Matthew McKague

Robust self-testing of the singlet

In this paper, we introduce a general framework to study the concept of robust self-testing which can be used to self-test maximally entangled pairs of qubits (EPR pairs) and local measurement operators. The result is based only on probabilities obtained from the experiment, with tolerance to experimental errors. In particular, we show that if the results of an experiment approach the Cirelson bound, or approximate the Mayers-Yao-type correlations, then the experiment must contain an approximate EPR pair. More specifically, there exist local bases in which the physical state is close to an EPR pair, possibly encoded in a larger environment or ancilla. Moreover, in these bases the measurements are close to the qubit operators used to achieve the Cirelson bound or the Mayers-Yao results.

Device-independent state estimation based on Bell's inequalities

The only information available about an alleged source of entangled quantum states is the amount S by which the Clauser-Horne-Shimony-Holt inequality is violated: nothing is known about the nature of the system or the measurements that are performed. We discuss how the quality of the source can be assessed in this black-box scenario, as compared to an ideal source that would produce maximally entangled states (more precisely, any state for which S=2 root 2). To this end, we present several inequivalent notions of fidelity, each one related to the use one can make of the source after having assessed it, and we derive quantitative bounds for each of them in terms of the violation S. We also derive a lower bound on the entanglement of the source as a function of S only.

Interactive Proofs for

Using the measurement-based quantum computation model, we construct interactive proofs with non-communicating quantum provers and a classical verifier. Our construction gives interactive proofs for all languages in BQP with a polynomial number of quantum provers, each of which, in the honest case, performs only a single measurement. Our techniques use self-tested graph states. In this regard we introduce two important improvements over previous work. Specifically, we derive new error bounds which scale polynomially with the size of the graph compared with exponential dependence on the size of the graph in previous work. We also extend the self-testing error bounds on measurements to a very general set which includes the adaptive measurements used for measurement-based quantum computation as a special case.

\mathsf{BQP}

We develop a means of simulating the evolution and measurement of a multipartite quantum state under discrete or continuous evolution using another quantum system with states and operators lying in a real Hilbert space. This extends previous results which were unable to simulate local evolution and measurements with local operators and was limited to discrete evolution. We also detail applications to Bell inequalities and self-testing of the quantum apparatus.

via Self-Tested Graph States

Self-tested quantum information processing provides a means for doing useful information processing with untrusted quantum apparatus. Previous work was limited to performing computations and protocols in real Hilbert spaces, which is not a serious obstacle if one is only interested in final measurement statistics being correct (for example, getting the correct factors of a large number after running Shors factoring algorithm). This limitation was shown by McKague et al. to be fundamental, since there is no way to experimentally distinguish any quantum experiment from a special simulation using states and operators with only real coefficients. In this paper, we show that one can still do a meaningful self-test of quantum apparatus with complex amplitudes. In particular, we define a family of simulations of quantum experiments, based on complex conjugation, with two interesting properties. First, we are able to define a self-test which may be passed only by states and operators that are equivalent to simulations within the family. This extends work of Mayers and Yao and Magniez et al. in self-testing of quantum apparatus, and includes a complex measurement. Second, any of the simulations in the family may be used to implement a secure 6-state QKD protocol, which was previously not known to be implementable in a self-tested framework.

Simulating quantum systems using real Hilbert spaces

Self-testing allows us to determine, through classical interaction only, whether some players in a non-local game share particular quantum states. Most work on self-testing has concentrated on developing tests for small states like one pair of maximally entangled qubits, or on tests where there is a separate player for each qubit, as in a graph state. Here we consider the case of testing many maximally entangled pairs of qubits shared between two players. Previously such a test was shown where testing is sequential, i.e., one pair is tested at a time. Here we consider the parallel case where all pairs are tested simultaneously, giving considerably more power to dishonest players. We derive sufficient conditions for a self-test for many maximally entangled pairs of qubits shared between two players and also two constructions for self-tests where all pairs are tested simultaneously.

Generalized self-testing and the security of the 6-state protocol

We give a construction for a self-test for any connected graph state. In other words, for each connected graph state we give a set of non-local correlations that can only be achieved (quantumly) by that particular graph state and certain local measurements. The number of correlations considered is small, being linear in the number of vertices in the graph. We also prove robustness for the test.

Self-testing in parallel

Device-independent self-testing offers the possibility of certifying the quantum state and measurements, up to local isometries, using only the statistics observed by querying uncharacterized local devices. In this paper we study parallel self-testing of two maximally entangled pairs of qubits; in particular, the local tensor product structure is not assumed but derived. We prove two criteria that achieve the desired result: a double use of the Clauser-Horne-Shimony-Holt inequality and the 3×3 magic square game. This demonstrate that the magic square game can only be perfectly won by measuring a two-singlet state. The tolerance to noise is well within reach of state-of-the-art experiments.

Self-Testing Graph States

We consider the power of various quantum complexity classes with the restriction that states and operators are defined over a real, rather than complex, Hilbert space. It is well known that a quantum circuit over the complex numbers can be transformed into a quantum circuit over the real numbers with the addition of a single qubit. This implies that BQP retains its power when restricted to using states and operations over the reals. We show that the same is true for QMA(k), QIP(k), QMIP and QSZK.

Device-independent parallel self-testing of two singlets

To ensure uninterrupted cryptographic security, it is important to begin planning the transition to post-quantum cryptography. In addition to creating post-quantum primitives, we must also plan how to adapt the cryptographic infrastructure for the transition, especially in scenarios such as public key infrastructures (PKIs) with many participants. The use of hybrids—multiple algorithms in parallel—will likely play a role during the transition for two reasons: “hedging our bets” when the security of newer primitives is not yet certain but the security of older primitives is already in question; and to achieve security and functionality both in post-quantum-aware and in a backwards-compatible way with not-yet-upgraded software. In this paper, we investigate the use of hybrid digital signature schemes. We consider several methods for combining signature schemes, and give conditions on when the resulting hybrid signature scheme is unforgeable. Additionally we address a new notion about the inability of an adversary to separate a hybrid signature into its components. For both unforgeability and non-separability, we give a novel security hierarchy based on how quantum the attack is. We then turn to three real-world standards involving digital signatures and PKI: certificates (X.509), secure channels (TLS), and email (S/MIME). We identify possible approaches to supporting hybrid signatures in these standards while retaining backwards compatibility, which we test in popular cryptographic libraries and implementations, noting especially the inability of some software to handle larger certificates.