Quantum discord

== Introduction == Discord happens to be one of the many quantities that can measure the non-classicality of a given quantum system.Discord is based on the measure of mutual information between two parts of a system. Its standard definition equates discord to the difference between two classically equivalent forms of mutual information. Entanglement was thought to be an indispensable resource for quantum information processing, which can outperform the corresponding classical information processing. Indeed, various quantum algorithms that exploit entanglement have been proposed and successfully tested. Separable (i.e. not entangled) states were considered insufficient to implement quantum information processing. That belief has changed since Ollivier and Zurek as well as Henderson and Vedral independently introduced a new measure of non-classical correlations named `discord'. It can be put in one-to-one correspondence with entanglement for pure states, but unlike entanglement it can be nonzero for separable mixed states. == Theory == === Conditional Entropy === In classical information theory the amount of information contained in a random variable $X$ is quantified as the Shannon entropy, :$\{\backslash mathcal\; H\}(X)=-\backslash sum\_\{x\}p\_\{x\}\backslash log\_\{2\}p\_\{x\}$ where $p\_\{x\}$ is the probability of occurrence of event $X$. When $\{\backslash mathcal\; H\}(X)=0$, the random variable $X$ is completely determined and no new information is gained by measuring it. Hence, Shannon entropy can be interpreted as either the uncertainty before measuring a random variable or the information gained upon measuring it. Consider a bipartite system containing two subsystems (or random variables), A and B. Conditional entropy of B quantifies the uncertainty in measurement of B when A is known, and is represented by $\{\backslash mathcal\; H\}(B|A)$. Using classical probability theory, it can be expressed as :$\{\backslash mathcal\; H\}(B|A)=\{\backslash mathcal\; H\}(A,B)-\{\backslash mathcal\; H\}(A)$ where $\{\backslash mathcal\; H\}(A,B)$ is the information content of the full system and $\{\backslash mathcal\; H\}(A)$ is the information content of the subsystem A. An equivalent way of defining the conditional entropy is :$\{\backslash mathcal\; H\}(B|A)\; =\; \backslash sum\_\{i\}p\_\{i\}^\{a\}\; \{\backslash mathcal\; H\}(B|a=i)$ where :$\{\backslash mathcal\; H\}(B|a=i)\; =\; -\backslash sum\_\{j\}p(b\_\{j\}|a\_\{i\})\backslash log\_2\{p(b\_\{j\}|a\_\{i\})\}$ and $p(b\_\{j\}|a\_\{i\})$ is the conditional probability of occurrence of event $b\_\{j\}$ given that event $a\_\{i\}$ has occurred. Unlike previous definition in, this definition involves measurement of one subsystem of a bipartite system. === Mutual Information: === It is the amount of information that is common to both the subsystems of a bipartite system, and is given by :$\{\backslash mathcal\; I\}(A:B)=\; \{\backslash mathcal\; H\}(A)\; +\; \{\backslash mathcal\; H\}(B)\; -\; \{\backslash mathcal\; H\}(A,B)$ This expression can be intuitively understood as follows. On the right hand side, the first two terms quantify the information content of subsystems A and B respectively. So the information common to both the subsystems is counted twice. Subtracting the information content of the combined system then gives the common (or mutual) information. The result is clearly symmetric, i.e. $\{\backslash mathcal\; I\}(A:B)\; =\; \{\backslash mathcal\; I\}(B:A)$. A classically equivalent form of mutual information, :$\{\backslash mathcal\; J\}(A:B)\; =\; \{\backslash mathcal\; H\}(B)\; -\; \{\backslash mathcal\; H\}(B|A)$ ::$=\{\backslash mathcal\; H\}(B)\; -\; \backslash sum\_\{i\}p\_\{i\}^\{a\}\{\backslash mathcal\; H\}(B|a=i)$ which removes from the information content of subsystem B the conditional contribution that is not contained in subsystem A. === Discord === In quantum information theory, the von Neumann entropy gives the information content of a density matrix, :$H(\backslash rho)\; =\; -\backslash sum\_\{x\}\backslash lambda\_\{x\}\backslash log\_\{2\_\{\}^\{\}\}\backslash lambda\_\{x\}$ where $\backslash lambda\_\{x\}$ are the eigenvalues of the density matrix $\backslash rho$. Although the two expressions of mutual information, $\{\backslash mathcal\; J\}(A:B)$ and $\{\backslash mathcal\; I\}(A:B)$, are equivalent in classical information theory, they are not so in quantum information theory. The reason for the difference is that the expression for mutual information given by $\{\backslash mathcal\; J\}(A:B)$ involves measurement and depends on its outcomes. Measurements in quantum theory are basis dependent and also change the state of the system. Henderson and Vedral have proved that the total classical correlation can be obtained as the largest value of, :$J(A:B)\; =\; H(B)\; -\; H(B|A\_\{\}^\{\})$ ::$=\; H(B)\; -\; \backslash sum\_\{i\}p\_\{i\}^\{a\}H(B|a=i)$ where the maximization is performed over all possible orthonormal measurement bases $\backslash \{\backslash Pi\_\{i\}^\{a\}\backslash \}$ for A, satisfying $\backslash sum\_\{i\}\backslash Pi\_\{i\}^\{a\}=I\_\{d\}$($I\_\{d\}=Identity\; matrix$) and $\backslash Pi\_\{i\}^\{a\}\backslash Pi\_\{j\}^\{a\}\; =\; \backslash delta\_\{ij\}\backslash Pi\_\{i\}^\{a\}$ .Therefore, the non-classical correlations can be quantified as the difference :$D(B|A)\; =\; I(A:B)\; -\; \backslash max\_\{\backslash \{\backslash Pi\_\{i\}^\{a\}\backslash \}\}J(A:B)$ Ollivier and Zurek named this difference `discord'. Zero discord states or `classical' states are the states in which the maximal information about a subsystem can be obtained without disturbing its correlations with the rest of the system. Discord is not a symmetric function in general, i.e. $D(B|A)$ and $D(A|B)$ can be different. Datta has proved that a state $\backslash rho\_\{AB\}$ satisfies $D(B|A)=0$ if and only if there exists a complete set of orthonormal measurement operators on $A$ such that :$\backslash rho\_\{AB\}\; =\; \backslash sum\_\{i\}p\_\{i\}^\{a\}\; \backslash Pi\_\{i\}^\{a\}\backslash otimes\backslash rho\_\{B|a=i\}$ When one part of a general bipartite system is measured, the resulting density matrix is of the form given by above equation. Since the state rendered upon measurement is a classical state, one can extract classical correlations from it. Thus for any quantum state and every orthonormal measurement basis, there exists a classically correlated state. Maximization of ,$J(A:B\_\{\}^\{\})$ captures the maximum classical correlation that can be extracted from the system, and whatever extra correlation that may remain is the quantum correlation. == Evaluation of Discord == Given a density matrix $\backslash rho\_\{AB\}^\{\}$, it is easy to construct the reduced density matrices $\backslash rho\_\{A\}^\{\}$ and $\backslash rho\_\{B\}^\{\}$, and then obtain the total correlation $I(A:B\_\{\}^\{\})$. Maximization of $J(A:B\_\{\}^\{\})$ to evaluate discord is non-trivial, however. The brute force method is to maximize $J(A:B\_\{\}^\{\})$ over as many orthonormal measurement bases as possible, taking into account all constraints and symmetries. For a general quantum state, a closed analytic formula for discord does not exist, but for certain special class of states analytical results are available \cite{girolami}. For example, Chen have described analytical evaluation of discord for two qubit X-states under specific circumstances . Luo has given an analytical formula for discord of the Bell-diagonal states that form a subset of the X-states . The two methods of measuring discord firstly by brute force method and then by Luo's method valid only for bell diagonal state is discussed below: === Extensive measurement method: === This method involves measurements over extensive sets of orthonormal basis vectors and maximization of $J(A:B\_\{\}^\{\})$. For measurement of a single qubit in a two qubit system, we use the orthonormal basis :$\backslash \{\backslash vert\; u\backslash rangle\; =\; \backslash cos\backslash theta\backslash vert\; 0\backslash rangle+\; e^\{i\backslash phi\}\backslash sin\backslash theta\backslash vert\; 1\backslash rangle\; ,\; \backslash vert\; v\backslash rangle\; =\; \backslash sin\backslash theta\backslash vert\; 0\backslash rangle\; -\; e^\{i\backslash phi\}\backslash cos\backslash theta\backslash vert\; 1\backslash rangle\; \backslash \}$ and let $\backslash cos\backslash theta\backslash in\; [-1,1]$ and $\backslash phi\backslash in[0,2\backslash pi)$ vary in small steps. For every choice of $\backslash theta\_\{\}^\{\}$ and $\backslash phi\_\{\}^\{\}$, we project the experimental density matrix obtained by tomography along the orthonormal basis. The post-projection density matrix is :$\backslash rho\text{'}\; =\; \backslash sum\_\{i=u,v\}\; \backslash Pi\_\{i\}^\{a\}\; \backslash rho\; \backslash Pi\_\{i\}^\{a\}\; =\; \backslash sum\_\{i=u,v\}\; p\_\{i\}^\{a\}\; \backslash Pi\_\{i\}^\{a\}\backslash otimes\backslash rho\_\{B|a=i\}$ with $p\_\{i\}^\{a\}\; =\; \{\backslash rm\; Tr\}[\backslash Pi\_\{i\}^\{a\}\backslash rho]$. Discord is then obtained from the conditional density matrix $\backslash rho\_\{B|a=i\}$. Strictly speaking, this method gives a lower bound on $J(A:B\_\{\}^\{\})$, since the direction maximizing $J(A:B\_\{\}^\{\})$ may not exactly match any of the points on the discrete $(\backslash theta\_\{\}^\{\},\backslash phi)$ grid. === Analytical method for the Bell-diagonal states: === As the name suggests, the Bell-diagonal states are diagonal in the Bell basis, given by :$\backslash vert\; \backslash psi^\{\backslash pm\}\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (\; \backslash vert\; 01\backslash rangle\; \backslash pm\; \backslash vert\; 10\backslash rangle\; )\; ~,~~\; \backslash vert\; \backslash phi^\{\backslash pm\}\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (\; \backslash vert\; 00\backslash rangle\; \backslash pm\; \backslash vert\; 11\backslash rangle\; ).$ The generic structure of a Bell-diagonal state is :$\backslash rho\_\{BD\}\; =\; \backslash lambda\_1\; \backslash vert\; \backslash psi^-\backslash rangle\backslash langle\; \backslash psi^-\; \backslash vert\; +\; \backslash lambda\_2\; \backslash vert\; \backslash phi^-\backslash rangle\backslash langle\; \backslash phi^-\; \backslash vert\; +\; \backslash lambda\_3\; \backslash vert\; \backslash phi^+\backslash rangle\backslash langle\; \backslash phi^+\; \backslash vert\; +\; \backslash lambda\_4\; \backslash vert\; \backslash psi^+\backslash rangle\backslash langle\; \backslash psi^+\; \backslash vert.$ With only local unitary operations (so as not to alter the correlations), all Bell-diagonal states can be transformed to the form :$\backslash rho\_\{BD\}\; =\; \backslash frac\{1\}\{4\}\; \backslash Big(\; I\_\{d\}\; +\; \backslash sum\_\{j=1\}^\{3\}r\_\{j\}\backslash sigma\_\{j\}\backslash otimes\backslash sigma\_\{j\}\; \backslash Big)\; ~,$ where the real numbers $r\_\{j\}^\{\}$ are constrained so that all eigenvalues of $\backslash rho\_\{BD^\{\}\}$ remain in $[0,1\_\{\}^\{\}]$. The symmetric form of $\backslash rho\_\{BD^\{\}\}$ also implies that it has symmetric discord, i.e. $D\_\{BD\}(B|A\_\{\}^\{\})\; =\; D\_\{BD\}(A|B\_\{\}^\{\})$. Luo choose the set of measurement basis as $\backslash \{V\backslash Pi\_\{k\}^\{a\}V^\{\backslash dagger\}\backslash \}$, where $\backslash Pi\_\{k\}^\{a\}=\backslash vert\; k\backslash rangle\backslash langle\; k\backslash vert$ are the projection operators for the standard basis states ($k=0,1\_\{\}^\{\}$), and $V\_\{\}^\{\}$ is an arbitrary $SU(2)\_\{\}^\{\}$ rotation matrix. A projective measurement yields the probabilities $p\_\{0\}=p\_\{1\}=\backslash frac\{1\}\{2\}$, and an analytical formula for the classical correlation, :$\backslash max\_\{\backslash \{\backslash Pi\_\{k\}^\{a\}\backslash \}\}\; J(A:B)\; =\; \backslash left(\backslash frac\{1-r\}\{2\}\backslash right)\backslash log\_\{2\}(1-r)\; +\; \backslash left(\backslash frac\{1+r\}\{2\}\backslash right)\backslash log\_\{2\}(1+r)\; ~$ with $r=\backslash max\backslash \{|r\_\{1\}|,|r\_\{2\}|,|r\_\{3\}^\{\}|\backslash \}$. For the Bell-diagonal states, the reduced density matrices are $\backslash rho\_A^\{\}\; =\; \backslash rho\_B\; =\; I\_\{d\}/2$, and the total correlation is :$I(A:B)\; =\; 2+\backslash sum\_\{i=1\}^\{4\}\backslash lambda\_\{i\}\backslash log\_2\backslash lambda\_\{i\}$ where the eigenvalues $\backslash lambda\_\{i\}^\{\}$ of $\backslash rho\_\{BD\}^\{\}$ are: :$\backslash lambda\_\{1\}^\{\}\; =\; (1-r\_\{1\}-r\_\{2\}-r\_\{3\})/4$ :$\backslash lambda\_\{2\}^\{\}\; =\; (1-r\_\{1\}+r\_\{2\}+r\_\{3\})/4$ :$\backslash lambda\_\{3\}^\{\}\; =\; (1+r\_\{1\}-r\_\{2\}+r\_\{3\})/4$ :$\backslash lambda\_\{4\}^\{\}\; =\; (1+r\_\{1\}+r\_\{2\}-r\_\{3\})/4$ Thus the analytical formula for discord is, :$\{D\}\_\{BD\}(B|A)\; =\; 2\; +\; \backslash sum\_\{i=1\}^\{4\}\backslash lambda\_\{i\}\backslash log\_2\backslash lambda\_\{i\}\; -\; \backslash left(\backslash frac\{1-r\}\{2\}\backslash right)\backslash log\_\{2\}(1-r)\; -\; \backslash left(\backslash frac\{1+r\}\{2\}\backslash right)\backslash log\_\{2\}(1+r)\; ~.$ == Geometric Discord == Since the maximization of $J(A:B\_\{\}^\{\})$ involved in calculating discord is a hard problem, Dakic ''et al''. introduced a more easily computable form of discord based on a geometric measure. For every quantum quantum state there is a set of post-measurement classical states, and the geometric discord is defined as the distance between the quantum state and the nearest classical state, :$DG(B|A)\; =\; \backslash min\_\{\backslash chi\; \backslash in\; \backslash Omega\_0\}\backslash |\backslash rho-\backslash chi\backslash |^2$ where $\backslash Omega\_0^\{\}$ represents the set of classical states, and $\backslash |X-Y\backslash |^2\; =\; \{\backslash rm\; Tr\}(X-Y)^2$ is the Hilbert-Schmidt quadratic norm. The two-qubit density matrix in the Bloch representation is :$\backslash rho\; =\; \backslash frac\{1\}\{4\}\; \backslash Big(\; I\_\{d\}\; \backslash otimes\; I\_\{d\}\; +\; \backslash sum\_\{i=1\}^\{3\}\; x\_\{i\}\backslash sigma\_\{i\}\backslash otimes\backslash otimes\; I\_\{d\}\; +\; \backslash sum\_\{i=1\}^\{3\}y\_\{i\}\; \backslash otimes\; I\_\{d\}\backslash otimes\backslash sigma\_\{i\}\; +\; \backslash sum\_\{i,j=1\}^\{3\}\; T\_\{ij\}\backslash sigma\_\{i\}\backslash otimes\backslash sigma\_\{j\}\; \backslash Big)$ where $x\_\{i\}^\{\}$ and $y\_\{i\}^\{\}$ represent the Bloch vectors for the two qubits, and $T\_\{ij\}^\{\}=\{\backslash rm\; Tr\}(\backslash rho(\backslash sigma\_\{i\}\backslash otimes\backslash sigma\_\{j\}))$ are the components of the correlation matrix. The geometric discord for such a state is :$DG(B|A)\; =\; \backslash frac\{1\}\{4\}(\backslash |x\backslash |^2\; +\; \backslash |T\backslash |^2\; -\; \backslash eta\_\{\backslash rm\; max\})$ where $\backslash |T\backslash |^2\; =\; \{\backslash rm\; Tr\}[T^\{T\}T]$, and $\backslash eta\_\{\backslash rm\; max\}$ is the largest eigenvalue of $\backslash vec\{x\}\backslash vec\{x\}^\{T\}\; +\; TT^\{T\}$. The Bloch parameters $x\_i^\{\}$, $y\_i^\{\}$ and $T\_\{ij\}^\{\}$ provide a complete description of any two-qubit state. So tomographic measurement of these parameters determines the geometric discord exactly. == References == *Quantum Computation and Quantum Information by Nielsen and Chuang. *H. Ollivier and W.H. Zurek, Phys. Rev. Lett. 88, 017901 (2002). *L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). This reference actually maximizes $J(A:B)$ over generalised POVMs, and not over orthonormal measurements. It has been shown that rank-1 POVMs suffice for this purpose . In case of two-dimensional Hilbert spaces that we are dealing with, rank-1 POVMs can be reduced to projective measurements. *B. Dakic, V. Vedral and C. Brukner, Phys. Rev. Lett. 105, 190502 (2010). *S. Luo, Phys. Rev. A 77, 042303 (2008). *H. Katiyar, S. S. Roy, T. S. Mahesh and A. Patel, Phys. Rev. A 86, 012309 (2012). [[Category:Handbook of Quantum Information]] [[Category: Quantum Discord]]