# ATESIT Sintesi della relazione

Project ID:
IST-2000-29681

Finanziato nell'ambito di:
FP5-IST

Paese:
United Kingdom

## Generalized measurements

The key realisation which lead to the emergence of the new field of quantum information processing is that quantum mechanics allows the processing of information in fundamentally new ways. But just as in classical information processing, errors occur in quantum information processing, and these have to be corrected. A fundamental breakthrough was the realisation that quantum error correction is in fact possible.

However most work so far has not been concerned with technological feasibility, but rather with proving that quantum error correction is possible in principle. Partner 4 described a error correction method for filtering out errors and ntanglement purification which can be experimentally implemented. - Partner 4 has also investigated what are the fundamental limitations in aligning the frames of reference of two distant parties, and signalling a direction when the frames are already aligned. Along similar lines, the problem of communicating chirality has also been investigated. Partner 4 discussed in detail this issue and introduced the natural concept of quantum gloves, i.e. rotationally invariant quantum states that encode as much as possible the concept of chirality. - Nonlocality is the basic aspect of quantum mechanics underlying quantum information and communication. The limits of quantum non-locality were investigated by Partner 4 by considering non local correlation in their full generality, not only those correlations that may be produced by quantum mechanics. What kinds of non-locality are present in quantum states? By studying the extremal points of the space of all multi-party probability distributions, in which all parties can make one of a pair of measurements, each with two possible outcomes, a necessary condition for classical non-local models to reproduce the statistics of all quantum states was found.

The condition generalizes and extends the results of Sveltichny and of Collins, Gisin, Popescu, Roberts and Scarani who showed that separable classical non-local models cannot reproduce the statistics of all multi-particle quantum states. This condition shows that the non-locality present in some entangled multi-particle quantum states is much stronger than previously thought.

It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are non-local, in the sense that a Bell inequality is violated. They cannot, however, be used for super-luminal signalling. It is also known that it is possible to write down sets of super-quantum correlations that are more non-local than is allowed by quantum mechanics, yet are still non-signalling. Viewed as an information theoretic resource, super-quantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks.

An intriguing question is why quantum mechanics does not allow these more powerful correlations. In order to shed light on the range of quantum possibilities by placing them within a wider context, Partner 4 investigated the set of correlations that are constrained only by the no-signalling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset.- Partner 4 considered the situation in which an observer internal to an isolated system wants to measure the total energy of the isolated system (this includes his own energy, that of the measuring device and clocks used, etc...). It was shown that he can do this in an arbitrarily short time, as measured by his own clock. This measurement is not subjected to a time-energy uncertainty relation.

The properties of such measurements were discussed in detail with particular emphasis on the relation between the duration of the measurement as measured by internal clocks versus external clocks. It has recently been shown that all causal correlations between two parties which output each one bit, a and b, when receiving each one bit, x and y, can be expressed as convex combinations of local correlations (i.e., correlations that can be simulated with local random variables) and non-local correlations of the form a+b=xy mod 2. Partner 4 demonstrated that a single instance of the latter elementary non-local correlation suffices to simulate exactly all possible projective measurements that can be performed on the singlet state of two qubits, with no communication needed at all.- Partner 4 investigated also the concentration of multi-party entanglement by focusing on simple family of three-partite pure states, superpositions of Greenberger-Horne-Zeilinger states and singlets.

Despite the simplicity of the states, it was shown that they cannot be reversibly concentrated by the standard entanglement concentration procedure, to which they seem ideally

suited.

However most work so far has not been concerned with technological feasibility, but rather with proving that quantum error correction is possible in principle. Partner 4 described a error correction method for filtering out errors and ntanglement purification which can be experimentally implemented. - Partner 4 has also investigated what are the fundamental limitations in aligning the frames of reference of two distant parties, and signalling a direction when the frames are already aligned. Along similar lines, the problem of communicating chirality has also been investigated. Partner 4 discussed in detail this issue and introduced the natural concept of quantum gloves, i.e. rotationally invariant quantum states that encode as much as possible the concept of chirality. - Nonlocality is the basic aspect of quantum mechanics underlying quantum information and communication. The limits of quantum non-locality were investigated by Partner 4 by considering non local correlation in their full generality, not only those correlations that may be produced by quantum mechanics. What kinds of non-locality are present in quantum states? By studying the extremal points of the space of all multi-party probability distributions, in which all parties can make one of a pair of measurements, each with two possible outcomes, a necessary condition for classical non-local models to reproduce the statistics of all quantum states was found.

The condition generalizes and extends the results of Sveltichny and of Collins, Gisin, Popescu, Roberts and Scarani who showed that separable classical non-local models cannot reproduce the statistics of all multi-particle quantum states. This condition shows that the non-locality present in some entangled multi-particle quantum states is much stronger than previously thought.

It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are non-local, in the sense that a Bell inequality is violated. They cannot, however, be used for super-luminal signalling. It is also known that it is possible to write down sets of super-quantum correlations that are more non-local than is allowed by quantum mechanics, yet are still non-signalling. Viewed as an information theoretic resource, super-quantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks.

An intriguing question is why quantum mechanics does not allow these more powerful correlations. In order to shed light on the range of quantum possibilities by placing them within a wider context, Partner 4 investigated the set of correlations that are constrained only by the no-signalling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset.- Partner 4 considered the situation in which an observer internal to an isolated system wants to measure the total energy of the isolated system (this includes his own energy, that of the measuring device and clocks used, etc...). It was shown that he can do this in an arbitrarily short time, as measured by his own clock. This measurement is not subjected to a time-energy uncertainty relation.

The properties of such measurements were discussed in detail with particular emphasis on the relation between the duration of the measurement as measured by internal clocks versus external clocks. It has recently been shown that all causal correlations between two parties which output each one bit, a and b, when receiving each one bit, x and y, can be expressed as convex combinations of local correlations (i.e., correlations that can be simulated with local random variables) and non-local correlations of the form a+b=xy mod 2. Partner 4 demonstrated that a single instance of the latter elementary non-local correlation suffices to simulate exactly all possible projective measurements that can be performed on the singlet state of two qubits, with no communication needed at all.- Partner 4 investigated also the concentration of multi-party entanglement by focusing on simple family of three-partite pure states, superpositions of Greenberger-Horne-Zeilinger states and singlets.

Despite the simplicity of the states, it was shown that they cannot be reversibly concentrated by the standard entanglement concentration procedure, to which they seem ideally

suited.