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Theoretical and Experimental Aspects of Quantum Discord and Related Measures

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Theoretical and Experimental Aspects of Quantum Discord and Related Measures Theoretical and experimental aspects of quantum discord and related measures Lucas C. Céleri,1, ∗ Jonas Maziero,1, y and Roberto M. Serra1, z 1Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, R. Santa Adélia 166, 09210-170 Santo André, São Paulo, Brazil Correlations are a very important tool in the study of multipartite systems, both for classical and quantum ones. The discussion about the quantum nature of correlations permeates Physics since Einstein, Podolski and Rosen published their famous article criticizing quantum mechanics. Here we provide a short review about the quantum nature of correlations, discussing both its theoretical and experimental aspects. We focus on quantum discord and related measures. After discussing their fundamental aspects (theoretically and experimentally), we proceed by analysing the dynamical behaviour of correlations under decoherence as well as some applications in different scenarios, such as quantum computation and relativity, passing through critical and biological systems. I. INTRODUCTION principle underlying quantum mechanics, known as non- classical (or quantum) correlations. As we shall see, this Correlations are ubiquitous in nature and have been is a very intricate issue for which we do not have a closed playing an important role in human life for a long time. theory yet. First we observe that the simplicity and gen- For instance, in economy correlations between price and erality of Eq. (1) allows its direct extension to the quan- demand are extremely important for a businessman (or tum domain as a measure of the total correlations in a even for a government) to decide his investment policy. bipartite system, In the field of biology, the genetic correlations are fun- I (A : B) ≡ S (A) + S (B) − S (A; B) ; (2) damental to follow individual traits. The relationship between income distribution and crime rate is just one where S (X) ≡ −TrρX logρX is the von Neumann entropy example coming from the Social Sciences. Broadly speak- [2]. The reduced density matrix of partition X, ρX , is ing, correlation is a quantity that describes the degree obtained from the entire density operator by means of of relationship between two variables. In the classical the partial trace operation ρX ≡ TrY (ρXY ). However, domain, such a quantity can be measured within the differently from the classical scenario, here we face the framework of information theory, developed by Shannon problem of distinguishing between the various types of in 1948 [1]. correlations. As we shall see below, such problem does Let us consider first the classical scenario with two dis- not have a clear-cut solution hitherto. tinct random variables, A and B, with well defined prob- Since the early years of quantum mechanics, non-local ability distributions fpAg and fpBg, respectively. We correlations have been at the core of a long standing de- can think of A and B in terms of probability sets for the bate about the foundations of the theory. We may say outcome of measurements performed on the joint system that the discussion started in 1935, when Einstein, Podol- or on the two different subsystems, usually named Al- sky and Rosen (EPR) published their famous article en- ice and Bob. To say that A and B are correlated in titled “Can quantum-mechanical description of physical some form means that the joint probability distribution reality be considered complete?” [3]. They considered two fpABg cannot be written in the product form fpA × pBg. spatially separated particles, A and B, having both per- To quantify the amount of correlations shared by the two fectly correlated positions and momenta. By defining a classical random variables, Shannon introduced the mu- complete theory as the one for which there is one element tual information [1]: of physical reality corresponding to each element of the theory and assuming local realism, i.e., shortly I (A : B) ≡ H (A) + H (B) − H (A; B) ; (1) arXiv:1107.3428v1 [quant-ph] 18 Jul 2011 locality P • - There is no action at a distance; with H (X) ≡ − x2X px log px being the well-known Shannon entropy. Here and throughout this article we • realism - “If, without disturbing a system, we can use the short-hand notation px ≡ pX=x, in which the predict with certainty the value of a physical quan- random variable X assumes the value x from the set X , tity, then there exists an element of physical reality and we use log to represent the binary logarithm. So, the corresponding to this physical quantity.”; correlations are quantified in bits. they argued that quantum mechanics would not be com- In this article we are interested in a very special kind plete (see [4] for a recent review about the EPR paradox). of correlation, the ones arising from the superposition In a sense, EPR implied that quantum mechanics should be extended, possibly through the introduction of hidden variables, making the new theory compatible with their ∗ [email protected] premise of local realism stated above. In other words, y [email protected] EPR demonstrated an inconsistency between local real- z [email protected] ism and the completeness of quantum mechanics. 2 Schrödinger noted that at the core of the paradox lies must be abandoned. A recent review about the theoreti- the very structure of the Hilbert space [5]. He pointed cal and the experimental aspects of this theorem can be out that the state vector of the entire system (including found in Ref. [4]. We observe here that the extension both particles A and B) is entangled. This means that of the quantum theory proposed by Bell is classical, in we cannot ascribe an individual state vector for each par- the sense that the hidden variables are local in nature. ticle separately, only the entire system can have such a However, more recently, Colbeck and Renner have shown, mathematical representation. This fact leads to the def- under the assumption that the measurement setting can inition of entanglement. A pure state is separable if and be freely chosen by the observer, that the quantum the- only if it can be written in the product form: ory cannot be extended at all [8]. With concern to correlations, the scenario regarding j ABi = j Ai ⊗ j Bi; (3) what was discussed until here is the following: The non- classical nature of correlations was attributed to states j i j i where A ( B ) is the state vector describing the sys- that violate some Bell inequality. This kind of correla- A B tem ( ) alone. Otherwise it is entangled. The argu- tion, also named non-local, cannot be explained by an ment of Schrödinger against EPR was that this entan- LHV model. However, this situation changed in 1989, A glement is degraded during the process by which and when Werner published an article showing that there are B become spatially separated [5], leading to the impos- mixed entangled states that do not violate any Bell in- Gedankenexperiment sibility of physically realizing the equality and therefore could be described by means of proposed by EPR. Although interesting, this argument an LHV model [9]. This was a surprising fact because it does not settle the paradox. indicates that there are non-separable (entangled) mixed Since then, entanglement has been at the core of the states that could be described in “classical terms” by an discussion about the non-local aspects of quantum the- LHV model. So, another kind of correlation arises, the ory, both in respect to its fundamental aspects as well one contained in an entangled state, but that do not vio- as for (more recently) technological applications. The ef- late any Bell inequality. This kind of correlation is indeed forts that lead us to the theory of entanglement have been non-classical since we disregard LHV models. This pic- mainly focused on two aspects of the problem: Given a ture remained unchanged until 2001, when it was iden- certain state, is it entangled? If yes, how much entangle- tified a new kind of quantum correlation, which may be ment is there? Besides many advances (both in the theo- present even in separable (non-entangled) states. retical and in the experimental fields) have been achieved, a complete theory describing entanglement is still lack- ing. We refer the reader to the reviews in Ref. [6] for II. QUANTUM DISCORD a more complete discussion about entanglement. Before we proceed, let us define the entanglement for more gen- eral states, the mixed states. For systems described by a This new scenario emerged from an nonequivalence be- density operator ρAB, the separability condition given in tween the classical and the quantum versions of informa- Eq. (3) becomes tion theory. We may explore this nonequivalence in sev- eral ways in order to reveal some non-classicality. In what X i i follows we will present a way to do it that we consider ρAB = piρA ⊗ ρB; (4) i very illustrative. From Bayes’ rule we can write the clas- sical conditional probability for obtaining the value a for i with fpig being a probability distribution and ρX a den- the random variable A when the value of the random vari- sity operator for the subsystem X. In the case of con- able B is known to be b, pajb, in the form pajb ≡ pa;b=pb; tinuous variables, we must replace the summation by an which leads to an equivalent expression for the classical integral over the entire probability space. The state (4) mutual information (1) is the most general bipartite state that Alice and Bob can create using local quantum operations and classical J(A : B) ≡ H(A) − H(AjB); (5) communication (LOCC). Therefore, an entangled state where the classical conditional entropy reads H(AjB) ≡ is that one that cannot be prepared by LOCC.
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