Pointing Out the Obstacle of Quantification of Quantum Discord

Composite quantum systems can be in generic states characterised not only by entanglement but also by more general quantum correlat ions .The inter-relation between these two quantities ensures of non-locality is still not completely understood. Last few years people are concentrating in studying the different aspects of non-locality of quantum mechanics. Many correlation measures have been introduced and well studied. Quantum d iscord is one of such correlation measure that creates new challenges among the physicists and mathemat icians. New quantificat ion of quantum discord is one of the fascinating area .The analytic formula has been introduced only two qubit X states only. So in this paper we study for the investigation of the difficu lties in finding the analytic expression of quantum discord in general two and higher dimension quantum systems.


Introduction
In the quantum information and computation theory, quantum entanglement, a non-classical correlation, is the key resource and a most remarkable feature. Quantum entanglement was first introduced by Einstein, Podolsky and Rosen (EPR) [1] and also Schrödinger [2]. There was a question in [1] by EPR, whether quantum mechanics is local and complete theory or not? In th is context, Bell [3] has given a very significant result : the well known Bell's Inequality and the consequent features of quantum mechanics are usually called non-local theory. It is accepted that quantum entanglement is responsible for the non-locality of quantum mechanics and the performances of so many information theoretic tasks like Teleportation, Dense coding, Cloning and many others [4][5][6]. Hence characterization and quantification of quantum entanglement are the most important tasks of quantum information and computation theory.

Now the most popular measure introduced by Olliver and
Zurek [17] and separately by Hendersen and Vedral [18] is quantum discord-which is much better than the other measures and can find out the non-classical correlat ions even in separable states. Quantum discord brought as an informat ion theoretic measure of the 'quantumness' of correlations [19] and is used to determine so me results in thermodynamics [20]. Characterizing correlations in terms of its quantum discord, it is proved that classical correlat ions leads to completely positive reduced dynamics and the induced maps can be co mpletely non-positive when quantum correlat ions is present [21] and co mpletely positive(CP) maps arise exclusively fro m the class of separable states with vanishing quantum discord [22]. Use of quantum discord for the characterization of correlat ions present in the quantum co mputational model DQC1, introduced by Knill and Laflamme reveals that non-zero values of discord indicates non-classical correlat ions whenever there is no entanglement between the t wo parts [23]. A large amount of discord is found but no entanglement in the experiment by the implementation of DQC1 in an all-optical arch itecture [24]. Also in the DQC1 model, it is proved that a non-zero quantum d iscord imp lies a non-zero shift under locally non-effective unitary operations(LNUs) [25]. In the dissipative dynamics of two-qubit quantum discord under Markovian environments, comparison of the dynamics of entanglement with that of quantum discord was made and shown that the entanglement suddenly disappears in all cases where quantum discord vanishes only in the asymptotic limit as the individual decoherence of the qubits, also in finite temperature. Which concludes that quantum discord is more robust than the entanglement against decoherence so that quantum algorith ms depending on the correlation 'quantum discord' may be mo re robust than those based on quantum entanglement [26]. Study of quantum d iscord for t wo-qubit states gives that for separable states, the entanglement of formation always vanishes but discord does not vanish implies the superiority o f quantum discord [27].For finding the monotonic nature of quantum discord ,a few discussion on incomparability and majorization is required. The monotonic nature of quantum discord through incomparab ility is also observed here.But our present discussion, we are concentrating in the difficult ies in the quantification of quantum discord.

Concept Of Quantum Discord
Now we know that a bipartite quantum state has both classical and quantum correlat ions. An information theoretic measure of a bipartite quantum state is 'quantum mutual informat ion'. I Mutual informat ion is the maximu m amount of informat ion that A can securely send to B if a co mposite correlated quantum state is used as the key for a one-time pad cryptographic system [28]. Quantum mutual information is the sum of classical correlation ( ) II.Quantum discord: Now, the mutual info rmation may be written as Then the quantum conditional entropy with respect to this measurement is given by And the associated quantum mutual informat ion of this measurement is defined as Classical correlation is given by [17,18,27,29]    Then it can be obtained from Nielsen's criterion that ψ and φ are incomparab le if and only if either

Review of incomparability under deterministic LOCC
All the above studies is for the deterministic transformat ion.

Analitical Approach in Quantification Procedure of Discord
It is briefly d iscussed and completely explained [30] that for t wo-qubit X-states quantum discord can be found. The method applied for finding quantum discord has required the use of the von-Neuman measurements for the subsystem B as

Monotonicity of Quantum discord under deterministic incomparability
In this section our attempt to observe the monotonic nature of quantum discord under determin istic incomparab ility LOCC.

Conclusions
Quantum discord is more powerfu l feature for realizing the non-locality aspect of quantum mechanics than quantum entanglement because some separable states have non-zero quantum d iscord. In this paper our aim is to find out the mathematical difficult ies in the calculating procedure of Quantum discord. We observe that even in 3 3 ⊗ , the large expressions of elements of the matrix are really hard to handle. So it obstacles us for finding the eigen values of the matrices. The next big problem is due to the optimization occurred in the expression of the Quantum discord. So finding the general expression of Quantum discord in this above mathematical process is really a great challenge to the people. Though many t ight bounds have been discovered and theoretical works have been done the analytic expression for this discord in higher dimension is not yet found.