Though it was proven that secure quantum sealing of a single classical bit is impossible in principle, here we propose an unconditionally secure quantum sealing protocol which seals a classical bit string. Any reader can obtain each bit of the sealed string with an arbitrarily small error rate, while reading the string is detectable. The protocol is simple and easy to be implemented. The possibility of using this protocol to seal a single bit in practical is also discussed.
This paper proposes a cheat sensitive quantum bit commitment (CSQBC) scheme based on single photons, in which Alice commits a bit to Bob. Here, Bob only can cheat the committed bit with probability close to $0$ with the increasing of used single photons' amount. And if Alice altered her committed bit after commitment phase, she will be detected with probability close to $1$ with the increasing of used single photons' amount. The scheme is easy to be realized with nowadays technology.
The model of the quantum protocols sealing a classical bit is studied. It is shown that there exist upper bounds on its security. For any protocol where the bit can be read correctly with the probability $α$, and reading the bit can be detected with the probability $β$, the upper bounds are $% β\leqslant 1/2$ and $α+β\leqslant 9/8$.
Quantum bit seal is a way to encode a classical bit quantum mechanically so that everyone can obtain non-zero information on the value of the bit. Moreover, such an attempt should have a high chance of being detected by an authorized verifier. Surely, a reader looks for a way to get the maximum amount of information on the sealed bit and at the same time to minimize her chance of being caught. And a verifier picks a sealing scheme that maximizes his chance of detecting any measurement of the sealed bit. Here, I report a strategy that passes all measurement detection procedures at least half of…
Unconditionally secure bit commitment and coin flipping are known to be impossible in the classical world. Bit commitment is known to be impossible also in the quantum world. We introduce a related new primitive - {\em quantum bit escrow}. In this primitive Alice commits to a bit $b$ to Bob. The commitment is {\em binding} in the sense that if Alice is asked to reveal the bit, Alice can not bias her commitment without having a good probability of being detected cheating. The commitment is {\em sealing} in the sense that if Bob learns information about the encoded bit, then if later on he is as…
It is well known that unconditionally secure bit commitment is impossible even in the quantum world. In this paper a weak variant of quantum bit commitment, introduced independently by Aharonov et al. [STOC, 2000] and Hardy and Kent [Phys. Rev. Lett. 92 (2004)] is investigated. In this variant, the parties require some nonzero probability of detecting a cheating, i.e. if Bob, who commits a bit b to Alice, changes his mind during the revealing phase then Alice detects the cheating with a positive probability (we call this property binding); and if Alice gains information about the committed bit…
The relationship between the quantum bit commitment (QBC) and quantum seal (QS) is studied. It is elaborated that QBC and QS are not equivalent, but QS protocols satisfying a stronger unconditional security requirement can lead to an unconditionally secure QBC. In this sense, QS is strictly stronger than QBC in security requirements. Based on an earlier proposal on sealing a single bit, a feasible QBC protocol is also put forward, which is secure in practice even if the cheater has a strong quantum computational power.
Security trade-offs have been established for one-way bit commitment in quant-ph/0106019. We study this trade-off in two superselection settings. We show that for an `abelian' superselection rule (exemplified by particle conservation) the standard trade-off between sealing and binding properties still holds. For the non-abelian case (exemplified by angular momentum conservation) the security trade-off can be more subtle, which we illustrate by showing that if the bit-commitment is forced to be ancilla-free an asymptotically secure quantum bit commitment is possible.
In this work, we propose a quantum bit string commitment protocol using polarization of mesoscopic coherent states. The protocol is described and its security against brute force and quantum cloning machine attack is analyzed.
This work uses a simple quantum computer model to discuss the randomness of bit strings originated from integer sequences. The considered quantum computer model has three elements: a processing unit responsible for a mathematical operation, an initial equally weighted superposition and a quantum state used as resource. The randomness depends on the last.
In this short note we present a new approach to non-classical correlations that is based on the compression rates for bit strings generated by Alice and Bob. We use normalised compression distance introduced by Cilibrasi and Vitanyi to derive information-theoretic inequalities that must be obeyed by classically correlated bit strings and that are violated by PR-boxes. We speculate about a violation of our inequalities by quantum mechanical correlations.
Quantum bit-string commitment[A.Kent, Phys.Rev.Lett., 90, 237901 (2003)] or QBSC is a variant of bit commitment (BC). In this paper, we propose a new QBSC protocol that can be implemented using currently available technology, and prove its security under the same security criteria as discussed by Kent. QBSC is a generalization of BC, but has slightly weaker requirements, and our proposed protocol is not intended to break the no-go theorem of quantum BC.
Several kinds of qubit-string-based(QS-based) bit commitment protocols are presented, and a definition of information-theoretic concealing is given. All the protocols presented here are proved to be secure under this definition. We suggest an attack algorithm to obtain the local unitary transformation introduced in no-go theorem, which is used to attack the binding condition, then study the security of our QS-based bit commitment protocols under this attack via introducing a new concept "physical security of protocol". At last we present a practical QS-based bit commitment scheme against chann…
Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and the quantum worlds. But when committing to a string of n bits at once, how far can we stretch the quantum limits? In this paper, we introduce a framework for quantum schemes where Alice commits a string of n bits to Bob in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are two-fold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the securi…
The first achievement of quantum supremacy has been claimed recently by Google for the random quantum circuit benchmark with 53 superconducting qubits. Here, we analyze the randomness of Google's quantum random-bit sampling. The heat maps of Google's random bit-strings show stripe patterns at specific qubits in contrast to the Haar-measure or classical random-bit strings. Google's data contains more bit 0 than bit 1, i.e., about 2.8\% difference, and fail to pass the NIST random number tests, while the Haar-measure or classical random-bit samples pass. Their difference is also illustrated by t…
Coin tossing is a cryptographic task in which two parties who do not trust each other aim to generate a common random bit. Using classical communication this is impossible, but non trivial coin tossing is possible using quantum communication. Here we consider the case when the parties do not want to toss a single coin, but many. This is called bit string generation. We report the experimental generation of strings of coins which are provably more random than achievable using classical communication. The experiment is based on the ``plug and play'' scheme developed for quantum cryptography, and…
Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? In this letter, we introduce a framework of quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are two-fold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the s…
We consider the cryptographic task of bit-string generation. This is a generalisation of coin tossing in which two mistrustful parties wish to generate a string of random bits such that an honest party can be sure that the other cannot have biased the string too much. We consider a quantum protocol for this task, originally introduced in Phys. Rev. A {\bf 69}, 022322 (2004), that is feasible with present day technology. We introduce security conditions based on the average bias of the bits and the Shannon entropy of the string. For each, we prove rigorous security bounds for this protocol in b…
An $n$-bit string is encoded as a sequence of non-orthogonal quantum states. The parity bit of that $n$-bit string is described by one of two density matrices, $ρ_0^{(n)}$ and $ρ_1^{(n)}$, both in a Hilbert space of dimension $2^n$. In order to derive the parity bit the receiver must distinguish between the two density matrices, e.g., in terms of optimal mutual information. In this paper we find the measurement which provides the optimal mutual information about the parity bit and calculate that information. We prove that this information decreases exponentially with the length of the string i…
We discuss the security implications of noise for quantum coin tossing protocols. We find that if quantum error correction can be used, so that noise levels can be made arbitrarily small, then reasonable security conditions for coin tossing can be framed so that results from the noiseless case will continue to hold. If, however, error correction is not available (as is the case with present day technology), and significant noise is present, then tossing a single coin becomes problematic. In this case, we are led to consider random n-bit string generation in the presence of noise, rather than s…
A bit string commitment protocol securely commits $N$ classical bits in such a way that the recipient can extract only $M<N$ bits of information about the string. Classical reasoning might suggest that bit string commitment implies bit commitment and hence, given the Mayers-Lo-Chau theorem, that non-relativistic quantum bit string commitment is impossible. Not so: there exist non-relativistic quantum bit string commitment protocols, with security parameters $ε$ and $M$, that allow $A$ to commit $N = N(M, ε)$ bits to $B$ so that $A$'s probability of successfully cheating when revealing any bit …
We prove the unconditional security of a quantum key distribution (QKD) protocol on a noisy channel against the most general attack allowed by quantum physics. We use the fact that in a previous paper we have reduced the proof of the unconditionally security of this QKD protocol to a proof that a corresponding Quantum String Oblivious Transfer (String-QOT) protocol would be unconditionally secure against Bob if implemented on top of an unconditionally secure bit commitment scheme. We prove a lemma that extends a security proof given by Yao for a (one bit) QOT protocol to this String-QOT protoc…
Imagine that Alice and Bob, unable to communicate, are both given a 16-bit string such that the strings are either equal, or they differ in exactly 8 positions. Both parties are then supposed to output a 4-bit string in such a way that these short strings are equal if and only if the original longer strings given to them were equal as well. It is known that this task can be fulfilled without failure and without communication if Alice and Bob share 4 maximally entangled quantum bits. We show that, on the other hand, they CANNOT win the same game with certainty if they only share classical bits,…
Random access code (RAC), a primitive for many information processing protocols, enables one party to encode n-bit string into one bit of message such that another party can retrieve partial information of that string. We introduce the multiparty version of RAC in which the n-bit string is distributed among many parties. For this task, we consider two distinct quantum communication scenarios: one allows shared quantum entanglement among the parties with classical communication, and the other allows communication through a quantum channel. We present several multiparty quantum RAC protocols tha…
We present a scheme for multi-bit quantum random number generation using a single qubit discrete-time quantum walk in one-dimensional space. Irrespective of the initial state of the qubit, quantum interference and entanglement of particle with the position space in the walk dynamics certifies high randomness in the system. Quantum walk in a position space of dimension $2^l+1$ ensures string of $(l+ 2)$-bits of random numbers from a single measurement. Bit commitment with the position space and control over the spread of the probability distribution in position space enable us with options to e…
We propose an efficient algorithm for partitioning Pauli strings into subgroups, which can be simultaneously measured in a single quantum circuit. Our partitioning algorithm drastically reduces the total number of measurements in a variational quantum eigensolver for a quantum chemistry, one of the most promising applications of quantum computing. The algorithm is based on the Ising model optimization problem, which can be quickly solved using an Ising machine. We develop an algorithm that is applicable to problems with sizes larger than the maximum number of variables that an Ising machine ca…
In this paper, we consider the secret-string-learning problem in the teacher-student setting: the teacher has a secret string $s\in {\{0,1\}^{n}}$, and the student wants to learn the secret $s$ by question-answer interactions with the teacher, where at each time, the student can ask the teacher with a pair $(x, q) \in {\{0,1\}^{n}}\times\{0,1,\cdots, n-1\}$ and the teacher returns a bit given by the oracle $f_{s}(x,q)$ that indicates whether the length of the longest common prefix of $s$ and $x$ is greater than $q$ or not. Our contributions are as follows. (i) We prove that any classical deter…
We propose an entanglement-based quantum bit string commitment protocol whose composability is proven in the random oracle model. This protocol has the additional property of preserving the privacy of the committed message. Even though this property is not resilient against man-in-the-middle attacks, this threat can be circumvented by considering that the parties communicate through an authenticated channel. The protocol remains secure (but not private) if we realize the random oracles as physical unclonable functions in the so-called bad PUF model with access before the opening phase.
String commitment schemes are similar to the well studied bit commitment schemes in cryptography with the difference that the committing party, say Alice, is supposed to commit a long string instead of a single bit, to another party say Bob. Similar to bit commitment schemes, such schemes are supposed to be binding, i.e Alice cannot change her choice after committing, and concealing i.e. Bob cannot find Alice's committed string before Alice reveals it. Ideal commitment schemes are known to be impossible. Even if some degrees of cheating is allowed, Buhrman, Christandl, Hayden, Lo and Wehner (q…
A long sequence of tosses of a classical coin produces an apparently random bit string, but classical randomness is an illusion: the algorithmic information content of a classically-generated bit string lies almost entirely in the description of initial conditions. This letter presents a simple argument that, by contrast, a sequence of bits produced by tossing a quantum coin is, almost certainly, genuinely (algorithmically) random. This result can be interpreted as a strengthening of Bell's no-hidden-variables theorem, and relies on causality and quantum entanglement in a manner similar to Bel…
We address the question whether quantum memory is more powerful than classical memory. In particular, we consider a setting where information about a random n-bit string X is stored in r classical or quantum bits, for r<n, i.e., the stored information is bound to be only partial. Later, a randomly chosen binary question F about X is asked, which has to be answered using only the stored information. The maximal probability of correctly guessing the answer F(X) is then compared for the cases where the storage device is classical or quantum mechanical, respectively. We show that, despite the fact…
The scheduling problem consists of finding a common 1 in two remotely located N bit strings. Denote the number of 1s in the string with the fewer 1s by epsilon*N. Classically, it needs at least O(epsilon*N) bits of communication to find the common 1 (ignoring logarithmic factors). The best known quantum algorithm would require O(sqrt(N)) qubits of communication. This paper gives a modified quantum algorithm to find the common 1 with only O(sqrt(epsilon*N)) qubits of communication.
Theoretically speaking, a photon can travel arbitrarily long before it enters into a detector, resulting a click. How much information can a photon carry? We study a bipartite asymmetric "two-way signaling" protocol as an extension of that proposed by Del Santo and Dakić. Suppose that Alice and Bob are distant from each other and each of them has an $n$-bit string. They are tasked to exchange the information of their local n-bit strings with each other, using only a single photon during the communication. It has been shown that the superposition of different spatial locations in a Mach-Zehnder…
We give a mathematical framework for manipulating indeterminate-length quantum bit strings. In particular, we define prefixes, fragments, tensor products and concatenation of such strings of qubits, and study their properties and relationships. The results are then used to define prefix-free Hilbert spaces in a more general way than in previous work, without assuming the existence of a basis of length eigenstates. We prove a quantum analogue of the Kraft inequality, illustrate the results with some examples and discuss the relevance of prefix-free Hilbert spaces for lossless compression.
We present two complementary ways in which Saraceno's symmetric version of the quantum baker's map can be written as a shift map on a string of quantum bits. One of these representations leads naturally to a family of quantizations of the baker's map.
Holographic Bit Threads from String-Diagrammatic Quantum Information Flow
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
Oblivious transfer (OT) is an important tool in cryptography. It serves as a subroutine to other complex procedures of both theoretical and practical significance. Common attribute of OT protocols is that one party (Alice) has to send a message to another party (Bob) and has to stay oblivious on whether Bob did receive the message. Specific (OT) protocols vary by exact definition of the task - in the all-or-nothing protocol Alice sends a single bit-string message, which Bob is able to read only with 50% probability, whereas in 1-out-of-2 OT protocol Bob reads one out of two messages sent by Al…
We introduce a new type of cryptographic primitive that we call hiding fingerprinting. A (quantum) fingerprinting scheme translates a binary string of length $n$ to $d$ (qu)bits, typically $d\ll n$, such that given any string $y$ and a fingerprint of $x$, one can decide with high accuracy whether $x=y$. Classical fingerprinting schemes cannot hide information very well: a classical fingerprint of $x$ that guarantees error at most $ε$ necessarily reveals $Ω(\log(1/ epsilon))$ bits about $x$. We call a scheme hiding if it reveals $o(\log(1/ε))$ bits; accordingly, no classical scheme is hiding. F…
We present a bit-string quantum oblivious transfer protocol based on single-qubit rotations. Our protocol is built upon a previously proposed quantum public-key protocol and its practical security relies on the laws of Quantum Mechanics. Practical security is reflected in the fact that, due to technological limitations, the receiver (Bob) of the transferred bit-string is restricted to performing only "few-qubit" coherent measurements. We also present a single-bit oblivious transfer based on the proposed bit-string protocol. The protocol can be implemented with current technology based on optic…
We propose a bit-oriented quantum public-key scheme which uses Boolean function as private-key and randomly changed pairs of quantum state and classical string as public-keys. Contrast to the typical classical public-key scheme, one private-key in our scheme corresponds to an exponential number of public-keys. The goal of our scheme is to achieve information-theoretic security, and the security analysis is also given.
Min-entropy sampling gives a bound on the min-entropy of a randomly chosen subset of a string, given a bound on the min-entropy of the whole string. König and Renner showed a min-entropy sampling theorem that holds relative to quantum knowledge. Their result achieves the optimal rate, but it can only be applied if the bits are sampled in blocks, and only gives weak bounds for the non-smooth min-entropy. We give two new quantum min-entropy sampling theorems that do not have the above weaknesses. The first theorem shows that the result by König and Renner also applies to bitwise sampling, and th…
PARITY is the problem of determining the parity of a string $f$ of $n$ bits given access to an oracle that responds to a query $x\in\{0,1,...,n-1\}$ with the $x^{\rm th}$ bit of the string, $f(x)$. Classically, $n$ queries are required to succeed with probability greater than 1/2 (assuming equal prior probabilities for all length $n$ bitstrings), but only $\lceil n/2\rceil$ quantum queries suffice to determine the parity with probability 1. We consider a generalization to strings $f$ of $n$ elements of $\Z_k$ and the problem of determining $\sum f(x)$. By constructing an explicit algorithm, we…
Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on \textit{at most} a bits? Let 0\leq a\leq n-1. We show that any deterministic protocol, as well as any error-free quantum protocol (C* version), for this problem requires at least n-2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly a bits. We also prove a lower bound of n/2-1 for error-free Q* quantum protocols. Our results are obtained by lower-bounding the ranks of the appropriate matrices.
We report the results of an experiment that searches for causal non-linear state-dependent terms in quantum field theory. Our approach correlates a binary macroscopic classical voltage with the outcome of a projective measurement of a quantum bit, prepared in a coherent superposition state. Measurement results are recorded in a bit string, which is used to control a voltage switch. Presence of a non-zero voltage reading in cases of no applied voltage is the experimental signature of a non-linear state-dependent shift of the electromagnetic field operator. We implement blinded measurement and d…
We study the problem of computing the Hamming weight of an $n$-bit string modulo $m$, for any positive integer $m \leq n$ whose only prime factors are 2 and 3. We show that the exact quantum query complexity of this problem is $\left\lceil n(1 - 1/m) \right\rceil$. The upper bound is via an iterative query algorithm whose core components are the well-known 1-query quantum algorithm (essentially due to Deutsch) to compute the Hamming weight a 2-bit string mod 2 (i.e., the parity of the input bits), and a new 2-query quantum algorithm to compute the Hamming weight of a 3-bit string mod 3. We sho…
We prove that the binary classifiers of bit strings generated by random wide deep neural networks with ReLU activation function are biased towards simple functions. The simplicity is captured by the following two properties. For any given input bit string, the average Hamming distance of the closest input bit string with a different classification is at least sqrt(n / (2π log n)), where n is the length of the string. Moreover, if the bits of the initial string are flipped randomly, the average number of flips required to change the classification grows linearly with n. These results are confir…
Pure quantum states are often approximately encoded as classical bit strings such as those representing probability amplitudes and those describing circuits that generate the quantum states. The crucial quantity is the minimum length of classical bit strings from which the original pure states are approximately reconstructible. We derive asymptotically tight bounds on the minimum bit length required for probabilistic encodings with which one can approximately reconstruct the original pure state as an ensemble of the quantum states encoded in classical strings. We also show that such a probabil…
