159 research outputs found
Compatibility of quantum measurements and inclusion constants for the matrix jewel
In this work, we establish the connection between the study of free
spectrahedra and the compatibility of quantum measurements with an arbitrary
number of outcomes. This generalizes previous results by the authors for
measurements with two outcomes. Free spectrahedra arise from matricial
relaxations of linear matrix inequalities. A particular free spectrahedron
which we define in this work is the matrix jewel. We find that the
compatibility of arbitrary measurements corresponds to the inclusion of the
matrix jewel into a free spectrahedron defined by the effect operators of the
measurements under study. We subsequently use this connection to bound the set
of (asymmetric) inclusion constants for the matrix jewel using results from
quantum information theory and symmetrization. The latter translate to new
lower bounds on the compatibility of quantum measurements. Among the techniques
we employ are approximate quantum cloning and mutually unbiased bases.Comment: v5: section 3.3 has been expanded significantly to incorporate the
generalization of the Cartesian product and the direct sum to matrix convex
sets. Many other minor modifications. Closed to the published versio
Maximal violation of steering inequalities and the matrix cube
In this work, we characterize the amount of steerability present in quantum
theory by connecting the maximal violation of a steering inequality to an
inclusion problem of free spectrahedra. In particular, we show that the maximal
violation of an arbitrary unbiased dichotomic steering inequality is given by
the inclusion constants of the matrix cube, which is a well-studied object in
convex optimization theory. This allows us to find new upper bounds on the
maximal violation of steering inequalities and to show that previously obtained
violations are optimal. In order to do this, we prove lower bounds on the
inclusion constants of the complex matrix cube, which might be of independent
interest. Finally, we show that the inclusion constants of the matrix cube and
the matrix diamond are the same. This allows us to derive new bounds on the
amount of incompatibility available in dichotomic quantum measurements in fixed
dimension.Comment: Slightly generalized Lemma 2.3 and Theorem 3.
Position-based cryptography: Single-qubit protocol secure against multi-qubit attacks
While it is known that unconditionally secure position-based cryptography is
impossible both in the classical and the quantum setting, it has been shown
that some quantum protocols for position verification are secure against
attackers which share a quantum state of bounded dimension. In this work, we
consider the security of two protocols for quantum position verification that
combine a single qubit with classical strings of total length : The qubit
routing protocol, where the classical information prescribes the qubit's
destination, and a variant of the BB84-protocol for position verification,
where the classical information prescribes in which basis the qubit should be
measured. We show that either protocol is secure for a randomly chosen function
if each of the attackers holds at most qubits. With this, we show for
the first time that there exists a quantum position verification protocol where
the ratio between the quantum resources an honest prover needs and the quantum
resources the attackers need to break the protocol is unbounded. The verifiers
need only increase the amount of classical resources to force the attackers to
use more quantum resources. Concrete efficient functions for both protocols are
also given -- at the expense of a weaker but still unbounded ratio of quantum
resources for successful attackers. Finally, we show that both protocols are
robust with respect to noise, making them appealing for applications.Comment: 26 pages, 4 figures. Content significantly expanded. In particular,
we have added the function BB84 protocol and prove its security in Section 4.
Finally, we give lower bounds for concrete functions in Section
On the simulation of quantum multimeters
In the quest for robust and universal quantum devices, the notion of
simulation plays a crucial role, both from a theoretical and from an applied
perspective. In this work, we go beyond the simulation of quantum channels and
quantum measurements, studying what it means to simulate a collection of
measurements, which we call a multimeter. To this end, we first explicitly
characterize the completely positive transformations between multimeters.
However, not all of these transformations correspond to valid simulations, as
evidenced by the existence of maps that always prepare the same multimeter
regardless of the input, which we call trash-and-prepare. We give a new
definition of multimeter simulations as transformations that are
triviality-preserving, i.e., when given a multimeter consisting of trivial
measurements they can only produce another trivial multimeter. In the absence
of a quantum ancilla, we then characterize the transformations that are
triviality-preserving and the transformations that are trash-and-prepare.
Finally, we use these characterizations to compare our new definition of
multimeter simulation to three existing ones: classical simulations,
compression of multimeters, and compatibility-preserving simulations
Continuity of quantum entropic quantities via almost convexity
Based on the proofs of the continuity of the conditional entropy by Alicki,
Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF)
method. This method allows us to prove a great variety of continuity bounds for
the derived entropic quantities. First, we apply the ALAFF method to the
Umegaki relative entropy. This way, we recover known almost tight bounds, but
also some new continuity bounds for the relative entropy. Subsequently, we
apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This
yields novel explicit bounds in particular for the BS-conditional entropy, the
BS-mutual and BS-conditional mutual information. On the way, we prove almost
concavity for the Umegaki relative entropy and the BS-entropy, which might be
of independent interest. We conclude by showing some applications of these
continuity bounds in various contexts within quantum information theory.Comment: 68 pages, 6 figure
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