Complete condition for nonzero quantum correlation in continuous variable systems

Quantum correlation provides a promising measure beyond entanglement. Here, we propose a necessary and sufficient condition for nonzero quantum correlation in continuous variable systems, which is simple and easy to perform in terms of a marker $Q_r$. In order to get this condition, we introduce continuous-variable local orthogonal bases of the operator space, which are generalized from the orthogonal basis sets in local operator space for discrete variables. Based on this, we obtain the marker $Q_r$ for all bipartite continuous variable states, and provide several examples including two-mode Gaussian and non-Gaussian states. Our result may provide a candidate for quantum correlation measures, and can be measured by designed quantum circuits.

It is believed that entanglement is an essential resource in almost all quantum computation and communication tasks. However, there are still certain quantum computing tasks which display the quantum advantage without much entanglement. One typical example is the deterministic quantum computation with one qubit (DQC1) [1]. In order to quantify the quantum correlations beyond entanglement, quantum discord was introduced by Ollivier and Zurek [2] and independently by Henderson and Vedral [3]. Interestingly, the quantum discord was proposed as a figure of merit for characterizing the nonclassical resources present in the DQC1 [4]. Thereafter, the quantum discord has found numerous applications [5] in addition to its initial motivation in pointer states [2]. For examples, its role in open dynamics [6], observing the operational significance of discord consumption [7], quantum discord as resource for remote state preparation [8], the local broadcasting of the quantum correlations [9], and the phase transitions [10].
The quantum discord, which provides a measure for the quantum correlation, is defined as (1) where S(̺) = −Tr(̺ log 2 ̺) is the von Neumann entropy, ̺ A (̺ B ) is the reduced density matrix of subsystem A (B), and the minimum is taken over all possible von Neumann measurements [2] or all possible positive operator-valued measures [3] {E k } on subsystem A with p k = Tr(E k ⊗ ½̺) and ̺ B|k = Tr A (E k ⊗ ½̺)/p k . These two definitions with two kinds of measurements produce in general different values but coincide in the case of zero quantum discord states where {|i A } is a set of orthonormal basis for subsystem A. Furthermore, the quantum discord has been generalized to continuous variable systems using Gaussian measurements for two-mode Gaussian states [11,12]. Since the quantum discord is an important measure of quantum correlations, it becomes a fundamental issue to decide whether a given state contains nonzero quantum discord or not for both discrete variables and continuous variables. For a given bipartite state in a finite-dimensional system, several conditions for nonzero quantum discord have been proposed [13][14][15][16][17] in the form of local commutativity, strong positive partial transpose, etc. In particular, Dakić, Vedral and Brukner [15] have proposed a necessary and sufficient condition for nonzero quantum discord in bipartite finite-dimensional system. Subsequently, Ref. [16] generalized the condition to continuous variable system. However, this condition in general requires calculating an infinite number of commutators for continuous variable states (since for an arbitrary d × d state there are d 2 (d 2 − 1)/2 commutation relations to be checked), which cannot be implemented efficiently. Therefore, the necessary and sufficient condition for nonzero quantum discord of bipartite continuous variable states still remains unsettled.
In this Letter we derive a necessary and sufficient criterion for nonvanishing quantum discord in continuous variable systems in terms of an expression Q designated as a marker. Unlike checking an infinite number of commutators as in Refs. [15,16], our criterion is simple and easy to perform. In order to obtain the condition, we introduce the complete orthogonal basis sets in local Hilbert-Schmidt spaces of operators for continuous variables. Based on this condition, we derive an explicit expression Q for all the two-mode states, and present examples including all the two-mode Gaussian states and several non-Gaussian states. Moreover, the criterion provides a candidate for quantum correlation measures, and it can be efficiently measured by a designed quantum cir-cuit.
Continuous-variable local operator basis.-In discrete variable systems, take a d × d system as an example, for each subsystem a complete set of orthogonal bases of the operator space {G k } consists of d 2 operators (which are not necessarily Hermitian) satisfying Tr(G † k G l ) = δ kl and ̺ = d 2 k=1 Tr(̺G k )G † k . The local orthogonal operator basis has been widely used in many problems of discrete variables, such as entanglement detection [18], necessary and sufficient condition for nonzero quantum discord [15], and quantifying quantum uncertainty based on skew information [19].
Consider a two-mode continuous-variable state, for each subsystem a complete set of orthogonal bases of the operator space contains an infinite number of operators G(λ) of this subsystem satisfying orthogonal relations and complete-set condition ̺ = G(λ) ̺ G † (λ)d 2 λ, where the index λ is an arbitrary complex number, d 2 λ := d Re(λ)d Im(λ), and δ (2) (λ − λ ′ ) := δ(Re(λ − λ ′ ))δ(Im(λ−λ ′ )) with Re(z) and Im(z) being the real and imaginary part of the complex number z, respectively. All the integrals throughout the paper are from −∞ to +∞. There are infinite complete sets of continuousvariable operator bases. For later use, we introduce a typical complete set of bases, where D(λ) is the Weyl displacement operator defined by D(λ) = exp(λâ † − λ * â ). Necessary and sufficient condition.-Consider a twomode continuous-variable state ̺ AB , let us choose the complete set of operator basis {G(λ)} in subsystem B. The density matrix ̺ AB of the two-mode system has a partial expression (2), one can show that the state ̺ AB is of zero discord if and only if there exists a von Neumann measurement {Π i = |i i|} such that Therefore, the necessary and sufficient condition becomes These infinite conditions can merge into a single condition: Eq. (5) is satisfied if and only if the expression Q = 0 where since on the one hand [̺ A (λ), ̺ A (λ ′ )] = 0 for all possible λ and λ ′ leads directly to Q = 0, and on the other hand For an arbitrary two-mode state ̺ AB , its characteristic function is defined as the expectation value of the two-mode Weyl displacement operator χ(λ 1 , λ 2 ) = Tr[̺ AB D 1 (λ 1 )D 2 (λ 2 )], and its Wigner function is defined as the Fourier transform of the characteristic function W (α 1 , Conversely, one can express ̺ AB using its Wigner function: Based on this, we arrive at the expression of Q using the Wigner function: quantum discord for all two-mode Gaussian states and several non-Gaussian states in the following sections. Two-mode Gaussian states.-Let us define the position and momentum operators asx = (â +â † )/2 andp = −i(â −â † )/2, respectively. The Wigner function of the two-mode Gaussian state is [20]: where the four-dimensional vector ξ has the quadrature pairs of all two-modes as its components ξ = ( There is a standard form for the covariance matrix V of two-mode Gaussian state, where A = diag{a, a}, B = diag{b, b} and C = diag{c 1 , c 2 } with a, b ≥ 1/4 and ab ≥ c 2 1 , c 2 2 . Under this standard form, it can be obtained that Therefore, for an arbitrary two-mode Gaussian state, it is easy to check whether or not it contains nonzero quantum discord using Eq. (9).
As an example of two-mode Gaussian states, let us consider the covariance matrix of two-mode symmetrical squeezed thermal state given by a = b = [(1 + 2n) cosh 2r]/4, c 1 = −c 2 = [(1 + 2n) sinh 2r]/4 with r and n being the squeezing parameter and average photon number, respectively. For n = 0, it is reduced to a pure two-mode squeezed vacuum state. Using Eq. (9), one can derive Q = 2 sinh 2r tanh 2r/[(1 + 2n) 3 (1 + 2n + 2n 2 )(3 + 8n + 8n 2 + cosh 4r)] for the two-mode squeezed thermal states. The covariance matrix of two-mode squeezed thermal state has the property c 1 = −c 2 . Actually, for all the two-mode Gaussian states with covariance matrix satisfying c 2 1 = c 2 2 ≡ c 2 , the expression of Q in Eq. When will Eq. (9) be exactly equal to zero for a general two-mode Gaussian state? In order to get Q = 0 one needs to check whether f : There are two cases: It means g is a monotonic decreasing function of χ 1 and χ 2 . The maximal value of g is ab 3 /16 when χ 1 = χ 2 = 0. Thus, f ≤ 0 holds.
Therefore, it can be concluded that f ≤ 0 holds for both cases and f = 0 if and only if χ 1 = χ 2 = 0, i.e., c 1 = c 2 = 0. Two-mode non-Gaussian states.-In general, one can also use Eq. (7) to detect the quantum discord of an arbitrary two-mode non-Gaussian state. Since non-Gaussian states have no standard form as Gaussian states, we show several examples to demonstrate how to detect quantum discord of non-Gaussian states based on Eq. (7).
As the first example, consider the photon-number mixed state ̺ = k|00 00| +k| + 1 +1| withk = 1 − k, |+ = (|0 + |1 )/ √ 2 and 0 ≤ k ≤ 1. This state is a simple non-Gaussian state in a finite dimension. Based on the definition of Wigner function, one can obtain its Wigner function as Wigner function of |00 , α i = x i + ip i and x i , p i are real parameters. After some algebra, one can get Q = k 2k2 /2 using Eq. (7). Therefore, this photon-number mixed state is of zero-discord if and only if Q = 0, i.e., k = 0 or 1. Actually, this finite-dimensional state can also be detected by the condition shown in Ref. [15]. We present it here to show that our necessary and sufficient condition is not only suitable for continuous-variable system but also for finite-dimensional system.
The second example of two-mode non-Gaussian state is a convex combination of vacuum state and the Gaussian state with covariance matrix satisfying c 2 1 = c 2 2 ≡ c 2 , i.e., ̺ = k̺ G +k|00 00| with 0 ≤ k ≤ 1. ̺ G is the two-mode Gaussian state satisfying c 2 1 = c 2 2 with its Wigner function having been previously introduced, and |00 is vacuum state with the Wigner function being W 0 (α 1 , α 2 ). In general, the convex combination of two Gaussian states may not be a Gaussian state. Based on Eq. (7), one can obtain that  1: (color online). Logarithm of explicit value Q with respect to base 10 versus parameters n and r for the single-photon-added two-mode symmetrical squeezed thermal states. All the states in the range except the ones with r = 0 can be detected as nonzero quantum discord states by the necessary and sufficient condition.
Discussions.-From all the examples of Gaussian states and non-Gaussian states above, one can see that the two terms in the integrand of Eq. (7) can always be analytically calculated since for Gaussian states they are Gaussian integrals like exp(−ax 2 + bx + c)dx with a > 0, and for non-Gaussian states they can be expressed as integrals of Gaussian function with a polynomial factor like exp(−ax 2 + bx + c)f (x)dx where f (x) = n k=0 a k x k and a > 0. Based on the identities +∞ −∞ exp(−ax 2 + bx + c)dx = π/a exp[b 2 /(4a) + c] and +∞ −∞ x n exp(−ax 2 + bx+ c)dx = π/a exp[b 2 /(4a)+ c] ⌊n/2⌋ k=0 n!b n−2k (2a) k−n /[2 k k!(n − 2k)!] where a > 0, n is a positive integer and the floor function ⌊x⌋ is the largest integer not greater than x, the analytical results of all the integrals can be obtained. On the other hand, the integral Eq. (7) can always be calculated since −Q is actually equal to the expectation value of the quantum discord witness shown in Ref. [22], which is always a real number.
Actually, Q is non-negative for all the two-mode states, since the integrand in the integral over λ and λ ′ in Eq. (6) is nonnegative. From this point of view, Q can be regarded as a candidate for quantum correlation measures, because it is always nonnegative, and Q = 0 if and only if the state is of zero quantum discord. If a zero-discord state σ 0 has experimental imperfections, for example, it mixes with an arbitrarily small amount of noise, i.e., σ ′ 0 = (σ 0 + ǫ̺)/(1 + ǫ) where ǫ is an arbitrarily small quantity and ̺ is a nonzero-discord state. One can see that Q(σ ′ 0 ) ∼ O(ǫ), that is to say Q(σ ′ 0 ) is also arbitrarily small, which means σ ′ 0 has arbitrarily small amount of quantum discord. In that sense, the marker Q is robust with respect to experimental imperfections. Furthermore, the marker Q can be rewritten as Q(̺) = −Tr(W ̺ ⊗4 ) where W is the quantum discord witness shown in Ref. [22]. Therefore, for an experimentally unknown state, Q can be easily measured by a designed quantum circuit in Ref. [22], since Q = σ 1 x − σ 2 x with σ i x being the local σ x -measurement on the i-th auxiliary qubit.
To summarize, we have derived a necessary and sufficient criterion of nonvanishing quantum discord for arbitrary two-mode states in continuous variable systems in terms of a marker Q. This criterion is simple and easy to perform without checking an infinite number of commutators. We have introduced the complete orthogonal basis sets in local Hilbert-Schmidt spaces of operators for continuous variables. The criterion provides a candidate for quantum correlation measures, and can be efficiently measured by a designed quantum circuit. This work is supported by the National Research Foundation and Ministry of Education, Singapore (Grant No. WBS: R-710-000-008-271), the National Natural Science Foundation of China (Grant No. 11075227).