard security analysis.

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Fig. 1
Fig. 2

Notes

  1. Alice and Bob must also share an authenticated classical channel, which can be totally insecure, to realize the key distribution.

  2. We can also use maximally entangled states to create a secret key shared by Alice and Bob without using teleportation and without sending qubits encoding classical bits from Alice to Bob [26, 27].

  3. If Eve uses different strategies, Alice and Bob can detect Eve by randomly choosing different samples of the raw key to check for security. This will lead to different error rates for different samples if Eve uses different attacks. Thus, different error rates mean that Eve tampered with the key distribution scheme. In this case Alice and Bob discard all the data and restart the key distribution protocol all over.

  4. Although we will be dealing with collective attacks, the following analysis remains valid for the more general class of coherent attacks due to the results of Refs. [36,37,38,39].

  5. We can always use more than one orthonormal basis to encode the classical bits, with non-orthogonal states encoding the same bit. If we use two such basis, we can see the present protocol as an additional security layer to the BB84 protocol. But the whole point of the GR10 protocol, which will be made clearer when we present its security analysis, is that it is secure even if we use only one orthogonal basis to encode the classical bits.

  6. Incidentally, it is worth mentioning that if in the BB84 protocol we accept all instances as a valid outcome, even when Alice and Bob use different preparation and measurement basis, we get an error rate of \(25\%\) in the ideal case (no Eve or noise). When the matching condition is not satisfied, the outcomes of Bob’s measurements are completely uncorrelated to the bit values encoded by Alice in the qubits sent to him. On the other hand, the modified GR10 protocol’s ideal error rate, \(1-2p\), depends on the entanglement of the quantum states shared between Alice and Bob (the values of \(n_1\) and \(n_2\), cf. Eq. (129)). As such, for the modified GR10 protocol we can tune this error rate as we wish and whenever \(p\ge 3/8\approx 0.375\) the error rate \(1-2p\) is lower than \(25\%\), approaching zero as \(n_1\) and \(n_2\) tend to one. Furthermore, due to the GR10 protocol’s teleportation-based operation, Bob’s measurement outcomes are not completely independent of Alice’s teleported qubits, including the instances in which Alice and Bob assign different bit values at a given run of the protocol. It is this entanglement-dependent ubiquitous correlation that allows the modified GR10 protocol to operate securely for not too low levels of entanglement.

  7. We have also employed a different entanglement-based representation designed to handle the \(p=1/2\) case alone. The secret-key fraction we obtained was the same as the one we got for \(p=1/2\) using the present entanglement-based representation with the constraint \(\lambda _1=\lambda _2\).

  8. We can also intuitively understand the security of the GR10 protocol noting that the states prepared by Alice are not sent from her to Bob, they are teleported, which prevents Eve from having a direct access to those states. This is a key difference from the BB84 protocol, where the states encoding the bits are directly sent to Bob. The teleportation of orthogonal states, instead of their direct transmission, together with the inherent random aspect of the measuring results of Alice during the implementation of the teleportation protocol forbid the undetected cloning of those states by Eve.

  9. The entanglement-based representation and all the calculations leading to the secret-key fraction r when we deal with these subensembles are equal to the ones shown in Sect. 4. The only change is in the value of p, Eq. (129). It still depends on \(n_1\) and \(n_2\) but has a different functional form.

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Acknowledgements

DL thanks CAPES (Brazilian Agency for the Improvement of Personnel of Higher Education) for funding, and GR thanks the Brazilian agencies CNPq (National Council for Scientific and Technological Development) and CNPq and FAPERJ (State of Rio de Janeiro Research Foundation) for financial support through the National Institute of Science and Technology for Quantum Information.

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Lima, D., Rigolin, G. Asymptotic security analysis of teleportation-based quantum cryptography. Quantum Inf Process 19, 201 (2020). https://doi.org/10.1007/s11128-020-02701-w

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