Project description a) What is quantum information theory with continuous variables?

Transcription

1 Project description a) What is quantum information theory with continuous variables? According to pre-quantum physics, any theory should display the property of locality defined in three almost equivalent ways as: impossibility of instantaneous communication, impossibility of action-at-a-distance, and impossibility of faster-thanlight propagation. In 1935 Einstein, Podolsky and Rosen [1] have noticed that two quantum systems, say two particles, which interacted at some time in their past and became separated later are still connected (correlated) in the following way: a measurement performed on one of the two particles instantaneously modifies the quantum state of the other one. This phenomenon clearly violates locality. It was called entanglement by Schroedinger [1] (entanglement is the English translation of the German word Verschraenkung ) who considered it an essential feature of quantum physics, almost a drawback [2] that somehow should be eliminated. While in 1935 such quantum paradoxes were discussed through gedankenexperiments,fifty years after,a series of ingenious experiments was performed [3], mainly on photons, but also on other particles, which showed us that non-locality and entanglement (termed also inseparability) are the principal features of composite quantum states. A significant piece of progress was then made by considering individual quantum systems as carriers of information whose processing is determined by the laws of quantum physics. The new field of information theory based on quantum principles completes classical information theory [4]. Before proceeding we have to note that most of the basic theory in this field has been developed for finite-dimensional systems, namely spin systems. The simplest one is the ½ spin termed as qubit. However, most of the experiments in the area of quantum information processing (teleportation [5], cryptography) have been performed with light either single photons, or beams of laser light. Therefore, when speaking about the continuous-variable quantum information one means the processing of the electromagnetic field. As the field is a quantum system we are inevitably dealing with 1

2 concepts and methods of quantum optics. Most of the work within this proposal lies at the intersection of quantum optics and information theory. Fortunately, in quantum optics detecting and quantifying non-classicality is a well defined and explored issue, at least in the one-mode case [6]. The two-mode field is the prototype of a bipartite continuousvariable system. Theoretically, it is a perfect test-bed for studying entanglement or other kinds of correlations between the two modes. Experimentally, it is the most accessible quantum system, especially when dealing with Gaussian states. Work in the area of quantum information processing has increased enormously in the last two decades. In any settings (discrete or continuous), the most interesting areas explored are quantum communication, quantum cryptography, quantum teleportation and quantum computation. For instance, the invention of quantum teleportation in 1993 [7] showed a way to achieve disembodied transport of a quantum state by exploiting the nonlocal character of an entangled bipartite state. By ingenious measurements of one side of the overall system a quantum state is transferred to the other side due to the non-locality property. In the continuous-variable setting teleportation is a noisy process which distorts the teleported state. The amount of noise proved to decrease for strongly entangled resource state [8]. From this example it is clear that in quantum information entanglement is viewed as a quantum resource. Thus it is necessary to be able to quantify the amount of entanglement contained in a quantum state (pure or mixed) [9,10]. b) Quantum correlations Until very recently it was believed that in quantum information we are dealing with only two quantifiable kinds of information, classical information contained in classical correlations and quantum entanglement as the potential of quantum states to exhibit correlations that cannot be accounted classically. Entanglement was largely viewed as a useful but rather unique resource in quantum information processing. All research concentrated on identifying and quantifying entanglement [9] in different systems that seemed promising for applications. Typical information processes as dense coding, 2

3 cryptography and computing were shown to better perform when using entangled resource states than in the classical settings [5]. In spite of these successes, there are several examples where the quantum advantage cannot be attributed to entanglement. Quite early, Meyer [11] presented a quantum search algorithm that uses no entanglement. In Ref. [12] an experimental scheme for quantum computing without entanglement is shown to properly work. It is thus relevant to study in what sense correlations present in separable states may exhibit a certain quantum character. An alternative classification for correlations based on quantum measurements has arisen in recent years [13], In particular, the quantum discord as a measure of quantum correlations beyond entanglement, initially introduced by Ollivier and Zurek [14] and by Henderson and Vedral [15], is attracting increasing interest. In Refs.[14,15] quantum discord is the difference between two natural quantum extensions of the classical mutual information. While this quantity is classically insensitive to the interchanging of the two reduced states of a bipartite system, its quantum extension is asymmetric with respect to a measurement performed on the subsystems. It was shown that quantum discord could be nonzero in separable states. Some authors [13] have considered this property as an expression of the quantumness arising from the non-commutativity of operators representing states, observables, and measurements while entanglement should be a consequence of the superposition principle in quantum mechanics. While the discord is a fundamental notion allowing for the description of the quantumness of the correlations present in the state of a quantum system, its evaluation requires an optimization procedure over the set of all measurements on a given subsystem. Therefore, evaluating quantum discord is almost as difficult as evaluating entanglement. Few examples of explicit results are known in the continuous-variable settings. The Gaussian states, the most important theoretical and experimental resources of continuous-variable quantum information, have sometimes been considered as essentially classical in view of the positivity of their Wigner distribution. They are particularly interesting for evaluating quantum discord. It was found that the Gaussian quantum discord is nonzero for all bipartite states excepting the product ones [16,17]. 3

4 c) Relations between entanglement and discord: monogamy properties About ten years ago, an important remark regarding entanglement distribution was done in Ref.[18]: Unlike classical correlations, quantum entanglement cannot be freely shared among many objects. If A is partly entangled with B, then A can have only a limited entanglement with C. This limitation in the distribution of entanglement was nicely called monogamy of quantum entanglement. When using entanglement of formation (EF) as a measure of entanglement, the monogamy expresses in fact a clear relation between the EF of the system AB and the discord in the system AC [19]. Very recently, quantum discord of a system partially measured by a von Neumann measurement was shown to create entanglement between apparatus and the system [20]. Finally, we remark that the monogamy properties are still to be explored and explicit relations between EF and discord are to be found. d) Our proposal motivation I am encouraged to formulate a project proposal on quantum correlations not only by the conceptual and even practical importance of the problems described above but also by several recent results that I have obtained together with my collaborators: 1) Tudor A. Marian : In the last ten years we invested a lot of effort in evaluating entanglement expressed by distance-type measures for two-mode Gaussian states [21]. To this end we had to investigate the structure of quantum Gaussian fidelity for twomode states (which is not a simple issue) and give formulas for Bures-distance entanglement. We also gave a treatment to the problem of EF in the Gaussian case [10]. What we want to do working for this proposal is to relate our entanglement findings to the discord issues via the monogamy relations. Preliminary investigations showed us that this new research in the field of Gaussian states is realizable. 4

5 2) Gunnar Bjork, Iulia Ghiu, Tudor A. Marian: Recently we opened together the possibility of evaluating a degree of polarization for two-mode states of the quantized electromagnetic field, a topic much required in quantum optics [22]. To do this we applied methods established in quantum information processing. Moreover, evaluating a distance-type degree of polarization for a two-mode state, is in our treatment an issue closely related to the higher-order correlations between excitation manifolds of a state whose polarization is measured by usual lossless linear optics. What we want to do here is to apply and extend the treatment exposed in Ref.[22] to other quantum states. 3) Madalina Boca, Iulia Ghiu, Tudor A. Marian: We have evaluated non-classicality of one-mode states. What we want to do together for this proposal is to investigate decoherence of entanglement and discord due to the interaction to some bosonic environments. We shall begin with a dissipative (Markovian) environment to watch the so-called entanglement and discord sudden death for Gaussian states. References [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [2] E. Schroedinger, Naturwiss. 23, (1935). [3] A. Aspect, J. Dalibard, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982). [4] Charles H. Bennett and Peter W. Shor, IEEE Transactions on Information Theory 44, 2724 (1998). [5] A. Furusawa et al., Science 282, 706 (1998). [6] Paulina Marian, Tudor A. Marian and Horia Scutaru, Phys. Rev. Lett. 88, (2002). [7] C. H. Bennett, et al., Phys. Rev. Lett. 70, 1895 (1993). [8] Paulina Marian and Tudor A. Marian, Phys. Rev.A 74, (2006) [9] R. Horodecki et al., Rev. Mod. Phys. 81, (2009). [10] Paulina Marian and Tudor A. Marian, Phys. Rev. Lett. 101, (2008). [11] D. A. Meyer, Phys. Rev. Lett. 85, 2014 (2000). 5

6 [12] B. P. Lanyon, M. Barbieri, M. P. Almeida and A. G. White, Phys. Rev. Lett. 101, (2008). [13] M. Piani, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 100, (2008); S. Luo, Phys. Rev. A 77, (2008). [14] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, (2001). [15] L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). [16] P. Giorda and M. G. A. Paris, Phys. Rev. Lett. 105, (2010). [17] G. Adesso and A. Datta, Phys. Rev. Lett. 105, (2010). [18] Valerie Coffman, Joydip Kundu, and William K. Wootters, Phys. Rev. A 61, (2000). [19] M. Koashi and A. Winter, Phys. Rev. A 69, (2004). [20] A. Streltsov, H. Kampermann and Dagmar Bruss, Phys. Rev. Lett. 106, (2011). [21] Paulina Marian, T. A. Marian, and H. Scutaru, Phys. Rev. A 68, (2003); Paulina Marian and T. A. Marian, The Eur. Phys. Journal Special Topics 160, 281 (2008); Paulina Marian and T. A. Marian, Phys. Rev. A 77, (2008). [22] Iulia Ghiu, Gunnar Bjork, Paulina Marian, and Tudor A. Marian, Probing light polarization with the quantum Chernoff bound, Physical Review A 82, (2010). 6