arXiv subjects

Teiko Heinosaari

Publications and source records attributed to Teiko Heinosaari.

At least 73 records · Page 4Linked to original sources

arXiv2015

Notes on Deterministic Programming of Quantum Observables and Channels

We study the limitations of deterministic programmability of quantum circuits, e.g., quantum computer. More precisely, we analyse the programming of quantum observables and channels via quantum multimeters. We show that the programming vectors for any two different sharp observables are necessarily orthogonal, whenever post-processing is not allowed. This result then directly implies that also any two different unitary channels require orthogonal programming vectors. This approach generalizes the well-known orthogonality result first proven by Nielsen and Chuang. In addition, we give size-bounds for a multimeter to be efficient in quantum programming.

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arXiv2015

Breaking Gaussian incompatibility on continuous variable quantum systems

We characterise Gaussian quantum channels that are Gaussian incompatibility breaking, that is, transform every set of Gaussian measurements into a set obtainable from a joint Gaussian observable via Gaussian postprocessing. Such channels represent local noise which renders measurements useless for Gaussian EPR-steering, providing the appropriate generalisation of entanglement breaking channels for this scenario. Understanding the structure of Gaussian incompatibility breaking channels contributes to the resource theory of noisy continuous variable quantum information protocols.

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arXiv2015

Incompatibility breaking quantum channels

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.

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arXiv2015

Expanding the principle of local distinguishability

The principle of local distinguishability states that an arbitrary physical state of a bipartite system can be determined by the combined statistics of local measurements performed on the subsystems. A necessary and sufficient requirement for the local measurements is that each one must be able to distinguish between all pairs of states of the respective subsystems. We show that if the task is changed into the determination of an arbitrary bipartite pure state, then at least in certain cases it is possible to restrict to local measurements which can distintinguish all pure states but not all states. This surprising fact gives evidence that the principle of local distinguishability may be expanded beyond its usual applicability.

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arXiv2015

How many orthonormal bases are needed to distinguish all pure quantum states?

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d=2 the answer is three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4 the answer is either three or four. Curiously, the exact number in d=4 seems to be an open problem.

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arXiv2014

Nonuniqueness of phase retrieval for three fractional Fourier transforms

We prove that, regardless of the choice of the angles $θ_1,θ_2,θ_3$, three fractional Fourier transforms $F_{θ_1}$, $F_{θ_2}$ and $F_{θ_3}$ do not solve the phase retrieval problem. That is, there do not exist three angles $θ_1$, $θ_2$, $θ_3$ such that any signal $ψ\in L^2(R)$ could be determined up to a constant phase by knowing only the three intensities $|F_{θ_1}ψ|^2$, $|F_{θ_2}ψ|^2$ and $|F_{θ_3}ψ|^2$. This provides a negative argument against a recent speculation by P. Jaming, who stated that three suitably chosen fractional Fourier transforms are good candidates for phase retrieval in infinite dimension. We recast the question in the language of quantum mechanics, where our result shows that any fixed triple of rotated quadrature observables $Q_{θ_1}$, $Q_{θ_2}$ and $Q_{θ_3}$ is not enough to determine all unknown pure quantum states. The sufficiency of four rotated quadrature observables, or equivalently fractional Fourier transforms, remains an open question.

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arXiv2014

Universality of Sequential Quantum Measurements

Unavoidable disturbance caused by a quantum measurement implies that the realizable subsequent measurements are getting limited after one performs some measurement. The obvious general limitation that one cannot circumvent by sequential or any other method is that the actually implemented measurements must be jointly measurable. In this work we show that any jointly measurable pair of observables can be obtained in a sequential measurement scheme, even if the second observable would be decided after the first measurement. This universality feature holds only for measurement schemes with a specific structure. As a supplementing result, we provide a characterization of all possible joint measurements obtained from a sequential measurement lacking universality.

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arXiv2014

Tasks and premises in quantum state determination

The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (POVMs) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.

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arXiv2014

Qualitative Noise-Disturbance Relation for Quantum Measurements

The inherent connection between noise and disturbance is one of the most fundamental features of quantum measurements. In the two well-known extreme cases a measurement either makes no disturbance but then has to be totally noisy or is as accurate as possible but then has to disturb so much that all subsequent measurements become redundant.Most of the measurements are, however, something between these two extremes.We derive a structural relation between observables and channels that properly explains the trade-off between noise and disturbance.

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arXiv2013

Maximally incompatible quantum observables

The existence of maximally incompatible quantum observables in the sense of a minimal joint measurability region is investigated. Employing the universal quantum cloning device it is argued that only infinite dimensional quantum systems can accommodate maximal incompatibility. It is then shown that two of the most common pairs of complementary observables (position and momentum; number and phase) are maximally incompatible.

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arXiv2013

Strongly Incompatible Quantum Devices

The fact that there are quantum observables without a simultaneous measurement is one of the fundamental characteristics of quantum mechanics. In this work we expand the concept of joint measurability to all kinds of possible measurement devices, and we call this relation compatibility. Two devices are incompatible if they cannot be implemented as parts of a single measurement setup. We introduce also a more stringent notion of incompatibility, strong incompatibility. Both incompatibility and strong incompatibility are rigorously characterized and their difference is demonstrated by examples.

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arXiv2013

An operational link between MUBs and SICs

We exhibit an operational connection between mutually unbiased bases and symmetric infomationally complete positive operator-valued measures. Assuming that the latter exists, we show that there is a strong link between these two structures in all prime power dimensions. We also demonstrate that a similar link cannot exists in dimension 6.

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arXiv2013

Coexistence of effects from an algebra of two projections

The coexistence relation of quantum effects is a fundamental structure, describing those pairs of experimental events that can be implemented in a single setup. Only in the simplest case of qubit effects an analytic characterization of coexistent pairs is known. We generalize the qubit coexistence characterization to all pairs of effects in arbitrary dimension that belong to the von Neumann algebra generated by two projections. We demonstrate the presented mathematical machinery by several examples, and show that it covers physically relevant classes of effect pairs.

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arXiv2013

Comparing the degrees of incompatibility inherent in probabilistic physical theories

We introduce a new way of quantifying the degrees of incompatibility of two ob- servables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This opens up a novel and flexible way of comparing probabilistic theories with respect to the nonclassical feature of incompatibility, raising many interesting questions, some of which will be answered here. We show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have if arbitrary pairs of observables are considered. If one adopts a more refined measure of the degree of incompatibility, for instance, by restricting the comparison to binary observables, it turns out that there are probabilistic theories whose inherent degree of incompatibility is greater than that of quantum mechanics.

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arXiv2013

A simple sufficient condition for the coexistence of quantum effects

Two quantum effects are considered coexistent if they can be measured together. It is known that commutativity and comparability are sufficient but not necessary for the coexistence of two effects. We unify those two conditions to a simple but more widely applicable sufficient condition.

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arXiv2012

Minimal covariant observables identifying all pure states

It has been recently shown that an observable that identifies all pure states of a d-dimensional quantum system has minimally 4d-4 outcomes or slightly less (the exact number depending on the dimension d). However, no simple construction of this type of observable with minimal number of outcomes is known. In this work we investigate the possibility to have a covariant observable that identifies all pure states and has minimal number of outcomes for this purpose. It is shown that the existence of these kind of observables depends on the dimension of the Hilbert space. The fact that these kind of observables fail to exist in some dimensions indicates that the dual pair of observables -- pure states lacks the symmetry that the dual pair of observables -- states has.

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arXiv2012

Extending quantum operations

For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian in- or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.

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arXiv2012

When do pieces determine the whole? Extremal marginals of a completely positive map

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extremal point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extremal, then the whole map is extremal. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.

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