arXiv subjects

Daniel Gottesman

Publications and source records attributed to Daniel Gottesman.

At least 55 records · Page 3Linked to original sources

arXiv2005

Quantum accuracy threshold for concatenated distance-3 codes

We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold epsilon_0. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, epsilon_0 > 2.73 \times 10^{-5} for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.

quant-ph↗

arXiv2005

The Minimum Distance Problem for Two-Way Entanglement Purification

Entanglement purification takes a number of noisy EPR pairs and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can achieve the rate k/n = 1 - (t/n) \log_2 3 - h(t/n) - O(1/n) (asymptotically reaching the quantum Hamming bound), where the EPP produces at least k good pairs out of n total pairs with up to t arbitrary errors, and h(x) = -x \log_2 x - (1-x) \log_2 (1-x) is the usual binary entropy. In contrast, the best known lower bound on the rate of QECCs is the quantum Gilbert-Varshamov bound k/n \geq 1 - (2t/n) \log_2 3 - h(2t/n). Indeed, in some regimes, the known upper bound on the asymptotic rate of good QECCs is strictly below our lower bound on the achievable rate of 2-EPPs.

quant-ph↗

arXiv2005

Quantum Error Correction and Fault-Tolerance

I give an overview of the basic concepts behind quantum error correction and quantum fault tolerance. This includes the quantum error correction conditions, stabilizer codes, CSS codes, transversal gates, fault-tolerant error correction, and the threshold theorem.

quant-ph↗

arXiv2005

Classicality in discrete Wigner functions

Gibbons et al. [Phys. Rev. A 70, 062101(2004)] have recently defined a class of discrete Wigner functions W to represent quantum states in a Hilbert space with finite dimension. We show that the only pure states having non-negative W for all such functions are stabilizer states, as conjectured by one of us [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of W for all definitions of W form a subgroup of the Clifford group. This means pure states with non-negative W and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.

quant-ph↗

arXiv2005

GHZ extraction yield for multipartite stabilizer states

Let $|Ψ>$ be an arbitrary stabilizer state distributed between three remote parties, such that each party holds several qubits. Let $S$ be a stabilizer group of $|Ψ>$. We show that $|Ψ>$ can be converted by local unitaries into a collection of singlets, GHZ states, and local one-qubit states. The numbers of singlets and GHZs are determined by dimensions of certain subgroups of $S$. For an arbitrary number of parties $m$ we find a formula for the maximal number of $m$-partite GHZ states that can be extracted from $|Ψ>$ by local unitaries. A connection with earlier introduced measures of multipartite correlations is made. An example of an undecomposable four-party stabilizer state with more than one qubit per party is given. These results are derived from a general theoretical framework that allows one to study interconversion of multipartite stabilizer states by local Clifford group operators. As a simple application, we study three-party entanglement in two-dimensional lattice models that can be exactly solved by the stabilizer formalism.

quant-ph↗

arXiv2005

Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes

It is a standard result in the theory of quantum error-correcting codes that no code of length n can fix more than n/4 arbitrary errors, regardless of the dimension of the coding and encoded Hilbert spaces. However, this bound only applies to codes which recover the message exactly. Naively, one might expect that correcting errors to very high fidelity would only allow small violations of this bound. This intuition is incorrect: in this paper we describe quantum error-correcting codes capable of correcting up to (n-1)/2 arbitrary errors with fidelity exponentially close to 1, at the price of increasing the size of the registers (i.e., the coding alphabet). This demonstrates a sharp distinction between exact and approximate quantum error correction. The codes have the property that any $t$ components reveal no information about the message, and so they can also be viewed as error-tolerant secret sharing schemes. The construction has several interesting implications for cryptography and quantum information theory. First, it suggests that secret sharing is a better classical analogue to quantum error correction than is classical error correction. Second, it highlights an error in a purported proof that verifiable quantum secret sharing (VQSS) is impossible when the number of cheaters t is n/4. More generally, the construction illustrates a difference between exact and approximate requirements in quantum cryptography and (yet again) the delicacy of security proofs and impossibility results in the quantum model.

quant-ph↗

arXiv2004

Uncloneable Encryption

Quantum states cannot be cloned. I show how to extend this property to classical messages encoded using quantum states, a task I call "uncloneable encryption." An uncloneable encryption scheme has the property that an eavesdropper Eve not only cannot read the encrypted message, but she cannot copy it down for later decoding. She could steal it, but then the receiver Bob would not receive the message, and would thus be alerted that something was amiss. I prove that any authentication scheme for quantum states acts as a secure uncloneable encryption scheme. Uncloneable encryption is also closely related to quantum key distribution (QKD), demonstrating a close connection between cryptographic tasks for quantum states and for classical messages. Thus, studying uncloneable encryption and quantum authentication allows for some modest improvements in QKD protocols. While the main results apply to a one-time key with unconditional security, I also show uncloneable encryption remains secure with a pseudorandom key. In this case, to defeat the scheme, Eve must break the computational assumption behind the pseudorandom sequence before Bob receives the message, or her opportunity is lost. This means uncloneable encryption can be used in a non-interactive setting, where QKD is not available, allowing Alice and Bob to convert a temporary computational assumption into a permanently secure message.

quant-ph↗

arXiv2004

Security of quantum key distribution with imperfect devices

We prove the security of the Bennett-Brassard (BB84) quantum key distribution protocol in the case where the source and detector are under the limited control of an adversary. Our proof applies when both the source and the detector have small basis-dependent flaws, as is typical in practical implementations of the protocol. We derive a general lower bound on the asymptotic key generation rate for weakly basis-dependent eavesdropping attacks, and also estimate the rate in some special cases: sources that emit weak coherent states with random phases, detectors with basis-dependent efficiency, and misaligned sources and detectors.

quant-ph↗

arXiv2003

Comment on "The black hole final state"

Horowitz and Maldacena have suggested that the unitarity of the black hole S-matrix can be reconciled with Hawking's semiclassical arguments if a final-state boundary condition is imposed at the spacelike singularity inside the black hole. We point out that, in this scenario, departures from unitarity can arise due to interactions between the collapsing body and the infalling Hawking radiation inside the event horizon. The amount of information lost when a black hole evaporates depends on the extent to which these interactions are entangling.

hep-th↗

arXiv2002

Proof of security of quantum key distribution with two-way classical communications

Shor and Preskill have provided a simple proof of security of the standard quantum key distribution scheme by Bennett and Brassard (BB84) by demonstrating a connection between key distribution and entanglement purification protocols with one-way communications. Here we provide proofs of security of standard quantum key distribution schemes, BB84 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 QKD. This is done by the explicit construction of a new protocol for (the error correction/detection and privacy amplification of) BB84 that can tolerate a bit error rate of up to 18.9%, which is higher than what any BB84 scheme with only one-way classical communications can possibly tolerate. Moreover, we demonstrate the advantage of the six-state scheme over BB84 by showing that the six-state scheme can strictly tolerate a higher bit error rate than BB84. 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 BB84 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.

quant-ph↗

arXiv2002

Secure Multi-party Quantum Computing

Secure multi-party computing, also called "secure function evaluation", has been extensively studied in classical cryptography. We consider the extension of this task to computation with quantum inputs and circuits. Our protocols are information-theoretically secure, i.e. no assumptions are made on the computational power of the adversary. For the weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to show how to perform any multi-party quantum computation as long as the number of dishonest players is less than n/6.

quant-ph↗

arXiv2002

Authentication of Quantum Messages

Authentication is a well-studied area of classical cryptography: a sender S and a receiver R sharing a classical private key want to exchange a classical message with the guarantee that the message has not been modified by any third party with control of the communication line. In this paper we define and investigate the authentication of messages composed of quantum states. Assuming S and R have access to an insecure quantum channel and share a private, classical random key, we provide a non-interactive scheme that enables S both to encrypt and to authenticate (with unconditional security) an m qubit message by encoding it into m+s qubits, where the failure probability decreases exponentially in the security parameter s. The classical private key is 2m+O(s) bits. To achieve this, we give a highly efficient protocol for testing the purity of shared EPR pairs. We also show that any scheme to authenticate quantum messages must also encrypt them. (In contrast, one can authenticate a classical message while leaving it publicly readable.) This has two important consequences: On one hand, it allows us to give a lower bound of 2m key bits for authenticating m qubits, which makes our protocol asymptotically optimal. On the other hand, we use it to show that digitally signing quantum states is impossible, even with only computational security.

quant-ph↗

arXiv2001

From Quantum Cheating to Quantum Security

For thousands of years, code-makers and code-breakers have been competing for supremacy. Their arsenals may soon include a powerful new weapon: quantum mechanics. We give an overview of quantum cryptology as of November 2000.

quant-ph↗

arXiv2001

Quantum Digital Signatures

We present a quantum digital signature scheme whose security is based on fundamental principles of quantum physics. It allows a sender (Alice) to sign a message in such a way that the signature can be validated by a number of different people, and all will agree either that the message came from Alice or that it has been tampered with. To accomplish this task, each recipient of the message must have a copy of Alice's "public key," which is a set of quantum states whose exact identity is known only to Alice. Quantum public keys are more difficult to deal with than classical public keys: for instance, only a limited number of copies can be in circulation, or the scheme becomes insecure. However, in exchange for this price, we achieve unconditionally secure digital signatures. Sending an m-bit message uses up O(m) quantum bits for each recipient of the public key. We briefly discuss how to securely distribute quantum public keys, and show the signature scheme is absolutely secure using one method of key distribution. The protocol provides a model for importing the ideas of classical public key cryptography into the quantum world.

quant-ph↗

arXiv2001

Measurability of Wilson loop operators

We show that the nondemolition measurement of a spacelike Wilson loop operator W(C) is impossible in a relativistic non-Abelian gauge theory. In particular, if two spacelike-separated magnetic flux tubes both link with the loop C, then a nondemolition measurement of W(C) would cause electric charge to be transferred from one flux tube to the other, a violation of relativistic causality. A destructive measurement of W(C) is possible in a non-Abelian gauge theory with suitable matter content. In an Abelian gauge theory, many cooperating parties distributed along the loop C can perform a nondemolition measurement of the Wilson loop operator if they are equipped with a shared entangled ancilla that has been prepared in advance. We also note that Abelian electric charge (but not non-Abelian charge) can be transported superluminally, without any accompanying transmission of information.

hep-th↗

arXiv2001

Causal and localizable quantum operations

We examine constraints on quantum operations imposed by relativistic causality. A bipartite superoperator is said to be localizable if it can be implemented by two parties (Alice and Bob) who share entanglement but do not communicate; it is causal if the superoperator does not convey information from Alice to Bob or from Bob to Alice. We characterize the general structure of causal complete measurement superoperators, and exhibit examples that are causal but not localizable. We construct another class of causal bipartite superoperators that are not localizable by invoking bounds on the strength of correlations among the parts of a quantum system. A bipartite superoperator is said to be semilocalizable if it can be implemented with one-way quantum communication from Alice to Bob, and it is semicausal if it conveys no information from Bob to Alice. We show that all semicausal complete measurement superoperators are semilocalizable, and we establish a general criterion for semicausality. In the multipartite case, we observe that a measurement superoperator that projects onto the eigenspaces of a stabilizer code is localizable.

quant-ph↗

arXiv2001

Encoding a qubit in an oscillator

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.

quant-ph↗

arXiv2000

Secure quantum key distribution using squeezed states

We prove the security of a quantum key distribution scheme based on transmission of squeezed quantum states of a harmonic oscillator. Our proof employs quantum error-correcting codes that encode a finite-dimensional quantum system in the infinite-dimensional Hilbert space of an oscillator, and protect against errors that shift the canonical variables p and q. If the noise in the quantum channel is weak, squeezing signal states by 2.51 dB (a squeeze factor e^r=1.34) is sufficient in principle to ensure the security of a protocol that is suitably enhanced by classical error correction and privacy amplification. Secure key distribution can be achieved over distances comparable to the attenuation length of the quantum channel.

quant-ph↗