In this paper we consider the problem of extracting secret key from an eavesdropped source $p_{XYZ}$ at a rate given by the conditional mutual information. We investigate this question under three different scenarios: (i) Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness with the eavesdropper Eve ($Z$), (ii) Alice and Bob are allowed one-way public communication, and (iii) Alice and Bob are allowed two-way public communication. Distributions having a key rate of the conditional mutual information are precisely those in which a "helping" Eve offers Alice and Bob no greater advantage for obtaining secret key than a fully adversarial one. For each of the above scenarios, strong necessary conditions are derived on the structure of distributions attaining a secret key rate of $I(X:Y|Z)$. In obtaining our results, we completely solve the problem of secret key distillation under scenario (i) and identify $H(S|Z)$ to be the optimal key rate using shared randomness, where $S$ is the Gács-Körner Common Information. We thus provide an operational interpretation of the conditional Gács-Körner Common Information. Additionally, we introduce simple example distributions in which the rate $I(X:Y|Z)$ is achievable if and only if two-way communication is allowed.
Our aim is to formulate continuous-variable quantum secret sharing as a continuous-variable ramp quantum secret sharing protocol, provide a certification procedure for it and explain the criteria for the certification. Here we introduce a technique for certifying continuous-variable ramp quantum secret-sharing schemes in the framework of quantum interactive-proof systems. We devise pseudocodes in order to represent the sequence of steps taken to solve the certification problem. Furthermore, we derive the expression for quantum mutual information between the quantum secret extracted by any mult…
A theorem of Davies states that for symmetric quantum states there exists a symmetric POVM maximizing the mutual information. To apply this theorem the representation of the symmetry group has to be irreducible. We obtain a similar yet weaker result for reducible representations. We apply our results to the double trines ensemble and show numerically that for this ensemble the pretty good measurement is optimal.
Secure communication protocols are often formulated in a paradigm where the message is encoded in measurement outcomes. In this work we propose a rather unexplored framework in which the message is encoded in measurement settings rather than in their outcomes. In particular, we study two different variants of such secure communication protocols in which the message alphabet corresponds to measurement settings of mutually unbiased bases.
We develop new methods of quantifying the impact of photon detector imperfections on achievable secret key rates in Time-Entanglement based Quantum Key Distribution (QKD). We address photon detection timing jitter, detector downtime, and photon dark counts and show how each may decrease the maximum achievable secret key rate in different ways. We begin with a standard Discrete Memoryless Channel (DMC) model to get a good bound on the mutual information lost due to the timing jitter, then introduce a novel Markov Chain (MC) based model to characterize the effect of detector downtime and show ho…
In order to avoid the risk of information leakage during the information mutual transmission between two authorized participants, i.e., Alice and Bob, a quantum dialogue protocol based on the entanglement swapping between any two Bell states and the shared secret Bell state is proposed. The proposed protocol integrates the ideas of block transmission, two-step transmission and unitary operation encoding together using the Bell states as the information carriers. Besides the entanglement swapping between any two Bell states, a shared secret Bell state is also used to overcome the information le…
Semi-quantum key distribution is traditionally based on two-level quantum systems. In this paper, an unconditional security of a semi quantum key distribution protocol based on higher-dimensional systems using various mutually unbiased bases is presented. We first consider the three dimensional case using three and four mutually unbiased bases and derive a lower bound for the key rate as a function of the quantum channel's noise. We then generalize the result to a semi-quantum key distribution protocol that employs different number of mutually unbiased bases for four-dimensional states. It is …
We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation sugge…
We study and solve the problem of distilling secret key from quantum states representing correlation between two parties (Alice and Bob) and an eavesdropper (Eve) via one-way public discussion: we prove a coding theorem to achieve the "wire-tapper" bound, the difference of the mutual information Alice-Bob and that of Alice-Eve, for so-called cqq-correlations, via one-way public communication. This result yields information--theoretic formulas for the distillable secret key, giving ``ultimate'' key rate bounds if Eve is assumed to possess a purification of Alice and Bob's joint state. Specialis…
We present an experimental study of higher-dimensional quantum key distribution protocols based on mutually unbiased bases, implemented by means of photons carrying orbital angular momentum. We perform (d+1) mutually unbiased measurements in a classical prepare and measure scheme and on a pair of entangled photons for dimensions ranging from d = 2 to 5. In our analysis, we pay attention to the detection efficiency and photon pair creation probability. As security measures, we determine from experimental data the average error rate, the mutual information shared between the sender and receiver …
We show how two distrustful parties, "Bob" and "Charlie", can share a secret key with the help of a mutually trusted "Alice", counterfactually - that is with no information-carrying particles travelling between any of the three parties.
Using an experimental setup that simulates a turbulent atmosphere, we study the secret key rate for quantum key distribution protocols in orbital angular momentum based free space quantum communication. The quantum key distribution protocols under consideration include the Ekert 91 protocol for different choices of mutually unbiased bases and the six-state protocol. We find that the secret key rate of these protocols decay to zero roughly at the same scale where the entanglement of formation decays to zero.
We introduce the concept of mutual independence -- correlations shared between distant parties which are independent of the environment. This notion is more general than the standard idea of a secret key -- it is a fully quantum and more general form of privacy. The states which possess mutual independence also generalize the so called private states -- those that possess private key. We then show that the problem of distributed compression of quantum information at distant sources can be solved in terms of mutual independence, if free entanglement between the senders and the receiver is avail…
A high-dimensional quantum key distribution (QKD) can improve error rate tolerance and the secret key rate. Many $d$-dimensional QKDs have used two mutually unbiased bases (MUBs), while $(d+1)$ MUBs enable a more robust QKD, especially against correlated errors. However, a scalable implementation has not been achieved because the setups have required $d$ devices even for two MUBs or a flexible convertor for a specific optical mode. Here, we propose a scalable and general implementation of $(d+1)$ MUBs using $\log_p d$ interferometers in prime power dimensions $d=p^N$. We implemented the setup …
This paper addresses multi-user quantum key distribution networks, in which any two users can mutually exchange a secret key without trusting any other nodes. The same network also supports conventional classical communications by assigning two different wavelength bands to quantum and classical signals. Time and code division multiple access (CDMA) techniques, within a passive star network, are considered. In the case of CDMA, it turns out that the optimal performance is achieved at a unity code weight. A listen-before-send protocol is then proposed to improve secret key generation rates in t…
Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel. Suppose further that they share many copies of a quantum state $ρ_{ABE}$, such that Alice possesses the $A$ systems and Bob the $BE$ systems. In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the $E$ systems. Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of hi…
In order to be practically useful, quantum cryptography must not only provide a guarantee of secrecy, but it must provide this guarantee with a useful, sufficiently large throughput value. The standard result of generalized privacy amplification yields an upper bound only on the average value of the mutual information available to an eavesdropper. Unfortunately this result by itself is inadequate for cryptographic applications. A naive application of the standard result leads one to incorrectly conclude that an acceptable upper bound on the mutual information has been achieved. It is the point…
The core of security proofs of quantum key distribution (QKD) is the estimation of a parameter that determines the amount of privacy amplification that the users need to apply in order to distill a secret key. To estimate this parameter using the observed data, one needs to apply concentration inequalities, such as random sampling theory or Azuma's inequality. The latter can be straightforwardly employed in a wider class of QKD protocols, including those that do not rely on mutually unbiased encoding bases, such as the loss-tolerant (LT) protocol. However, when applied to real-life finite-leng…
In this paper we present finite-key security analysis for quantum key distribution protocol based on weak coherent (in particular phase-coded) states using a fully quantum asymptotic equipartition property technique. This work is the extension of the proof for non-orthogonal states on the coherent states. Below we consider two types of attacks each of them maximizes either Alice-Eve or Eve-Bob mutual information. The cornerstone of this paper is that we do assume the possibility of crucial intercept-resend attack based on errorless unambiguous state discrimination measurement. We demonstrate t…
Intrinsic entanglement (IE) is a quantity which aims at quantifying bipartite entanglement carried by a quantum state as an optimal amount of the intrinsic information that can be extracted from the state by measurement. We investigate in detail the properties of a Gaussian version of IE, the so-called Gaussian intrinsic entanglement (GIE). We show explicitly how GIE simplifies to the mutual information of a distribution of outcomes of measurements on a conditional state obtained by a measurement on a purifying subsystem of the analyzed state, which is first minimized over all measurements on …
We present a detailed security analysis of a d-dimensional quantum key distribution protocol based on two and three mutually unbiased bases (MUBs) both in an asymptotic and finite key length scenario. The finite secret key rates are calculated as a function of the length of the sifted key by (i) generalizing the uncertainly relation-based insight from BB84 to any d-level 2-MUB QKD protocol and (ii) by adopting recent advances in the second-order asymptotics for finite block length quantum coding (for both d-level 2- and 3-MUB QKD protocols). Since the finite and asymptotic secret key rates inc…
Quantum key distribution (QKD) allows two remote users to establish a secret key in the presence of an eavesdropper. The users share quantum states prepared in two mutually-unbiased bases: one to generate the key while the other monitors the presence of the eavesdropper. Here, we show that a general $d$-dimension QKD system can be secured by transmitting only a subset of the monitoring states. In particular, we find that there is no loss in the secure key rate when dropping one of the monitoring states. Furthermore, it is possible to use only a single monitoring state if the quantum bit error …
A significant number of servers that constitute the Internet are to provide private data via private communication channels to mutually anonymous registered users. Such are the servers of banks, hospitals that provide cloud storage and many others. Replacing communication channels by maximally entangled states is a promising idea for the quantum-secured Internet (QI). While it is an important idea for large distances secure communication, for the case of the mentioned class of servers pure entanglement based solution is not only unnecessary but also opens a threat. A crack stimulating a node t…
We investigate the error tolerance of quantum cryptographic protocols using $d$-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper boun…
Security against simple eavesdropping attacks is demonstrated for a recently proposed quantum key distribution protocol which uses the Fibonacci recursion relation to enable high-capacity key generation with entangled photon pairs. No transmitted pairs need to be discarded in reconciliation; the only pairs not used for key generation are those used for security-checking. Although the proposed approach does not allow eavesdropper-induced errors to be detected on single trials, it can nevertheless reveal the eavesdropper's action on the quantum channel by detecting changes in the distribution of…
We proposed a scheme of continuous-variable quantum key distribution, in which the bright Einstein-Podolsky-Rosen entangled optical beams are utilized. The source of the entangled beams is placed inside the receiving station, where half of the entangled beams are transmitted with round trip and the other half are retained by the receiver. The amplitude and phase signals modulated on the signal beam by the sender are simultaneously extracted by the authorized receiver with the scheme of the dense-coding correlation measurement for continuous quantum variables, thus the channel capacity is signi…
We investigate the security bounds of quantum cryptographic protocols using $d$-level systems. In particular, we focus on schemes that use two mutually unbiased bases, thus extending the BB84 quantum key distribution scheme to higher dimensions. Under the assumption of general coherent attacks, we derive an analytic expression for the ultimate upper security bound of such quantum cryptography schemes. This bound is well below the predictions of optimal cloning machines. The possibility of extraction of a secret key beyond entanglement distillation is discussed. In the case of qutrits we argue …
We propose the use of intra-particle entanglement to enhance the security of a practical implementation of the Bennett-Brassard-1984 (BB84) quantum key distribution scheme. Intra-particle entanglement is an attractive resource since it can be easily generated using only linear optics. Security is studied under a simple model of incoherent attack for protocols involving two or all five mutually unbiased bases. In terms of efficiency of secret key generation and tolerable error rate, the latter is found to be superior to the former. We find that states that allow secrecy distillation are necessa…
We study quantum soft covering and privacy amplification against quantum side information. The former task aims to approximate a quantum state by sampling from a prior distribution and querying a quantum channel. The latter task aims to extract uniform and independent randomness against quantum adversaries. For both tasks, we use trace distance to measure the closeness between the processed state and the ideal target state. We show that the minimal amount of samples for achieving an $\varepsilon$-covering is given by the $(1-\varepsilon)$-hypothesis testing information (with additional logarit…
We analyse a central broadcast continuous variable quantum key distribution protocol in which a beam produced by a thermal source is used to create a secret key between two parties, Alice and Bob. A beam splitter divides the initial beam into a pair of output beams, which are sent to Alice and Bob, with Eve intercepting Bob's beam. We investigate the protocol in detail, calculating mutual informations through a pair of analytic methods and comparing the results to the outputs of a Monte Carlo simulation of the protocol. In a lossless system, we find that a lower bound on the key rate remains p…
The quantum states of two laser pulses---coherent states---are never mutually orthogonal, making perfect discrimination impossible. Even so, coherent states can achieve the ultimate quantum limit for capacity of a classical channel, the Holevo capacity. Attaining this requires the receiver to make joint-detection measurements on long codeword blocks, optical implementations of which remain unknown. We report the first experimental demonstration of a joint-detection receiver, demodulating quaternary pulse-position-modulation (PPM) codewords at a word error rate of up to 40% (2.2 dB) below that …
It is known that the maximum classical mutual information that can be achieved between measurements on a pair of quantum systems can drastically underestimate the quantum mutual information between those systems. In this article, we quantify this distinction between classical and quantum information by demonstrating that after removing a logarithmic-sized quantum system from one half of a pair of perfectly correlated bitstrings, even the most sensitive pair of measurements might only yield outcomes essentially independent of each other. This effect is a form of information locking but the defi…
We outline a straightforward approach for obtaining a secret key rate using only no-signaling constraints and linear programming. Assuming an individual attack, we consider all possible joint probabilities. Initially, we study only the case where Eve has binary outcomes, and we impose constraints due to the no-signaling principle and given measurement outcomes. Within the remaining space of joint probabilities, by using linear programming, we get bound on the probability of Eve correctly guessing Bob's bit. We then make use of an inequality that relates this guessing probability to the mutual …
Reference-frame-independent quantum key distribution (RFI QKD) protocol can reduce the requirement on the alignment of reference frames in practical systems. However, comparing with the Bennett-Brassard (BB84) QKD protocol, the main drawback of RFI QKD is that Alice needs to prepare six encoding states in the three mutually unbiased bases (X, Y, and Z), and Bob also needs to measures the quantum state with such three bases. Here, we show that the RFI QKD protocol can be secured in the case where Alice sends fewer states. In particular, we find that transmitting three states (two eigenstates of…
