R Muthuganesan, V K Chandrasekar,Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur—613 401, Tamil Nadu, India
Abstract Measurement-induced nonlocality (MIN), a quantum correlation measure for bipartite systems, is an indicator of maximal global effects due to locally invariant von Neumann projective measurements. It is originally defined as the maximal square of the Hilbert–Schmidt norm of the difference between pre- and post-measurement states. In this article, we propose a new form of MIN based on affinity. This quantity satisfies all the criteria of a bona fide measure of quantum correlation measures. This quantity is evaluated for both arbitrary pure and 2×n dimensional (qubit–qudit) mixed states. The operational meaning of the proposed quantity is interpreted in terms of the interferometric power of the quantum state. We apply these results on two-qubit mixed states, such as the Werner, isotropic and Bell diagonal states. Keywords：affinity;nonlocality;quantum correlation;local measurements
Quantum correlation, a fundamental aspect of quantum mechanics, significantly makes the departure from the classical regime. It is a useful physical resource for various types of quantum information processing, such as teleportation, super dense coding, communication and quantum algorithms [1]. Quantification of correlation between its local constituents of a system is a formidable task within the framework of quantum information theory. In this regard, entanglement has been believed to be a valid and advantageous resource for quantum information theory since the early 19th century [2, 3]. In light of the seminal work of Werner [4] and the presence of non-zero quantum correlation, namely, discord (beyond entanglement) [5], it is believed that entanglement alone is not responsible for the advantages over the classical algorithm. Recently, various measures have been proposed to capture quantumness which go beyond entanglement, such as measurement-induced disturbance (MID) [6], geometric discord (GD) [7, 8], measurement-induced nonlocality (MIN) [9] and uncertainty-induced nonlocality (UIN) [10].
Local measurements or operations are important tools for probing nonlocal aspects of a quantum state. In particular, locally invariant von Neumann projective measurements can induce global or nonlocal effects. Using local eigenprojectors of marginal states various measures have been studied extensively [6, 8–10]. The distance between quantum states in state space is useful in quantification of quantum correlation. Geometric measures are easy to compute and experimentally realizable [7]. In this context, several distance measures have been introduced; a few such measures are trace distance, the Hilbert–Schmidt norm, the Jensen–Shannon divergence, the fidelity-induced metric and the Hellinger distance [11, 12]. Recently, measurement-based nonclassical correlations (a family of discord-like measures) have been characterized using various distance measures [13].
MIN, which characterizes the nonlocality of a quantum state from the perspective of locally invariant projective measurements, is more general than the Bell nonlocality. It is also a bona fide measure of quantum correlation for the bipartite state. This quantity provides a novel classification scheme for bipartite states, and is a quantum resource quite different from entanglement. It may be useful in the quantitative study of quantum state steering [14, 15], remote state control, general quantum dense coding and cryptography [16–22]. MIN has also been investigated based on relative entropy [23], von Neumann entropy, skew information [24], trace distance [13, 25] and fidelity [26]. In this article, we introduce a new variant of MIN based on affinity. It is shown that this quantity is a remedy for the local ancilla problem of the Hilbert–Schmidt norm based MIN. This measure satisfies all the criteria of a bona fide measure of quantum correlation of a bipartite system. Affinity-based MIN is a valid resource for quantum communication, cryptography and dense coding. Further, we evaluate the affinity-based MIN analytically for an arbitrary pure state. We obtain an upper bound for an m×n dimensional mixed state, and a closed formula for the proposed version of MIN for a 2×n dimensional mixed state. We study the quantumness of a well-known family of two-qubit m×m dimensional mixed states. Finally, we study the dynamical behavior of affinity-based measure under a noisy quantum channel.
2. Measurement-induced nonlocality
Measurement-induced nonlocality, which captures nonlocal or global effects of a quantum state due to von Neumann projective measurements, is originally defined as the maximal square of the Hilbert–Schmidt norm of the difference between pre- and post-measurement states. Mathematically, it is defined as [9]$ \begin{eqnarray}N(\rho )={\,}_{{{\rm{\Pi }}}^{A}}^{\max }\parallel \rho -{{\rm{\Pi }}}^{A}(\rho ){\parallel }^{2},\end{eqnarray}$here, the maximum is taken over the von Neumann projective measurements on subsystem A, ${{\rm{\Pi }}}^{A}(\rho )={\sum }_{k}({{\rm{\Pi }}}_{k}^{A}\otimes {{\mathbb{1}}}^{B})\rho ({{\rm{\Pi }}}_{k}^{A}\,\otimes {{\mathbb{1}}}^{B})$, and ${{\rm{\Pi }}}^{A}=\{{{\rm{\Pi }}}_{k}^{A}\}=\{| k\rangle \langle k| \}$ being the projective measurements on the subsystem A, which do not change the marginal state ρA locally, i.e. ΠA(ρA)=ρA. The dual of this quantity is the GD of the given state ρ formulated as [8]$ \begin{eqnarray}D(\rho )={\,}_{{{\rm{\Pi }}}^{A}}^{\min }\parallel \rho -{{\rm{\Pi }}}^{A}(\rho ){\parallel }^{2}.\end{eqnarray}$For nondegenerate ρA, the optimization is not required. Hence, both MIN and GD are equal. Due to the computability and easy accessibility of experimentation, researchers have paid considerable attention to the both GD and MIN. However, both quantities have an unwanted property of quantum correlation measures [27]; it may change rather arbitrarily through some trivial and uncorrelated action on the unmeasured party B.
Consider a simple mapping ${{\rm{\Gamma }}}^{\sigma }:X\to X\otimes \sigma $, i.e. the map adding a noisy ancillary state to the party B. Under such an operation$ \begin{eqnarray}\parallel X\parallel \to \parallel {{\rm{\Gamma }}}^{\sigma }X\parallel =\parallel X\parallel \sqrt{\mathrm{Tr}{\sigma }^{2}}.\end{eqnarray}$Since the Hilbert–Schmidt norm is multiplicative on tensor products. After the addition of local ancilla ρC, the MIN of the resultant state is computed as $ \begin{eqnarray*}N({\rho }^{A:{BC}})=N({\rho }^{{AB}})\mathrm{Tr}{\left({\rho }^{C}\right)}^{2},\end{eqnarray*}$ implying that MIN differs arbitrarily due to local ancilla C as long as ρC is mixed, since the optimization over the projective measurements on A is unaffected by the presence of the uncorrelated ancilla state on B (unmeasured party). Thus, adding or removing local ancilla is a local and reversible operation. Due to this operation, a factor, namely purity of the ancillary state, is added to the original MIN.
A natural way to resolve this local ancilla problem is to redefine the MIN equation (1) as [27]$ \begin{eqnarray}\tilde{N}(\rho )={\,}_{{{\rm{\Lambda }}}_{B}}^{\max }\,\,N{\left({\rho }^{{AB}}\right)}_{{{\rm{\Lambda }}}_{B}},\end{eqnarray}$where the maximum is over the channel ΛB. On the other hand, Luo and Fu remedied this local ancilla problem by replacing the density matrix by its square root [28]. Further, the contractive distance measures such as trace distance, the Hellinger distance and the fidelity-induced metric are also useful for resolving this local ancilla problem. In what follows, we define a new variant of MIN using an affinity-induced metric.
3. α-affinity and MIN
Metrics in state space, which quantify the closeness of or similarity between two states in the state space, play a central role in the classification of states and the quantification of resources [29] in information theory. They are also associated with geometric measures. Further, they are useful for quantifying how precisely a quantum channel can transmit information. Affinity, like fidelity [12], characterizes the closeness of two quantum states. Here, we define a metric in state space and introduce a new quantum correlation measure using the affinity between pre- and post-measurement states. Classically, affinity is defined as [30]$ \begin{eqnarray}{ \mathcal A }(g,h)=\sum _{x}(\sqrt{g(x)}\sqrt{h(x)}),\end{eqnarray}$where g and h are discrete probability distributions. This definition is like the Bhattacharyya coefficient between two probability distributions (discrete or continuous) in classical probability theory [31]. Classical affinity quantifies the closeness of two probability distributions. Extending the same notion in the quantum regime, one can replace the probability distribution with the density matrix and the summation by the trace operator. Hence, the affinity of two quantum states is defined as$ \begin{eqnarray}{ \mathcal A }(\rho ,\sigma )=\mathrm{Tr}\left(\sqrt{\rho }\sqrt{\sigma }\right).\end{eqnarray}$Quantum affinity, similar to fidelity [12], describes how close two quantum states are. It also possess all the properties of fidelity. This quantity is more useful in quantum detection and estimation theory. The notion of affinity has been extended to α-affinity (0<α<1), which is defined as$ \begin{eqnarray}{{ \mathcal A }}_{\alpha }(\rho ,\sigma )=\mathrm{Tr}({\rho }^{\alpha }{\sigma }^{1-\alpha })\end{eqnarray}$with $\alpha \in (0,1)$. The α-affinity satisfies the following properties:α-affinity is bounded i.e. $0\leqslant {{ \mathcal A }}_{\alpha }(\rho ,\sigma )\leqslant 1$ and ${{ \mathcal A }}_{\alpha }(\rho ,\sigma )=1$ if, and only if, ρ=σ for all values of α. Monotonic $({{ \mathcal A }}_{\alpha }({\rm{\Phi }}(\rho ),{\rm{\Phi }}(\sigma ))\geqslant {{ \mathcal A }}_{\alpha }(\rho ,\sigma ))$ under a completely positive and trace preserving (CPTP) map. Joint concavity ${{ \mathcal A }}_{\alpha }({\sum }_{i}{p}_{i}{\rho }_{i},{\sum }_{i}{p}_{i}{\sigma }_{i})\geqslant {\sum }_{i}{p}_{i}{{ \mathcal A }}_{\alpha }({\rho }_{i},{\sigma }_{i})$.
In general, affinity itself is not a metric. Due to monotonicity and concavity properties of affinity [30], one can define any monotonically decreasing function of affinity as a metric in state space. One such affinity-based metric is$ \begin{eqnarray}{d}_{{{ \mathcal A }}_{\alpha }}(\rho ,\sigma )=\sqrt{1-{{ \mathcal A }}_{\alpha }(\rho ,\sigma )}.\end{eqnarray}$Further, it is easy to show that the above-defined metric satisfies all the axioms of a valid distance measure in state space. This metric is also useful in quantifications in quantum resources, such as entanglement [30], nonclassical correlation [32] and quantum coherence. To define affinity-based MIN, we set α=1/2. For simplicity, from here onwards we drop the subscript α and we denote the 1/2−affinity $({{ \mathcal A }}_{1/2})$ as ${ \mathcal A }$. Defining MIN in terms of affinity using the above-mentioned metric as$ \begin{eqnarray}\begin{array}{rcl}{N}_{{ \mathcal A }}(\rho ) & = & {\,}_{{{\rm{\Pi }}}^{A}}^{\max }\,{d}_{{ \mathcal A }}^{2}(\rho ,{{\rm{\Pi }}}^{A}(\rho ))\\ & = & 1-{\,}_{{{\rm{\Pi }}}^{A}}^{\min }\,\mathrm{Tr}\left(\sqrt{\rho }\sqrt{{{\rm{\Pi }}}^{A}(\rho )}\right),\end{array}\end{eqnarray}$where the maximum/minimum is taken over von Neumann projective measurements. In principle, one can generalize this definition for the multipartite scenario. Using the identity ${{\rm{\Pi }}}^{A}f(\rho ){{\rm{\Pi }}}^{A}\,=\,f({{\rm{\Pi }}}^{A}\rho {{\rm{\Pi }}}^{A})$ [33], one can rewrite the definition of MIN using affinity as$ \begin{eqnarray}{N}_{{ \mathcal A }}(\rho )=\,1{-}_{{{\rm{\Pi }}}^{A}}^{\min }\mathrm{Tr}\left[\sqrt{\rho }{{\rm{\Pi }}}^{A}(\sqrt{\rho })\right].\end{eqnarray}$It is worth mentioning that this quantity satisfies all the necessary axioms of quantum correlation measures. Here, we demonstrate some interesting properties of affinity-based MIN:${N}_{{ \mathcal A }}(\rho )$ is non-negative, i.e. ${N}_{{ \mathcal A }}(\rho )\geqslant 0$. ${N}_{{ \mathcal A }}(\rho )=0$ for any product state ρ=ρA⨂ρB and the classical quantum state in the form $\rho ={\sum }_{k}{p}_{k}| k\rangle \langle k| \otimes {\rho }_{k}$ with the nondegenerate marginal state ${\rho }^{A}={\sum }_{k}{p}_{k}| k\rangle \langle k| $. ${N}_{{ \mathcal A }}(\rho )$ is locally unitary invariant, i.e. ${N}_{{ \mathcal A }}(U\otimes V)\rho {(U\otimes V)}^{\dagger })={N}_{{ \mathcal A }}(\rho )$ for any local unitary operators U and V. For any m×n dimensional pure maximally entangled state with m≤n, ${N}_{{ \mathcal A }}(\rho )$ has the maximal value of $\tfrac{m-1}{m}$. For nondegenerate ρA, the affinity-based MIN ${N}_{{ \mathcal A }}(\rho )\,={d}_{{ \mathcal A }}^{2}(\rho ,{{\rm{\Pi }}}^{A}(\rho ))$. ${N}_{{ \mathcal A }}(\rho )$ is invariant under the addition of any local ancilla to the unmeasured party. To prove this invariant property, we show that affinity is unaltered by the addition of uncorrelated ancilla to the unmeasured party B. First, we recall the multiplicative property of affinity and this is given as$ \begin{eqnarray}\begin{array}{rcl}{ \mathcal A }({\rho }_{1}\otimes {\rho }_{2},{\sigma }_{1}\otimes {\sigma }_{2}) & = & { \mathcal A }({\rho }_{1}\otimes {\sigma }_{1})\\ & & \cdot \,{ \mathcal A }({\rho }_{2}\otimes {\sigma }_{2}).\end{array}\end{eqnarray}$After the addition of local ancilla to the unmeasured party B, the affinity between the pre- and post-measurement states is $ \begin{eqnarray*}\begin{array}{c}\begin{array}{l}{ \mathcal A }\left({\rho }^{A:{BC}},{{\rm{\Pi }}}^{A}({\rho }^{A:{BC}})\right)\\ \,=\,{ \mathcal A }\left({\rho }^{{AB}}\otimes {\rho }^{C},{{\rm{\Pi }}}^{A}({\rho }^{{AB}})\otimes {\rho }^{C}\right).\end{array}\end{array}\end{eqnarray*}$ Using the multiplicativity property of the affinity equation (11), we have$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{ \mathcal A }\left({\rho }^{A:{BC}},{{\rm{\Pi }}}^{A}({\rho }^{A:{BC}}\right)\\ \,=\,{ \mathcal A }\left({\rho }^{{AB}},{{\rm{\Pi }}}^{A}({\rho }^{{AB}}\right)\cdot { \mathcal A }({\rho }^{C},{\rho }^{C}),\\ \,=\,{ \mathcal A }\left({\rho }^{{AB}},{{\rm{\Pi }}}^{A}({\rho }^{{AB}})\right),\end{array}\end{array}\end{eqnarray}$which completes the proof of property (vi). Hence, ${N}_{{ \mathcal A }}(\rho )$ is a good measure of nonlocal correlation or quantumness in a given system.
4. MIN for the pure state
for any pure bipartite state with Schmidt decomposition $| {\rm{\Psi }}\rangle ={\sum }_{i}\sqrt{{s}_{i}}| {\alpha }_{i}\rangle \otimes | {\beta }_{i}\rangle $,$ \begin{eqnarray}{N}_{{ \mathcal A }}(| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| )=1-\sum _{k}{s}_{k}^{2}.\end{eqnarray}$
Noting that $ \begin{eqnarray*}\sqrt{\rho }=| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| =\sum _{{ij}}\sqrt{{s}_{i}{s}_{j}}| {\alpha }_{i}\rangle \langle {\alpha }_{j}| \otimes | {\beta }_{i}\rangle \langle {\beta }_{j}| .\end{eqnarray*}$ The local projective measurements on the subsystem A can be expressed as ${{\rm{\Pi }}}^{A}=\{{{\rm{\Pi }}}_{k}^{A}\otimes {\mathbb{1}}\}=\{| {\alpha }_{k}\rangle \langle {\alpha }_{k}| \otimes | {\mathbb{1}}\}$, which do not alter the marginal state ρA. The marginal state ${\rho }^{A}={{\rm{\Pi }}}^{A}({\rho }^{A})={\sum }_{k}{{\rm{\Pi }}}_{k}^{A}{\rho }^{A}{{\rm{\Pi }}}_{k}^{A}$ is written as $ \begin{eqnarray*}{\rho }^{A}=\sum _{k}U| {\alpha }_{k}\rangle \langle {\alpha }_{k}| {U}^{\dagger }{\rho }^{A}U| {\alpha }_{k}\rangle \langle {\alpha }_{k}| {U}^{\dagger }.\end{eqnarray*}$ The spectral decomposition of the marginal state ρA in the orthonormal bases $\{U| {\alpha }_{k}\rangle \}$ can be written as $ \begin{eqnarray*}{\rho }^{A}=\sum _{k}\langle {\alpha }_{k}| {U}^{\dagger }{\rho }^{A}U| {\alpha }_{k}\rangle U| {\alpha }_{k}\rangle \langle {\alpha }_{k}| {U}^{\dagger },\end{eqnarray*}$ where $\langle {\alpha }_{k}| {U}^{\dagger }{\rho }^{A}U| {\alpha }_{k}\rangle ={s}_{k}$ are the eigenvalues of state ρA. After a straightforward calculation and simplification, we show that$ \begin{eqnarray}\begin{array}{l}\sum _{k}\mathrm{Tr}[\sqrt{\rho }({{\rm{\Pi }}}_{k}^{A}\otimes {\mathbb{1}})\sqrt{\rho }({{\rm{\Pi }}}_{k}^{A}\otimes {\mathbb{1}})]\\ \quad =\sum _{k}{\left(\langle {\alpha }_{k}| {U}^{\dagger }{\rho }^{A}U| {\alpha }_{k}\rangle \right)}^{2}=\sum _{k}{s}_{k}^{2}.\end{array}\end{eqnarray}$Substituting equation (10) into equation (14), we obtain the affinity-based GD for the pure state as,$ \begin{eqnarray}{N}_{{ \mathcal A }}(| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| )=1-\sum _{k}{s}_{k}^{2}\end{eqnarray}$and hence the theorem is proved. It is worth mentioning that for a pure state, the affinity-based measure is equal to earlier quantities such as skew information, Hilbert–Schmidt norm based MINs, geometric measure of entanglement and the remedied version of MIN. For a pure m×n dimensional entangled state with m≤n, the quantity ${\sum }_{k}{s}_{k}^{2}$ is bounded by 1/m. Then,$ \begin{eqnarray}{N}_{{ \mathcal A }}\left(| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| \right)\leqslant \displaystyle \frac{m-1}{m}\end{eqnarray}$and the equality holds for the maximally entangled state.
5. MIN for the mixed state
Let $\{{X}_{i}:i=0,1,2,\cdots ,{m}^{2}-1\}\in { \mathcal L }({{ \mathcal H }}^{A})$ be a set of orthonormal operators for the state space ${{ \mathcal H }}^{A}$ with the operator inner product $\langle {X}_{i}| {X}_{j}\rangle =\mathrm{Tr}({X}_{i}^{\dagger }{X}_{j})$. Similarly, one can define $\{{Y}_{j}:j=0,1,2,\cdots ,{n}^{2}-1\}\in { \mathcal L }({{ \mathcal H }}^{B})$ for the state space ${{ \mathcal H }}^{B}$. The operators Xi and Yj are satisfying the conditions $\mathrm{Tr}({X}_{k}^{\dagger }{X}_{l})=\mathrm{Tr}({Y}_{k}^{\dagger }{Y}_{l})={\delta }_{{kl}}$. With this, one can construct a set of orthonormal operators $\{{X}_{i}\otimes {Y}_{j}\}\in { \mathcal L }({{ \mathcal H }}^{A}\otimes {{ \mathcal H }}^{B})$ for the composite system. Consequently, an arbitrary m×n dimensional state of a bipartite composite system can be written as$ \begin{eqnarray}\sqrt{\rho }=\sum _{i,j}{\gamma }_{{ij}}{X}_{i}\otimes {Y}_{j},\end{eqnarray}$where ${\gamma }_{{ij}}=\mathrm{Tr}(\sqrt{\rho }\,{X}_{i}\otimes {Y}_{j})$ are real elements of the matrix Γ. After straightforward algebra, one can compute the affinity between the pre- and post-measurement state as$ \begin{eqnarray}{ \mathcal A }(\rho ,{{\rm{\Pi }}}^{A}(\rho ))=\mathrm{Tr}(R{\rm{\Gamma }}{{\rm{\Gamma }}}^{t}{R}^{t}).\end{eqnarray}$The affinity-based MIN is$ \begin{eqnarray}{N}_{{ \mathcal A }}(\rho )=1-{\,}_{R}^{\min }\,\mathrm{Tr}(R{\rm{\Gamma }}{{\rm{\Gamma }}}^{t}{R}^{t}),\end{eqnarray}$where the matrix $R=({r}_{{ki}})=\mathrm{Tr}(| k\rangle \langle k| {X}_{i})$. Now we have, $ \begin{eqnarray*}\sum _{i=0}^{{m}^{2}-1}{r}_{{ki}}{r}_{{k}^{{\prime} }i}=\mathrm{Tr}\left(| k\rangle \langle k| {k}^{{\prime} }\rangle \langle {k}^{{\prime} }| \right)={\delta }_{{{kk}}^{{\prime} }}\end{eqnarray*}$with ${r}_{k0}=1/\sqrt{m}$. For $k={k}^{{\prime} }$$ \begin{eqnarray}\sum _{i=1}^{{m}^{2}-1}{r}_{{ki}}^{2}=\displaystyle \frac{m-1}{m}\end{eqnarray}$and for $k\ne {k}^{{\prime} }$$ \begin{eqnarray}\sum _{i=1}^{{m}^{2}-1}{r}_{{ki}}{r}_{{k}^{{\prime} }i}=-\displaystyle \frac{1}{m}.\end{eqnarray}$From equations (20) and (21) we can write the matrix RRt as $ \begin{eqnarray*}{{RR}}^{t}=\displaystyle \frac{1}{m}\left(\begin{array}{cccc}m-1 & -1 & \cdots & -1\\ -1 & m-1 & \cdots & -1\\ \vdots & \vdots & \ddots & \vdots \\ -1 & -1 & \cdots & m-1\end{array}\right),\end{eqnarray*}$ which is a square matrix of order m with eigenvalues 0 and 1 (with multiplicity of m−1). For this symmetric matrix, we have the similarity transformation RRt=U D Ut with the real unitary operator U and diagonal matrix D. Now, constructing the m×m2 matrix B as $ \begin{eqnarray*}B={U}^{t}R=\left(\begin{array}{c}{R}_{0}\\ 0\end{array}\right),\end{eqnarray*}$ where R0 is a $(m-1)\times {m}^{2}$ matrix, such that ${R}_{0}{R}_{0}^{t}={{\mathbb{1}}}_{m-1}$ and we have, $ \begin{eqnarray*}{}_{R}^{\min }\,\mathrm{Tr}\,(R{\rm{\Gamma }}{{\rm{\Gamma }}}^{t}{R}^{t})\,={\,}_{{R}_{0}}^{\min }\,\mathrm{Tr}\,({R}_{0}{\rm{\Gamma }}{{\rm{\Gamma }}}^{t}{R}_{0}^{t}).\end{eqnarray*}$ Then$ \begin{eqnarray}{N}_{{ \mathcal A }}(\rho )=\,1-{\,}_{{R}_{0}}^{\min }\,\mathrm{Tr}({R}_{0}{\rm{\Gamma }}{{\rm{\Gamma }}}^{t}{R}_{0}^{t}).\end{eqnarray}$Since $ \begin{eqnarray*}{}_{{R}_{0}:{R}_{0}{R}_{0}^{t}=\,{{\mathbb{1}}}_{m-1}}^{\min }\mathrm{Tr}({R}_{0}{\rm{\Gamma }}{{\rm{\Gamma }}}^{t}{R}_{0}^{t})=\sum _{i=1}^{m-1}{\mu }_{i},\end{eqnarray*}$ where μi are eigenvalues of the matrix ΓΓt listed in increasing order, we have the following tight upper bound for affinity-based MIN as$ \begin{eqnarray}{N}_{{ \mathcal A }}(\rho )\leqslant 1-\sum _{i=m}^{{m}^{2}-1}{\mu }_{i}.\end{eqnarray}$
Next, we compute the closed formula of MIN for a 2×n dimensional state. The projective measurement operator ${{\rm{\Pi }}}_{k}^{A}$ can be expressed in Bloch sphere representation,$ \begin{eqnarray}{{\rm{\Pi }}}_{k}^{A}=\displaystyle \frac{1}{2}\left({{\mathbb{1}}}_{2}+\vec{r}\cdot \vec{\sigma }\right),\end{eqnarray}$where ${{\mathbb{1}}}_{2}$ is a 2×2 unit matrix $\vec{r}\cdot \vec{\sigma }={\sum }_{i=1}^{3}{r}_{k}^{i}{\sigma }_{i}$ with ${\sum }_{i=1}^{3}{\left({r}_{k}^{i}\right)}^{2}=1$. Substituting equation (24) into equation (10), one can arrive at$ \begin{eqnarray}{N}_{{ \mathcal A }}(\rho )=1{-}_{{r}_{k}^{i}}^{\min }\,\sum _{i,j}{r}_{k}^{i}{T}_{{ij}}{r}_{k}^{j}=1-{\lambda }_{\min },\end{eqnarray}$where λmin is the minimal eigenvalue matrix T with matrix elements ${T}_{{ij}}=\mathrm{Tr}\left[\sqrt{\rho }({\sigma }_{i}\otimes {\mathbb{1}})\sqrt{\rho }({\sigma }_{j}\otimes {\mathbb{1}})\right]$. Therefore, ${N}_{{ \mathcal A }}(\rho )$ can be analytically solved for any qubit–qudit states, which is different from other nonclassical correlations such as quantum discord, which has no analytical formula, even for two-qubit states. Interestingly, for this qubit–qudit case, the quantum correlation happens to be the local quantum uncertainty as the minimum skew information achievable on a single local measurement.
It is important to note the operational meaning of the proposed measure. The interferometric power of a quantum state is defined as [34]$ \begin{eqnarray}{IP}({\rho }^{{AB}})={\,}_{| k \rangle \langle k| \otimes {\mathbb{1}}}^{\,min\,}F({\rho }^{{AB}},H\otimes {\mathbb{1}})\end{eqnarray}$by considering an interferometric setup and minimizing the quantum Fisher information over all the possible generators of a phase rotation on one party. Here, F(.) is quantum Fisher information (QFI) defined via symmetric logarithm derivatives, and the minimum is intended over all fixed observables H with a nondegenerate spectrum { μi} . Considering the spectral decomposition of the bipartite quantum state ${\rho }^{{AB}}={\sum }_{k}{\lambda }_{k}| {\psi }_{k} \rangle \langle {\psi }_{k}| $ with ${\lambda }_{{kl}}=\tfrac{{\left({\lambda }_{k}-{\lambda }_{l}\right)}^{2}}{({\lambda }_{k}+{\lambda }_{l})}$. Then, the QFI is $ \begin{eqnarray*}\begin{array}{rcl}{IP}({\rho }^{{AB}}) & = & {\,}_{{U}^{A}}^{min}F\left({\rho }^{{AB}},\displaystyle \sum _{i}{U}^{A}| i \rangle \langle i| {U}^{A}\otimes {\mathbb{1}}\right)\\ & = & {\,}_{{U}^{A}}^{min}\displaystyle \sum _{k\leqslant l}{\lambda }_{{kl}}| \langle {\psi }_{k}| \displaystyle \sum _{i}{\mu }_{i}{U}^{A}| i \rangle \langle i| {U}^{A\dagger }\otimes {\mathbb{1}}| {\psi }_{k} \rangle {| }^{2}\\ & \leqslant & {\,}_{{U}^{A}}^{min}\displaystyle \sum _{k\leqslant l}{\lambda }_{{kl}}\displaystyle \sum _{i}| {\mu }_{i}| | \langle {\psi }_{k}| {U}^{A}| i \rangle \langle i| {U}^{A\dagger }\otimes {\mathbb{1}}| {\psi }_{k} \rangle {| }^{2}\\ & = & {}_{{U}^{A}}^{min}\displaystyle \sum _{i}| {\mu }_{i}| F({\rho }^{{AB}},{U}^{A}| i \rangle \langle i| {U}^{A\dagger }\otimes {\mathbb{1}}).\end{array}\end{eqnarray*}$ The relation between skew information and the QFI is given as [35] $ \begin{eqnarray*}\begin{array}{c}\begin{array}{l}F({\rho }^{{AB}},{U}^{A}| i \rangle \langle i| {U}^{A\dagger }\otimes {\mathbb{1}})\\ \,\leqslant \,2{I}_{{\rm{WY}}}({\rho }^{{AB}},{U}^{A}| i \rangle \langle i| {U}^{A\dagger }\otimes {\mathbb{1}}),\end{array}\end{array}\end{eqnarray*}$ and the interferometric power of the state is $ \begin{eqnarray*}\begin{array}{rcl}{IP}({\rho }^{{AB}}) & = & {\,}_{{U}^{A}}^{min}F\left({\rho }^{{AB}},\displaystyle \sum _{i}{U}^{A}| i \rangle \langle i| {U}^{A}\otimes {\mathbb{1}}\right)\\ & \leqslant & {\,}_{{U}^{A}}^{min}2\displaystyle \sum _{i}| {\mu }_{i}| {I}_{{\rm{WY}}}({\rho }^{{AB}},{U}^{A}| i \rangle \langle i| {U}^{A\dagger }).\end{array}\end{eqnarray*}$ By choosing von Neumann projective measurement on $\{{{\rm{\Pi }}}^{A}\}=\{{U}^{A}| i \rangle \langle i| {U}^{A}\}$, it is easy to show that ${N}_{{ \mathcal A }}({\rho }^{{AB}})\,={I}_{{\rm{WY}}}({\rho }^{{AB}},{U}^{A}| i \rangle \langle i| {U}^{A\dagger })$. Hence, the interferometric power is$ \begin{eqnarray}{IP}({\rho }^{{AB}})\leqslant {N}_{{ \mathcal A }}({\rho }^{{AB}})\end{eqnarray}$where the minimum is taken over all local unitary operations UA. The affinity-based measure is interpreted as the upper bound of interferometric power of a quantum state.
Next, we identify the relation between local quantum uncertainty (LQU) [38] and affinity-based measures. The LQU is defined as$ \begin{eqnarray}{ \mathcal U }({\rho }^{{AB}})={}_{\{{{\rm{\Pi }}}_{k}^{A}\}}^{min}\,{I}_{{\rm{WY}}}({\rho }^{{AB}},{{\rm{\Pi }}}_{k}^{A}\otimes {\mathbb{1}})\end{eqnarray}$where minimization is taken over all von Neumann projective measurements on system A and ${I}_{{\rm{WY}}}({\rho }^{{AB}},{{\rm{\Pi }}}_{k}^{A}\otimes {\mathbb{1}}\,={ \mathcal A }({\rho }^{{AB}},{{\rm{\Pi }}}^{A}({\rho }^{{AB}}))$. Then, the relation between MIN and LQU is$ \begin{eqnarray}{N}_{{ \mathcal A }}({\rho }^{{AB}})\geqslant { \mathcal U }({\rho }^{{AB}}).\end{eqnarray}$
6. Examples
We compute the proposed quantity for the well-known two-qubit Bell diagonal state, Werner state and isotropic state and compare with the original version of the Hilbert–Schmidt distance-based MIN [36].
Bell diagonal state: the Bloch representation of the state can be expressed as$ \begin{eqnarray}{\rho }^{{BD}}=\displaystyle \frac{1}{4}\left[{\mathbb{1}}\otimes {\mathbb{1}}+\sum _{i=1}^{3}{c}_{i}({\sigma }^{i}\otimes {\sigma }^{i})\right],\end{eqnarray}$where the vector $\vec{c}=({c}_{1},{c}_{2},{c}_{3})$ is a three-dimensional vector composed of correlation coefficients such that $-1\leqslant {c}_{i}\,=\langle {\sigma }^{i}\otimes {\sigma }^{i}\rangle \leqslant 1$ completely specify the quantum state and λa, b; here, $a,b\in \{0,1\}$ denotes the eigenvalues of the Bell diagonal state which are given by $ \begin{eqnarray*}{\lambda }_{a,b}=\displaystyle \frac{1}{4}\left[1+{\left(-1\right)}^{a}{c}_{1}-{\left(-1\right)}^{a+b}{c}_{2}+{\left(-1\right)}^{b}{c}_{3}\right],\end{eqnarray*}$ and $| {\beta }_{{ab}}\rangle =\tfrac{1}{\sqrt{2}}[| 0,b\rangle +{\left(-1\right)}^{a}| 1,1+b\rangle ]$ are the Bell states. If ρBD describes a valid physical state, then 0≤λa, b≤1 and ${\sum }_{a,b}{\lambda }_{a,b}=1$. Under this constraint, the vector $\vec{c}\,=({c}_{1},{c}_{2},{c}_{3})$ must be restricted to the tetrahedron whose vertices are (1, 1,−1), (−1,−1,−1), (1,−1, 1) and (−1, 1, 1). The vertices are easily identified as Bell states, for which the correlation measures are maximum. Further, the measures are vanishing for the correlation vector $\vec{c}=(0,0,0)$, at which the state ${\rho }^{{BD}}={\mathbb{1}}/4$ is a maximally mixed state.
To compute the affinity-based measure, we first calculate the square root of the state ρBD as $ \begin{eqnarray*}\sqrt{{\rho }^{{BD}}}=\displaystyle \frac{1}{4}\left[h{\mathbb{1}}\otimes {\mathbb{1}}+\sum _{i=1}^{3}{d}_{i}({\sigma }^{i}\otimes {\sigma }^{i})\right],\end{eqnarray*}$ where $h=\mathrm{Tr}(\sqrt{{\rho }^{{BD}}})={\sum }_{{ab}}\sqrt{{\lambda }_{{ab}}}$ and$ \begin{eqnarray}\begin{array}{rcl}{d}_{1} & = & \sqrt{{\lambda }_{00}}-\sqrt{{\lambda }_{01}}+\sqrt{{\lambda }_{10}}-\sqrt{{\lambda }_{11}},\\ {d}_{2} & = & -\sqrt{{\lambda }_{00}}+\sqrt{{\lambda }_{01}}+\sqrt{{\lambda }_{10}}-\sqrt{{\lambda }_{11}},\\ {d}_{3} & = & \sqrt{{\lambda }_{00}}+\sqrt{{\lambda }_{01}}-\sqrt{{\lambda }_{10}}-\sqrt{{\lambda }_{11}}.\end{array}\end{eqnarray}$From the definition of MIN, we compute the affinity-based MIN for the Bell diagonal state as$ \begin{eqnarray}{N}_{{ \mathcal A }}({\rho }^{{BD}})=1-\displaystyle \frac{1}{4}({h}^{2}+{\min }_{j}\{{d}_{j}^{2}\}).\end{eqnarray}$In particular, if c1=c2=c3=−p, then the Bell diagonal state is reduced to the two-qubit Werner state, $ \begin{eqnarray*}{\rho }^{{BD}}=\displaystyle \frac{1-p}{4}{\mathbb{1}}+p| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,\,\,\,\,\,\,\,\,\,\,p\in [-1/3,1],\end{eqnarray*}$with $| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| $. The MINs of the Werner state are computed as$ \begin{eqnarray}\begin{array}{l}{N}_{{ \mathcal A }}({\rho }^{{BD}})=\displaystyle \frac{1}{4}[1+p-\sqrt{(1-p)(1+3p)}],\\ \qquad \times \,N({\rho }^{{BD}})=\displaystyle \frac{{p}^{2}}{2}.\end{array}\end{eqnarray}$
Werner state: we consider the $m\times m-$ dimensional Werner state, which is defined as [4]$ \begin{eqnarray}\omega =\displaystyle \frac{m-x}{{m}^{3}-m}{\mathbb{1}}+\displaystyle \frac{mx-1}{{m}^{3}-m}F,\,\,\,\,\,\mathrm{with}\,\,\,\,x\in [-1,1],\end{eqnarray}$with $F={\sum }_{{kl}}| {kl}\rangle \langle {kl}| $. The affinity-based MIN is computed as $ \begin{eqnarray*}{N}_{{ \mathcal A }}(\omega )=\displaystyle \frac{1}{2}\left(\displaystyle \frac{m-x}{m+1}-\sqrt{\displaystyle \frac{m-1}{m+1}(1-{x}^{2})}\right),\end{eqnarray*}$ and the Hilbert–Schmidt norm based MIN [36] $ \begin{eqnarray*}N(\omega )=\displaystyle \frac{{\left(mx-1\right)}^{2}}{m\left(m-1\right){\left(m+1\right)}^{2}}.\end{eqnarray*}$ It is observed that ${N}_{{ \mathcal A }}(\omega )=N(\omega )=0$, if, and only if, x=1/m. In the asymptotic limit $m\to \infty $, $ \begin{eqnarray*}\begin{array}{l}\mathop{\mathrm{lim}}\limits_{m\to \infty }{N}_{{ \mathcal A }}(\omega )=\displaystyle \frac{1}{2}(1-\sqrt{1-{x}^{2}}),\\ \qquad \times \,\mathop{\mathrm{lim}}\limits_{m\to \infty }N(\omega )=0.\end{array}\end{eqnarray*}$ The above equation suggests that the affinity-based measure is more robust in higher dimensions than the Hilbert–Schmidt norm based MIN.
Figure 1.
New window|Download| PPT slide Figure 1.The Hilbert–Schmidt norm and affinity-based MIN for 2 ⨯ 2 isotropic states.
Isotropic state: the m×mdimensional isotropic state is defined as [37]$ \begin{eqnarray}\rho =\displaystyle \frac{1-x}{{m}^{2}-1}{\mathbb{1}}+\displaystyle \frac{{m}^{2}x-1}{{m}^{2}-1}| {{\rm{\Psi }}}^{+}\rangle \langle {{\rm{\Psi }}}^{+}| \,\mathrm{with}\,x\in [0,1],\end{eqnarray}$where $| {{\rm{\Psi }}}^{+}\rangle =\tfrac{1}{\sqrt{m}}{\sum }_{i}| {ii}\rangle $. The affinity-based MIN is computed as $ \begin{eqnarray*}{N}_{{ \mathcal A }}(\rho )=\displaystyle \frac{1}{m}{\left(\sqrt{(m-1)x}-\sqrt{\displaystyle \frac{1-x}{m+1}}\right)}^{2},\end{eqnarray*}$ and the original MIN is [36] $ \begin{eqnarray*}N(\rho )=\displaystyle \frac{{\left({m}^{2}x-1\right)}^{2}}{m\left(m-1\right){\left(m+1\right)}^{2}}.\end{eqnarray*}$ We see that ${N}_{{ \mathcal A }}(\rho )=N(\rho )=0$ if x=1/m2. In the asymptotic limit, $ \begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{m\to \infty }{N}_{{ \mathcal A }}(\rho )=x,\,\,\,\,\,\,\,\,\,\,\mathop{\mathrm{lim}}\limits_{m\to \infty }N(\rho )={x}^{2}.\end{eqnarray*}$
Our result for m=2 is plotted in figure 2, which shows the consistency of the affinity-based correlation with the earlier one.
Figure 2.
New window|Download| PPT slide Figure 2.The Hilbert–Schmidt norm and affinity-based MIN for 2×2 isotropic states.
7. Conclusions
In this article, we have proposed a new variant of MIN using an affinity metric as a measure of quantum correlation for the bipartite state. It is shown that, in addition to capturing global nonlocal effects of a state due to von Neumann projective measurements, this quantity can be used to remedy the local ancilla problem of the Hilbert–Schmidt norm based MIN. We have presented a closed formula of affinity-based MIN for an arbitrary 2×n dimensional mixed state, with an upper bound for an m×n dimensional system. The physical meaning of the proposed quantity is interpreted as the upper bound of interferometric power of the quantum state. Further, we have computed the proposed version of MIN for m×m dimensional Werner and isotropic states.
Acknowledgments
This work is supported by the CSIR EMR Grant No. 03(1444)/18/EMR-II.
,Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, 
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