Quantum Privacy Amplification Research Articles

Overcoming noise limitations in quantum key distribution with quantum privacy amplification

Overcoming noise limitations in quantum key distribution with quantum privacy amplification

Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification

The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s\in (1,2]$ </tex-math></inline-formula> , which has not been addressed previously in the quantum setting.

Precise Evaluation of Leaked Information with Secure Randomness Extraction in the Presence of Quantum Attacker

We treat secret key extraction when the eavesdropper has correlated quantum states. We propose quantum privacy amplification theorems different from Renner’s, which are based on quantum conditional Rényi entropy of order 1 + s. Using those theorems, we derive an exponential rate of decrease for leaked information and the asymptotic equivocation rate, which have not been derived hitherto in the quantum setting.

Public Quantum Communication and Superactivation

Is there a meaningful quantum counterpart to public communication? We argue that the symmetric-side channel -- which distributes quantum information symmetrically between the receiver and the environment -- is a good candidate for a notion of public quantum communication in entanglement distillation and quantum error correction. This connection is partially motivated by [Brand\~ao and Oppenheim, arXiv:1004.3328], where it was found that if a sender would like to communicate a secret message to a receiver through an insecure quantum channel using a shared quantum state as a key, then the insecure quantum channel is only ever used to simulate a symmetric-side channel, and can always be replaced by it without altering the optimal rate. Here we further show, in complete analogy to the role of public classical communication, that assistance by a symmetric-side channel makes equal the distillable entanglement, the recently-introduced mutual independence, and a generalization of the latter, which quantifies the extent to which one of the parties can perform quantum privacy amplification. Symmetric-side channels, and the closely related erasure channel, have been recently harnessed to provide examples of superactivation of the quantum channel capacity. Our findings give new insight into this non-additivity of the channel capacity and its relation to quantum privacy. In particular, we show that single-copy superactivation protocols with the erasure channel, which encompasses all examples of non-additivity of the quantum capacity found to date, can be understood as a conversion of mutual independence into distillable entanglement.

Quantum secure direct communication and deterministic secure quantum communication

In this review article, we review the recent development of quantum secure direct communication (QSDC) and deterministic secure quantum communication (DSQC) which both are used to transmit secret message, including the criteria for QSDC, some interesting QSDC protocols, the DSQC protocols and QSDC network, etc. The difference between these two branches of quantum communication is that DSQC requires the two parties exchange at least one bit of classical information for reading out the message in each qubit, and QSDC does not. They are attractive because they are deterministic, in particular, the QSDC protocol is fully quantum mechanical. With sophisticated quantum technology in the future, the QSDC may become more and more popular. For ensuring the safety of QSDC with single photons and quantum information sharing of single qubit in a noisy channel, a quantum privacy amplification protocol has been proposed. It involves very simple CHC operations and reduces the information leakage to a negligible small level. Moreover, with the one-party quantum error correction, a relation has been established between classical linear codes and quantum one-party codes, hence it is convenient to transfer many good classical error correction codes to the quantum world. The one-party quantum error correction codes are especially designed for quantum dense coding and related QSDC protocols based on dense coding.

Quantum Privacy Amplification for a Sequence of Single Qubits

We present a scheme for quantum privacy amplification (QPA) for asequence of single qubits. The QPA procedure uses a unitaryoperation with two controlled-not gates and a Hadamard gate. Everytwo qubits are performed with the unitary gate operation, and ameasurement is made on one photon and the other one is retained.The retained qubit carries the state information of the discardedone. In this way, the information leakage is reduced. Theprocedure can be performed repeatedly so that the informationleakage is reduced to any arbitrarily low level. With this QPAscheme, the quantum secure direct communication with single qubitscan be implemented with arbitrarily high security. We also exploitthis scheme to do privacy amplification on the single qubits inquantum information sharing for long-distance communication with quantum repeaters.

Effects of single-qubit quantum noise on entanglement purification

We study the stability under quantum noise effects of the quantum privacy amplification protocol for the purification of entanglement in quantum cryptography. We assume that the E91 protocol is used by two communicating parties (Alice and Bob) and that the eavesdropper Eve uses the isotropic Bu\v{z}ek-Hillery quantum copying machine to extract information. Entanglement purification is then operated by Alice and Bob by means of the quantum privacy amplification protocol and we present a systematic numerical study of the impact of all possible single-qubit noise channels on this protocol. We find that both the qualitative behavior of the fidelity of the purified state as a function of the number of purification steps and the maximum level of noise that can be tolerated by the protocol strongly depend on the specific noise channel. These results provide valuable information for experimental implementations of the quantum privacy amplification protocol.

Conversion of a general quantum stabilizer code to an entanglement distillation protocol

We show how to convert a quantum stabilizer code to a one-way or two-way entanglement distillation protocol. The proposed conversion method is a generalization of those of Shor-Preskill and Nielsen-Chuang. The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2,1]] stabilizer codes. We also give an example of a two-way protocol converted from a stabilizer better than the recurrence protocol and the quantum privacy amplification protocol. The distillable entanglement by the class of one-way protocols converted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the channel corresponding to the distilled state, and they can distill asymptotically more entanglement from a very noisy Werner state than the hashing protocol.

On the analytical convergence of the QPA procedure

We present an analytical proof of the convergence of the “quantum privacy amplification” procedure proposed by Deutsch et al. [Phys. Rev. Lett. 77 (1996) 2818]. The proof specifies the range of states which can be purified by this method.

Erratum: Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels [Phys. Rev. Lett. 77, 2818 (1996)

Received 9 December 1997DOI:https://doi.org/10.1103/PhysRevLett.80.2022©1998 American Physical Society

Multiple-particle interference and quantum error correction

The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than 1 - 2 H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).

Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels.

Existing quantum cryptographic schemes are not, as they stand, operable in the presence of noise on the quantum communication channel. Although they become operable if they are supplemented by classical privacy-amplification techniques, the resulting schemes are difficult to analyse and have not been proved secure. We introduce the concept of quantum privacy amplification and a cryptographic scheme incorporating it which is provably secure over a noisy channel. The scheme uses an `entanglement purification' procedure which, because it requires only a few quantum Controlled-Not and single-qubit operations, could be implemented using technology that is currently being developed. The scheme allows an arbitrarily small bound to be placed on the information that any eavesdropper may extract from the encrypted message.