30 research outputs found
Self-testing of binary observables based on commutation
We consider the problem of certifying binary observables based on a Bell
inequality violation alone, a task known as self-testing of measurements. We
introduce a family of commutation-based measures, which encode all the distinct
arrangements of two projective observables on a qubit. These quantities by
construction take into account the usual limitations of self-testing and since
they are "weighted" by the (reduced) state, they automatically deal with
rank-deficient reduced density matrices. We show that these measures can be
estimated from the observed Bell violation in several scenarios and the proofs
rely only on standard linear algebra. The trade-offs turn out to be tight and,
in particular, they give non-trivial statements for arbitrarily small
violations. On the other extreme, observing the maximal violation allows us to
deduce precisely the form of the observables, which immediately leads to a
complete rigidity statement. In particular, we show that for all the
-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the
-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit
measurements on every party. Our results imply that any pair of projective
observables on a qubit can be certified in a truly robust manner. Finally, we
show that commutation-based measures give a convenient way of expressing
relations among more than two observables.Comment: 5 + 4 pages. v2: published version; v3: formatting errors fixe
Relativistic quantum cryptography
In this thesis we explore the benefits of relativistic constraints for
cryptography. We first revisit non-communicating models and its applications in
the context of interactive proofs and cryptography. We propose bit commitment
protocols whose security hinges on communication constraints and investigate
its limitations. We explain how some non-communicating models can be justified
by special relativity and study the limitations of such models. In particular,
we present a framework for analysing security of multiround relativistic
protocols. The second part of the thesis is dedicated to analysing specific
protocols. We start by considering a recently proposed two-round quantum bit
commitment protocol. We propose a fault-tolerant variant of the protocol,
present a complete security analysis and report on an experimental
implementation performed in collaboration with an experimental group at the
University of Geneva. We also propose a new, multiround classical bit
commitment protocol and prove its security against classical adversaries. This
demonstrates that in the classical world an arbitrarily long commitment can be
achieved even if the agents are restricted to occupy a finite region of space.
Moreover, the protocol is easy to implement and we report on an experiment
performed in collaboration with the Geneva group.Comment: 123 pages, 9 figures, many protocols, a couple of theorems, certainly
not enough commas. PhD thesis supervised by Stephanie Wehner at Centre for
Quantum Technologies, Singapor
A weak form of self-testing
The concept of self-testing (or rigidity) refers to the fact that for certain
Bell inequalities the maximal violation can be achieved in an essentially
unique manner. In this work we present a family of Bell inequalities which are
maximally violated by multiple inequivalent quantum realisations. We completely
characterise the quantum realisations achieving the maximal violation and we
show that each of them requires a maximally entangled state of two qubits. This
implies the existence of a new, weak form of self-testing in which the maximal
violation allows us to identify the state, but does not fully determine the
measurements. From the geometric point of view the set of probability points
that saturate the quantum bound is a line segment. We then focus on a
particular member of the family and show that the self-testing statement is
robust, i.e. that observing a non-maximal violation allows us to make a
quantitative statement about the unknown state. To achieve this we present a
new construction of extraction channels and analyse their performance. For
completeness we provide two independent approaches: analytical and numerical.
The noise robustness, i.e. the amount of white noise at which the bound becomes
trivial, of the analytical bound is rather small (~0.06%), but the numerical
method takes us into an experimentally-relevant regime (~5%). We conclude by
investigating the amount of randomness that can be certified using these Bell
violations. Perhaps surprisingly, we find that the qualitative behaviour
resembles the behaviour of rigid inequalities like the
Clauser-Horne-Shimony-Holt inequality. This shows that at least for some
device-independent applications rigidity is not a necessary ingredient.Comment: 5 + 10 pages, 3 figures, comments welcome; accepted manuscrip
Entropic uncertainty from effective anti-commutators
We investigate entropic uncertainty relations for two or more binary
measurements, for example spin- or polarisation measurements. We
argue that the effective anti-commutators of these measurements, i.e. the
anti-commutators evaluated on the state prior to measuring, are an expedient
measure of measurement incompatibility. Based on the knowledge of pairwise
effective anti-commutators we derive a class of entropic uncertainty relations
in terms of conditional R\'{e}nyi entropies. Our uncertainty relations are
formulated in terms of effective measures of incompatibility, which can be
certified device-independently. Consequently, we discuss potential applications
of our findings to device-independent quantum cryptography. Moreover, to
investigate the tightness of our analysis we consider the simplest (and very
well-studied) scenario of two measurements on a qubit. We find that our results
outperform the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60,
1103 (1988)] and provide a new analytical expression for the minimum
uncertainty which also outperforms some recent bounds based on majorisation.Comment: 5 pages, 3 figures (excluding Supplemental Material), revte
Secure bit commitment from relativistic constraints
We investigate two-party cryptographic protocols that are secure under
assumptions motivated by physics, namely relativistic assumptions
(no-signalling) and quantum mechanics. In particular, we discuss the security
of bit commitment in so-called split models, i.e. models in which at least some
of the parties are not allowed to communicate during certain phases of the
protocol. We find the minimal splits that are necessary to evade the
Mayers-Lo-Chau no-go argument and present protocols that achieve security in
these split models. Furthermore, we introduce the notion of local versus global
command, a subtle issue that arises when the split committer is required to
delegate non-communicating agents to open the commitment. We argue that
classical protocols are insecure under global command in the split model we
consider. On the other hand, we provide a rigorous security proof in the global
command model for Kent's quantum protocol [Kent 2011, Unconditionally Secure
Bit Commitment by Transmitting Measurement Outcomes]. The proof employs two
fundamental principles of modern physics, the no-signalling property of
relativity and the uncertainty principle of quantum mechanics.Comment: published version, IEEE format, 18 pages, 8 figure
Quantum preparation uncertainty and lack of information
The quantum uncertainty principle famously predicts that there exist
measurements that are inherently incompatible, in the sense that their outcomes
cannot be predicted simultaneously. In contrast, no such uncertainty exists in
the classical domain, where all uncertainty results from ignorance about the
exact state of the physical system. Here, we critically examine the concept of
preparation uncertainty and ask whether similarly in the quantum regime, some
of the uncertainty that we observe can actually also be understood as a lack of
information (LOI), albeit a lack of quantum information. We answer this
question affirmatively by showing that for the well known measurements employed
in BB84 quantum key distribution, the amount of uncertainty can indeed be
related to the amount of available information about additional registers
determining the choice of the measurement. We proceed to show that also for
other measurements the amount of uncertainty is in part connected to a LOI.
Finally, we discuss the conceptual implications of our observation to the
security of cryptographic protocols that make use of BB84 states.Comment: 7+15 pages, 4 figures. v2: expanded "Discussion" section, "Methods"
section moved before "Results" section, published versio
Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding
Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert
spaces, and the corresponding rank-1 projective measurements are ubiquitous in
quantum information theory. In this work, we study a recently introduced
generalization of MUBs called mutually unbiased measurements (MUMs). These
measurements inherit the essential property of complementarity from MUBs, but
the Hilbert space dimension is no longer required to match the number of
outcomes. This operational complementarity property renders MUMs highly useful
for device-independent quantum information processing. It has been shown that
MUMs are strictly more general than MUBs. In this work we provide a complete
proof of the characterization of MUMs that are direct sums of MUBs. We then
proceed to construct new examples of MUMs that are not direct sums of MUBs. A
crucial technical tool for these construction is a correspondence with
quaternionic Hadamard matrices, which allows us to map known examples of such
matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that --
in stark contrast with MUBs -- the number of MUMs for a fixed outcome number is
unbounded. Next, we focus on the use of MUMs in quantum communication. We
demonstrate how any pair of MUMs with d outcomes defines a d-dimensional
superdense coding protocol. Using MUMs that are not direct sums of MUBs, we
disprove a recent conjecture due to Nayak and Yuen on the rigidity of
superdense coding for infinitely many dimensions. The superdense coding
protocols arising in the refutation reveal how shared entanglement may be used
in a manner heretofore unknown.Comment: v2: Added some references and related discussion. v1: 20 pages.
Comments welcome
Robust self-testing of two-qubit states
It is well-known that observing nonlocal correlations allows us to draw
conclusions about the quantum systems under consideration. In some cases this
yields a characterisation which is essentially complete, a phenomenon known as
self-testing. Self-testing becomes particularly interesting if we can make the
statement robust, so that it can be applied to a real experimental setup. For
the simplest self-testing scenarios the most robust bounds come from the method
based on operator inequalities. In this work we elaborate on this idea and
apply it to the family of tilted CHSH inequalities. These inequalities are
maximally violated by partially entangled two-qubit states and our goal is to
estimate the quality of the state based only on the observed violation. For
these inequalities we have reached a candidate bound and while we have not been
able to prove it analytically, we have gathered convincing numerical evidence
that it holds. Our final contribution is a proof that in the usual formulation,
the CHSH inequality only becomes a self-test when the violation exceeds a
certain threshold. This shows that self-testing scenarios fall into two
distinct classes depending on whether they exhibit such a threshold or not.Comment: 8 pages, 1 figure, accepted manuscrip
Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems
Bell inequalities are an important tool in device-independent quantum
information processing because their violation can serve as a certificate of
relevant quantum properties. Probably the best known example of a Bell
inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined
in the simplest scenario involving two dichotomic measurements and whose all
key properties are well understood. There have been many attempts to generalise
the CHSH Bell inequality to higher-dimensional quantum systems, however, for
most of them the maximal quantum violation---the key quantity for most
device-independent applications---remains unknown. On the other hand, the
constructions for which the maximal quantum violation can be computed, do not
preserve the natural property of the CHSH inequality, namely, that the maximal
quantum violation is achieved by the maximally entangled state and measurements
corresponding to mutually unbiased bases. In this work we propose a novel
family of Bell inequalities which exhibit precisely these properties, and whose
maximal quantum violation can be computed analytically. In the simplest
scenario it recovers the CHSH Bell inequality. These inequalities involve
measurements settings, each having outcomes for an arbitrary prime number
. We then show that in the three-outcome case our Bell inequality can
be used to self-test the maximally entangled state of two-qutrits and three
mutually unbiased bases at each site. Yet, we demonstrate that in the case of
more outcomes, their maximal violation does not allow for self-testing in the
standard sense, which motivates the definition of a new weak form of
self-testing. The ability to certify high-dimensional MUBs makes these
inequalities attractive from the device-independent cryptography point of view.Comment: 19 pages, no figures, accepted in Quantu