== 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,

$${\mathcal H}(X)=-\sum_{x}p_{x}\log_{2}p_{x}$$

where p_x is the probability of occurrence of event X. When ${\mathcal H}(X)=0$, 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 ${\mathcal H}(B|A)$. Using classical probability theory, it can be expressed as

$${\mathcal H}(B|A)={\mathcal H}(A,B)-{\mathcal H}(A)$$

where ${\mathcal H}(A,B)$ is the information content of the full system and ${\mathcal H}(A)$ is the information content of the subsystem A. An equivalent way of defining the conditional entropy is

$${\mathcal H}(B|A) = \sum_{i}p_{i}^{a} {\mathcal H}(B|a=i)$$
where

$${\mathcal H}(B|a=i) = -\sum_{j}p(b_{j}|a_{i})\log_2{p(b_{j}|a_{i})}$$
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

$${\mathcal I}(A:B)= {\mathcal H}(A) + {\mathcal H}(B) - {\mathcal H}(A,B)$$

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. ${\mathcal I}(A:B) = {\mathcal I}(B:A)$. A classically equivalent form of mutual information,

$${\mathcal J}(A:B) = {\mathcal H}(B) - {\mathcal H}(B|A)$$

$={\mathcal H}(B) - \sum_{i}p_{i}^{a}{\mathcal H}(B|a=i)$

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(ρ) =  − ∑_xλ_xlog₂λ_x

where λ_x are the eigenvalues of the density matrix ρ. Although the two expressions of mutual information, ${\mathcal J}(A:B)$ and ${\mathcal I}(A:B)$, 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 ${\mathcal J}(A:B)$ 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) − ∑_ip_i^aH(B∣a = i)

where the maximization is performed over all possible orthonormal measurement bases {Π_i^a} for A, satisfying ∑_iΠ_i^a = I_d(I_d = Identitymatrix) and Π_i^aΠ_j^a = δ_ijΠ_i^a .Therefore, the non-classical correlations can be quantified as the difference

D(B∣A) = I(A : B) − max_{Πi^a}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 ρ_AB satisfies D(B∣A) = 0 if and only if there exists a complete set of orthonormal measurement operators on A such that

ρ_AB = ∑_ip_i^aΠ_i^a ⊗ ρ_{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 ρ_AB, it is easy to construct the reduced density matrices ρ_A and ρ_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

{|u⟩ = cosθ|0⟩ + e^iϕsinθ|1⟩, |v⟩ = sinθ|0⟩ − e^iϕcosθ|1⟩}

and let cosθ ∈ [ − 1, 1] and ϕ ∈ [0, 2π) vary in small steps. For every choice of θ and ϕ, we project the experimental density matrix obtained by tomography along the orthonormal basis. The post-projection density matrix is

ρʹ = ∑_{i = u, v}Π_i^aρΠ_i^a = ∑_{i = u, v}p_i^aΠ_i^a ⊗ ρ_{B∣a = i}

with $p_{i}^{a} = {\rm Tr}[\Pi_{i}^{a}\rho]$. Discord is then obtained from the conditional density matrix ρ_{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 (θ, ϕ) grid.

Analytical method for the Bell-diagonal states:

As the name suggests, the Bell-diagonal states are diagonal in the Bell basis, given by

$$\vert \psi^{\pm}\rangle = \frac{1}{\sqrt{2}} ( \vert 01\rangle \pm \vert 10\rangle ) ~,~~ \vert \phi^{\pm}\rangle = \frac{1}{\sqrt{2}} ( \vert 00\rangle \pm \vert 11\rangle ).$$

The generic structure of a Bell-diagonal state is

ρ_BD = λ₁|ψ⁻⟩⟨ψ⁻| + λ₂|ϕ⁻⟩⟨ϕ⁻| + λ₃|ϕ⁺⟩⟨ϕ⁺| + λ₄|ψ⁺⟩⟨ψ⁺|.
With only local unitary operations (so as not to alter the correlations), all Bell-diagonal states can be transformed to the form

$$\rho_{BD} = \frac{1}{4} \Big( I_{d} + \sum_{j=1}^{3}r_{j}\sigma_{j}\otimes\sigma_{j} \Big) ~,$$
where the real numbers r_j are constrained so that all eigenvalues of ρ_BD remain in [0, 1]. The symmetric form of ρ_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 {VΠ_k^aV^†}, where Π_k^a = |k⟩⟨k| 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 $p_{0}=p_{1}=\frac{1}{2}$, and an analytical formula for the classical correlation,

$$\max_{\{\Pi_{k}^{a}\}} J(A:B) = \left(\frac{1-r}{2}\right)\log_{2}(1-r) + \left(\frac{1+r}{2}\right)\log_{2}(1+r) ~$$
with r = max{∣r₁∣, ∣r₂∣, ∣r₃∣}.

For the Bell-diagonal states, the reduced density matrices are ρ_A = ρ_B = I_d/2, and the total correlation is

I(A : B) = 2 + ∑_i = 1⁴λ_ilog₂λ_i

where the eigenvalues λ_i of ρ_BD are:

λ₁ = (1 − r₁ − r₂ − r₃)/4

λ₂ = (1 − r₁ + r₂ + r₃)/4

λ₃ = (1 + r₁ − r₂ + r₃)/4

λ₄ = (1 + r₁ + r₂ − r₃)/4

Thus the analytical formula for discord is,

$${D}_{BD}(B|A) = 2 + \sum_{i=1}^{4}\lambda_{i}\log_2\lambda_{i} - \left(\frac{1-r}{2}\right)\log_{2}(1-r) - \left(\frac{1+r}{2}\right)\log_{2}(1+r) ~.$$

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) = min_{χ ∈ Ω0}∥ρ − χ∥²
where Ω₀ represents the set of classical states, and $\|X-Y\|^2 = {\rm Tr}(X-Y)^2$ is the Hilbert-Schmidt quadratic norm. The two-qubit density matrix in the Bloch representation is

$$\rho = \frac{1}{4} \Big( I_{d} \otimes I_{d} + \sum_{i=1}^{3} x_{i}\sigma_{i}\otimes\otimes I_{d} + \sum_{i=1}^{3}y_{i} \otimes I_{d}\otimes\sigma_{i} + \sum_{i,j=1}^{3} T_{ij}\sigma_{i}\otimes\sigma_{j} \Big)$$

where x_i and y_i represent the Bloch vectors for the two qubits, and $T_{ij}^{}={\rm Tr}(\rho(\sigma_{i}\otimes\sigma_{j}))$ are the components of the correlation matrix. The geometric discord for such a state is

$$DG(B|A) = \frac{1}{4}(\|x\|^2 + \|T\|^2 - \eta_{\rm max})$$

where $\|T\|^2 = {\rm Tr}[T^{T}T]$, and $\eta_{\rm max}$ is the largest eigenvalue of x⃗x⃗^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