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The effect of quantum steering arises from the judicious combination of an entangled state with a set of incompatible measurements. Recently, it was shown that this form of quantum correlations can be quantified in terms of a dimension, leading to the notion of genuine high-dimensional steering. While this naturally connects to the dimensionality of entanglement (Schmidt number), we show that this effect also directly connects to a notion of dimension for measurement incompatibility. More generally, we present a general connection between the concepts of steering and measurement incompatibility, when quantified in terms of dimension. From this connection, we propose a novel twist on the problem of simulating quantum correlations. Specifically, we show how the correlations of certain high-dimensional entangled states can be exactly recovered using only shared randomness and lower-dimensional entanglement. Finally, we derive criteria for testing the dimension of measurement incompatibility, and discuss the extension of these ideas to quantum channels.

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High-dimensional entangled states are of significant interest in quantum science as they increase the information content per photon and can remain entangled in the presence of significant noise. The authors develop the analytical theory and show experimentally that the noise tolerance of high-dimensional entanglement can be significantly increased by a modest increase in the size of the Hilbert space. For example, doubling the size of a Hilbert space with a local dimension of d = 300 leads to a reduction in the threshold detector efficiencies required for entanglement certification by two orders of magnitude. This work is developed in the context of spatial entanglement in the few-photon limit, but it can easily be translated to photonic states entangled in different degrees of freedom. The authors also demonstrate that knowledge of a single parameter, the signal-to-noise ratio, precisely links measures of entanglement to a range of experimental parameters quantifying noise in a quantum communication system, enabling accurate predictions of its performance. This work serves to answer a simple question: “Is high-dimensional photonic entanglement robust to noise?” Here, the authors show that the answer is more nuanced than a simple “yes” or “no” and involves a complex interplay between the noise characteristics of the state, channel, and detection system.

Noise can be considered the natural enemy of quantum information. An often implied benefit of high-dimensional entanglement is its increased resilience to noise. However, manifesting this potential in an experimentally meaningful fashion is challenging and has never been done before. In infinite dimensional spaces, discretization is inevitable and renders the effective dimension of quantum states a tunable parameter. Owing to advances in experimental techniques and theoretical tools, we demonstrate an increased resistance to noise by identifying two pathways to exploit high-dimensional entangled states. Our study is based on two separate experiments utilizing canonical spatiotemporal properties of entangled photon pairs. Following these different pathways to noise resilience, we are able to certify entanglement in the photonic orbital-angular-momentum and energy-time degrees of freedom up to noise conditions corresponding to a noise fraction of 72% and 92%, respectively. Our work paves the way toward practical quantum communication systems that are able to surpass current noise and distance limitations, while not compromising on potential device independence.

Resource theories can be used to formalize the quantification and manipulation of resources in quantum information processing such as entanglement, asymmetry and coherence of quantum states, and incompatibility of quantum measurements. Given a certain state or measurement, one can ask whether there is a task in which it performs better than any resourceless state or measurement. Using conic programming, we prove that any general robustness measure (with respect to a convex set of free states or measurements) can be seen as a quantifier of such outperformance in some discrimination task. We apply the technique to various examples, e.g., joint measurability, positive operator valued measures simulable by projective measurements, and state assemblages preparable with a given Schmidt number.

In the context of a physical theory, two devices, A and B, described by the
theory are called incompatible if the theory does not allow the existence of a
third device C that would have both A and B as its components. Incompatibility
is a fascinating aspect of physical theories, especially in the case of quantum
theory. The concept of incompatibility gives a common ground for several famous
impossibility statements within quantum theory, such as ``no-cloning'' and ``no
information without disturbance''; these can be all seen as statements about
incompatibility of certain devices. The purpose of this paper is to give a
concise overview of some of the central aspects of incompatibility.

A typical bipartite quantum protocol, such as EPR-steering, relies on two
quantum features, entanglement of states and incompatibility of measurements.
Noise can delete both of these quantum features. In this work we study the
behavior of incompatibility under noisy quantum channels. The starting point
for our investigation is the observation that compatible measurements cannot
become incompatible by the action of any channel. We focus our attention to
channels which completely destroy the incompatibility of various relevant sets
of measurements. We call such channels incompatibility breaking, in analogy to
the concept of entanglement breaking channels. This notion is relevant
especially for the understanding of noise-robustness of the local measurement
resources for steering.

Einstein-Podolsky-Rosen steering is a form of inseparability in quantum
theory commonly acknowledged to be intermediate between entanglement and Bell
nonlocality. However, this statement has so far only been proven for a
restricted class of measurements, namely projective measurements. Here we prove
that entanglement, one-way steering, two-way steering and nonlocality are
genuinely different considering general measurements, i.e. single round
positive-operator-valued-measures. Finally, we show that the use of sequences
of measurements is relevant for steering tests, as they can be used to reveal
"hidden steering".

The existence of quantum correlations that allow one party to steer the
quantum state of another party is a counterintuitive quantum effect that has
been described already at the beginning of the past century. It has been shown
that steering occurs if entanglement can be proven, but with the extra
difficulty that the description of the measurements on one party is not known,
while the other side is fully characterized. We introduce the concept of
steering maps that allow to unlock the sophisticated techniques developed in
regular entanglement detection to be used for certifying steerability. As an
application we show that this allows to go even beyond the canonical steering
scenario, enabling a generalized dimension-bounded steering where one only
assumes the Hilbert space dimension on the characterized side, but no
description of the measurements. Surprisingly this does not weaken the
detection strength of very symmetric scenarios that have recently been carried
out in experiments.

The fact that not all measurements can be carried out simultaneously is a
peculiar feature of quantum mechanics and responsible for many key phenomena in
the theory, such as complementarity or uncertainty relations. For the special
case of projective measurements quantum behavior can be characterized by the
commutator but for generalized measurements it is not easy to decide whether
two measurements can still be understood in classical terms or whether they
show already quantum features. We prove that generalized measurements which do
not fulfill the notion of joint measurability are nonclassical, as they can be
used for the task of quantum steering. This shows that the notion of joint
measurability is, among several definitions, the proper one to characterize
quantum behavior. Moreover, the equivalence allows to derive novel steering
inequalities from known results on joint measurability and new criteria for
joint measurability from known results on the steerability of states.

Using well known duality between quantum maps and states of composite systems we introduce the notion of Schmidt number for
a quantum channel. It enables one to define classes of quantum channels which partially break quantum entanglement.These classes
generalize the well known class of entanglement breaking channels.

We study the nonlocal properties of states resulting from the mixture of an arbitrary entangled state rho of two d-dimensional systems and completely depolarized noise, with respective weights p and 1-p. We first construct a local model for the case in which rho is maximally entangled and p at or below a certain bound. We then extend the model to arbitrary rho. Our results provide bounds on the resistance to noise of the nonlocal correlations of entangled states. For projective measurements, the critical value of the noise parameter p for which the state becomes local is at least asymptotically log(d) larger than the critical value for separability.

We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. We show that $k$-positive maps witness Schmidt number, in the same way that positive maps witness entanglement. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled state that is mixed in. We show that Schmidt number {\it does not necessarily increase} when taking tensor copies of a density matrix $\rho$; we give an example of a density matrix for which the Schmidt numbers of $\rho$ and $\rho \otimes \rho$ are both 2. Comment: 5 pages RevTex, 1 typo in Proof Lemma 1 corrected

We investigate the compression of quantum information with respect to a given set M of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from M and an arbitrary quantum state ρ are recovered exactly by first compressing ρ into a lower-dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set M is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative example involving mutually unbiased bases, and develop a method based on semidefinite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.

High-dimensional quantum entanglement can give rise to stronger forms of nonlocal correlations compared to qubit systems, offering significant advantages for quantum information processing. Certifying these stronger correlations, however, remains an important challenge, in particular in an experimental setting. Here we theoretically formalize and experimentally demonstrate a notion of genuine high-dimensional quantum steering. We show that high-dimensional entanglement, as quantified by the Schmidt number, can lead to a stronger form of steering, provably impossible to obtain via entanglement in lower dimensions. Exploiting the connection between steering and incompatibility of quantum measurements, we derive simple two-setting steering inequalities, the violation of which guarantees the presence of genuine high-dimensional steering, and hence certifies a lower bound on the Schmidt number in a one-sided device-independent setting. We report the experimental violation of these inequalities using macropixel photon-pair entanglement certifying genuine high-dimensional steering. In particular, using an entangled state in dimension d=31, our data certifies a minimum Schmidt number of n=15.

Quantum measurements based on mutually unbiased bases are commonly used in quantum information processing, as they are generally viewed as being maximally incompatible and complementary. Here we quantify precisely the degree of incompatibility of mutually unbiased bases (MUB) using the notion of noise robustness. Specifically, for sets of k MUB in dimension d, we provide upper and lower bounds on this quantity. Notably, we get a tight bound in several cases, in particular for complete sets of k=d+1 MUB (using the standard construction for d being a prime power). On the way, we also derive a general upper bound on the noise robustness for an arbitrary set of quantum measurements. Moreover, we prove the existence of sets of k MUB that are operationally inequivalent, as they feature different noise robustness, and we provide a lower bound on the number of such inequivalent sets up to dimension 32. Finally, we discuss applications of our results for Einstein-Podolsky-Rosen steering.

The term Einstein-Podolsky-Rosen steering refers to a quantum correlation intermediate between entanglement and Bell nonlocality, which has been connected to another fundamental quantum property: measurement incompatibility. In the finite-dimensional case, efficient computational methods to quantify steerability have been developed. In the infinite-dimensional case, however, less theoretical tools are available. Here, we approach the problem of steerability in the continuous variable case via a notion of state-channel correspondence, which generalizes the well-known Choi-Jamio\l{}kowski correspondence. Via our approach we are able to generalize the connection between steering and incompatibility to the continuous variable case and to connect the steerability of a state with the incompatibility breaking property of a quantum channel, e.g., noisy NOON states and amplitude damping channels. Moreover, we apply our methods to the Gaussian steering setting, proving, among other things, that canonical quadratures are sufficient for steering Gaussian states.

Quantum steering refers to a situation where two parties share a bipartite system onto which one of the parties applies measurements that changes the state of the other party in a way that can not be explained by classical means. Quantum steering has been interpreted as the task of entanglement detection in a one-sided device independent way and has received a lot of attention since then. Here we address the characterisation of quantum steering through semidefinite programming, including steering detection, quantification and applications. We also give a brief overview of some results on quantum steering not directly related to semidefinite programming. Finally, we make available a list of semidefinite programming codes that can be used to study the topics discussed in this article.

We develop a technique to find simultaneous measurements for noisy quantum observables in finite-dimensional Hilbert spaces. We use the method to derive lower bounds for the noise needed to make incompatible measurements jointly measurable. Using our strategy together with recent devel- opments in the field of one-sided quantum information processing we show that the attained lower bounds are tight for various symmetric sets of quantum measurements. We use this characterisation to prove the existence of so called 4-Specker sets in the qubit case.

Quantum steering refers to the possibility for Alice to remotely steer Bob's
state by performing local measurements on her half of a bipartite system. Two
necessary ingredients for steering are entanglement and incompatibility of
Alice's measurements. In particular, it has been recently proven that for the
case of pure states of maximal Schmidt rank the problem of steerability for
Bob's assemblage is equivalent to the problem of joint measurability for Alice
observables. We show that such an equivalence holds in general, namely, the
steerability of any assemblage can always be formulated as a joint
measurability problem, and vice versa. We use this connection to introduce
steering inequalities from joint measurability criteria and develop quantifiers
for the incompatibility of measurements.

Einstein-Podolsky-Rosen steering is a manifestation of quantum correlations
exhibited by quantum systems, that allows for entanglement certification when
one of the subsystems is not characterized. Detecting steerability of quantum
states is essential to assess their suitability for quantum information
protocols with partially trusted devices. We provide a hierarchy of sufficient
conditions for the steerability of bipartite quantum states of any dimension,
including continuous variable states. Previously known steering criteria are
recovered as special cases of our approach. The proposed method allows us to
derive optimal steering witnesses for arbitrary families of quantum states, and
provides a systematic framework to analytically derive non-linear steering
criteria. We discuss relevant examples and, in particular, provide an optimal
steering witness for a lossy single-photon Bell state; the witness can be
implemented just by linear optics and homodyne detection, and detects steering
with a higher loss tolerance than any other known method. Our approach is
readily applicable to multipartite steering detection and to the
characterization of joint measurability.

A robustness measure for incompatibility of quantum devices in the lines of
the robustness of entanglement is proposed. The concept of general robustness
measures is first introduced in general convex-geometric settings and these
ideas are then applied to measure how incompatible a given pair of quantum
devices is. The robustness of quantum incompatibility is calculated in three
special cases: a pair of Fourier-coupled rank-1 sharp observables, a pair of
decodable channels, where decodability means left-invertibility by a channel,
and a pair consisting of a rank-1 sharp observable and a decodable channel.

We investigate the relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e., incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.

Einstein-Podolsky-Rosen steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It allows for entanglement certification when the measurements performed by one of the parties are not characterized (or are untrusted) and has applications in quantum key distribution. Despite its foundational and applied importance, Einstein-Podolsky-Rosen steering lacks a quantitative assessment. Here we propose a way of quantifying this phenomenon and use it to study the steerability of several quantum states. In particular, we show that every pure entangled state is maximally steerable and the projector onto the antisymmetric subspace is maximally steerable for all dimensions; we provide a new example of one-way steering and give strong support that states with positive-partial transposition are not steerable.

Einstein-Podolsky-Rosen steering is a form of quantum nonlocality exhibiting an inherent asymmetry between the observers, Alice and Bob. A natural question is then whether there exist entangled states which are one-way steerable, that is, Alice can steer Bob's state, but it is impossible for Bob to steer the state of Alice. So far, such a phenomenon has been demonstrated for continuous variable systems, but with a strong restriction on allowed measurements, namely, considering only Gaussian measurements. Here we present a simple class of entangled two-qubit states which are one-way steerable, considering arbitrary projective measurements. This shows that the nonlocal properties of entangled states can be fundamentally asymmetrical.

We show that there are informationally complete joint measurements of two
conjugated observables on a finite quantum system, meaning that they enable to
identify all quantum states from their measurement outcome statistics. We
further demonstrate that it is possible to implement a joint observable as a
sequential measurement. If we require minimal noise in the joint measurement,
then the joint observable is unique. If the dimension d is odd, then this
observable is informationally complete. But if d is even, then the joint
observable is not informationally complete and one has to allow more noise in
order to obtain informational completeness.

A state of a composite quantum system is called classically correlated if it can be approximated by convex combinations of product states, and Einstein-Podolsky-Rosen correlated otherwise. Any classically correlated state can be modeled by a hidden-variable theory and hence satisfies all generalized Bell’s inequalities. It is shown by an explicit example that the converse of this statement is false.

The concept of steering was introduced by Schrödinger in 1935 as a generalization of the Einstein-Podolsky-Rosen paradox for arbitrary pure bipartite entangled states and arbitrary measurements by one party. Until now, it has never been rigorously defined, so it has not been known (for example) what mixed states are steerable (that is, can be used to exhibit steering). We provide an operational definition, from which we prove (by considering Werner states and isotropic states) that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. For arbitrary bipartite Gaussian states we derive a linear matrix inequality that decides the question of steerability via Gaussian measurements, and we relate this to the original Einstein-Podolsky-Rosen paradox.

This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given.
Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.

We present a local-hidden-variable model for positive-operator-valued measurements (an LHVPOV model) on a class of entangled generalized Werner states, thus demonstrating that such measurements do not always violate a Bell-type inequality. We also show that, in general, if the state $\rho'$ can be obtained from $\rho$ with certainty by local quantum operations without classical communication then an LHVPOV model for the state $\rho$ implies the existence of such a model for $\rho'$. Comment: 4 pages, no figures. Title changed to accord with Phys. Rev. A version. Journal reference added

Juha-Pekka Pellonpää, and Roope Uola. Incompatible measurements in quantum information science

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Testing the hilbert space dimension

- Nicolas Brunner
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Nicolas Brunner, Stefano Pironio, Antonio Acin, Nicolas Gisin, André Allan Méthot, and Valerio Scarani.
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