Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of nonlocal correlations. The uncertainty principle states that there exist pairs of incompatible measurements on quantum systems such that their outcomes cannot both be predicted. On the other hand, nonlocal correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in the presence of entanglement. Here, we show that these two fundamental quantum effects are quantitatively related. Namely, we provide an entropic uncertainty relation for the outcomes of two binary measurements, where the lower bound on the uncertainty is quantified in terms of the maximum Clauser–Horne–Shimony–Holt value that can be achieved with these measurements. We discuss applications of this uncertainty relation in quantum cryptography, in particular, to certify quantum sources using untrusted devices.
Device-independent quantum key distribution @cite_36 @cite_52 @cite_22 @cite_56 @cite_23 and randomness generation @cite_58 usually bases security on a relation between nonlocality and the randomness of the outcomes relative to some (quantum) adversary. Our result allows to split the security analysis of these protocols into two parts: the nonlocality of the measured correlations first gives a bound on the uncertainty of local measurement outcomes, which in turn can be used to ensure security. The two parts can be analyzed independently and thus our methods can be used to simplify such an analysis and, potentially, reduce the required assumptions.
@cite_22 @cite_36 @cite_52 @cite_56 @cite_23 @cite_58
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In this paper, we provide a proof of unconditional security for a semi-quantum key distribution protocol introduced in a previous work. This particular protocol demonstrated the possibility of using X basis states to contribute to the raw key of the two users (as opposed to using only direct measurement results) even though a semi-quantum participant cannot directly manipulate such states. In this work, we provide a complete proof of security by deriving a lower bound of the protocol's key rate in the asymptotic scenario. Using this bound, we are able to find an error threshold value such that for all error rates less than this threshold, it is guaranteed that A and B may distill a secure secret key; for error rates larger than this threshold, A and B should abort. We demonstrate that this error threshold compares favorably to several fully quantum protocols. We also comment on some interesting observations about the behavior of this protocol under certain noise scenarios. Standard security proofs of quantum-key-distribution (QKD) protocols often rely on symmetry arguments. In this paper, we prove the security of a three-state protocol that does not possess rotational symmetry. The three-state QKD protocol we consider involves three qubit states, where the first two states @math and @math can contribute to key generation, and the third state @math is for channel estimation. This protocol has been proposed and implemented experimentally in some frequency-based QKD systems where the three states can be prepared easily. Thus, by founding on the security of this three-state protocol, we prove that these QKD schemes are, in fact, unconditionally secure against any attacks allowed by quantum mechanics. The main task in our proof is to upper bound the phase error rate of the qubits given the bit error rates observed. Unconditional security can then be proved not only for the ideal case of a single-photon source and perfect detectors, but also for the realistic case of a phase-randomized weak coherent light source and imperfect threshold detectors. Our result in the phase error rate upper bound is independent of the loss in the channel. Also, we compare the three-state protocol with the Bennett-Brassard 1984 (BB84) protocol. For the single-photon source case, our result proves that the BB84 protocol strictly tolerates a higher quantum bit error rate than the three-state protocol, while for the coherent-source case, the BB84 protocol achieves a higher key generation rate and secure distance than the three-state protocol when a decoy-state method is used. We present a technique for proving the security of quantum-key-distribution (QKD) protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel. We present the optimal collective attack on a quantum key distribution protocol in the "device-independent" security scenario, where no assumptions are made about the way the quantum key distribution devices work or on what quantum system they operate. Our main result is a tight bound on the Holevo information between one of the authorized parties and the eavesdropper, as a function of the amount of violation of a Bell-type inequality. We consider quantum key distribution in the device-independent scenario, i.e., where the legitimate parties do not know (or trust) the exact specification of their apparatus. We show how secure key distribution can be realized against the most general attacks by a quantum adversary under the condition that measurements on different subsystems by the honest parties commute. 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. In this paper we consider a three-state variant of the BB84 quantum key distribution (QKD) protocol. We derive a new lower-bound on the key rate of this protocol in the asymptotic scenario and use mismatched measurement outcomes to improve the channel estimation. Our new key rate bound remains positive up to an error rate of @math , exactly that achieved by the four-state BB84 protocol. Quantum key distribution is widely thought to offer unconditional security in communication between two users. Unfortunately, a widely accepted proof of its security in the presence of source, device, and channel noises has been missing. This long-standing problem is solved here by showing that, given fault-tolerant quantum computers, quantum key distribution over an arbitrarily long distance of a realistic noisy channel can be made unconditionally secure. The proof is reduced from a noisy quantum scheme to a noiseless quantum scheme and then from a noiseless quantum scheme to a noiseless classical scheme, which can then be tackled by classical probability theory. We investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret-key rate of those QKD protocols involving only entropies of two-qubit density operators. As an illustration of our results, we determine new bounds for the Bennett-Brassard 1984, the 6-state, and the Bennett 1992 protocols. We show that in all these cases the first classical processing that the legitimate partners should apply consists in adding noise. Semi-quantum key distribution (SQKD) can share secret keys by using less quantum resource than its fully quantum counterparts, and this likely makes SQKD become more practical and realizable. In this paper, we present a new SQKD protocol by introducing the idea of B92 into semi-quantum key distribution and prove its unconditional security. In this protocol, the sender Alice just sends one qubit to the classical Bob and Bob just prepares one state in the preparation process. Indeed the classical user's measurement is not necessary either. This protocol can reduce some quantum communication and make it easier to be implemented. It can be seen as the semi-quantum version of B92 protocol, comparing to the protocol BKM2007 as the semi-quantum version of BB84 in fully quantum cryptography. We verify it has higher key rate and therefore is more efficient. Specifically we prove it is unconditionally secure by computing a lower bound of the key rate in the asymptotic scenario from information theory aspect. Then we can find a threshold value of errors such that for all error rates less than this value, the secure key can be established between the legitimate users definitely. We make an illustration of how to compute the threshold value in case of the reverse channel is a depolarizing one with parameter @math . Though the threshold value is a little smaller than those of some existed SQKD protocols, it can be comparable to the B92 protocol in fully quantum cryptography. Quantum key distribution, first proposed by C.H. Bennett and G. Brassard (1984), provides a possible key distribution scheme whose security depends only on the quantum laws of physics. So far the protocol has been proved secure even under channel noise and detector faults of the receiver but is vulnerable if the photon source used is imperfect. In this paper we propose and give a concrete design for a new concept, self-checking source, which requires the manufacturer of the photon source to provide certain tests; these tests are designed such that, if passed, the source is guaranteed to be adequate for the security of the quantum key distribution protocol, even though the testing devices may not be built to the original specification. The main mathematical result is a structural theorem which states that, for any state in a Hilbert space, if certain EPR-type equations are satisfied, the state must be essentially the orthogonal sum of EPR pairs. 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. Device-independent quantum key distribution aims to distribute cryptographic keys without requiring assumptions about the quantum devices in the protocol. Here, a general security proof is reported for a class of quantum key distribution protocols, which could aid the development of highly secure encryption. Secure key distribution among two remote parties is impossible when both are classical, unless some unproven (and arguably unrealistic) computation-complexity assumptions are made, such as the difficulty of factorizing large numbers. On the other hand, a secure key distribution is possible when both parties are quantum. What is possible when only one party (Alice) is quantum, yet the other (Bob) has only classical capabilities? Recently, a semi-quantum key distribution protocol was presented (Boyer, Kenigsberg and Mor, Physical Review Letters, 2007), in which one of the parties (Bob) is classical, and yet, the protocol is proven to be completely robust against an eavesdropping attempt. Here we extend that result much further. We present two protocols with this constraint, and prove their robustness against attacks: we prove that any attempt of an adversary to obtain information (and even a tiny amount of information) necessarily induces some errors that the legitimate parties could notice. One protocol presented here is identical to the one referred to above, however, its robustness is proven here in a much more general scenario. The other protocol is very different as it is based on randomization. Shor and Preskill (see Phys. Rev. Lett., vol.85, p.441, 2000) have provided a simple proof of security of the standard quantum key distribution scheme by Bennett and Brassard (1984) by demonstrating a connection between key distribution and entanglement purification protocols (EPPs) with one-way communications. Here, we provide proofs of security of standard quantum key distribution schemes, Bennett and Brassard and the six-state scheme, against the most general attack, by using the techniques of two-way entanglement purification. We demonstrate clearly the advantage of classical post-processing with two-way classical communications over classical post-processing with only one-way classical communications in quantum key distribution (QKD). This is done by the explicit construction of a new protocol for (the error correction detection and privacy amplification of) Bennett and Brassard that can tolerate a bit error rate of up to 18.9 , which is higher than what any Bennett and Brassard scheme with only one-way classical communications can possibly tolerate. Moreover, we demonstrate the advantage of the six-state scheme over Bennett and Brassard by showing that the six-state scheme can strictly tolerate a higher bit error rate than Bennett and Brassard. In particular, our six-state protocol can tolerate a bit error rate of 26.4 , which is higher than the upper bound of 25 bit error rate for any secure Bennett and Brassard protocol. Consequently, our protocols may allow higher key generation rate and remain secure over longer distances than previous protocols. Our investigation suggests that two-way entanglement purification is a useful tool in the study of advantage distillation, error correction, and privacy amplification protocols. Recent advances in theoretical and experimental quantum computing bring us closer to scalable quantum computing devices. This makes the need for protocols that verify the correct functionality of quantum operations timely and has led to the field of quantum verification. In this paper we address key challenges to make quantum verification protocols applicable to experimental implementations. We prove the robustness of the single server verifiable universal blind quantum computing protocol of Fitzsimons and Kashefi (2012 arXiv:1203.5217) in the most general scenario. This includes the case where the purification of the deviated input state is in the hands of an adversarial server. The proved robustness property allows the composition of this protocol with a device-independent state tomography protocol that we give, which is based on the rigidity of CHSH games as proposed by (2013 Nature 496 456–60). The resulting composite protocol has lower round complexity for the verification of entangled quantum servers with a classical verifier and, as we show, can be made fault tolerant. We study quantum key distribution with standard weak coherent states and show, rather counter-intuitively, that the detection events originated from vacua can contribute to secure key generation rate, over and above the best prior art result. Our proof is based on a communication complexity quantum memory argument. The construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver. We show that if only near-perfect security is required, the size of the key can be reduced by a factor of two. More specifically, we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d2 operators required to randomize perfectly. Aside from the private quantum channel, variations of this construction can be applied to many other tasks in quantum information processing. We show, for instance, that it can be used to construct LOCC data hiding schemes for bits and qubits that are much more efficient than any others known, allowing roughly log d qubits to be hidden in 2 log d qubits. The method can also be used to exhibit the existence of quantum states with locked classical correlations, an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key. Our construction also provides the basic building block for a method of remotely preparing arbitrary d-dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement. Randomness extraction is of fundamental importance for information-theoretic cryptography. It allows to transform a raw key about which an attacker has some limited knowledge into a fully secure random key, on which the attacker has essentially no information. Up to date, only very few randomness-extraction techniques are known to work against an attacker holding quantum information on the raw key. This is very much in contrast to the classical (non-quantum) setting, which is much better understood and for which a vast amount of different techniques are known and proven to work. We prove a new randomness-extraction technique, which is known to work in the classical setting, to be secure against a quantum attacker as well. Randomness extraction is done by xor'ing a so-called δ-biased mask to the raw key. Our result allows to extend the classical applications of this extractor to the quantum setting. We discuss the following two applications. We show how to encrypt a long message with a short key, information-theoretically secure against a quantum attacker, provided that the attacker has enough quantum uncertainty on the message. This generalizes the concept of entropically-secure encryption to the case of a quantum attacker. As second application, we show how to do errorcorrection without leaking partial information to a quantum attacker. Such a technique is useful in settings where the raw key may contain errors, since standard error-correction techniques may provide the attacker with information on, say, a secret key that was used to obtain the raw key. We prove that the 1984 protocol of Bennett and Brassard (BB84) for quantum key distribution is secure. We first give a key distribution protocol based on entanglement purification, which can be proven secure using methods from Lo and Chau's proof of security for a similar protocol. We then show that the security of this protocol implies the security of BB84. The entanglement purification based protocol uses Calderbank-Shor-Steane codes, and properties of these codes are used to remove the use of quantum computation from the Lo-Chau protocol.