Necessary and sufficient conditions for vanishing Lüders-type quantum discord and the operational meaning of the normalized POVM coherence functional

Yao–Dong–Xiao–Li–Sun (Phys. Rev. A 96, 052322 (2017)) connected the resource theory of quantum coherence with quantum discord through the measurement process: for a fixed observable $R$ on subsystem $B$, the Lüders-type (measurement-dependent) discord $\delta^{A|B}(R,\rho^{AB})$, i.e. the loss of quantum mutual information under the local Lüders operation of $R$, equals the local coherence of $\rho^{AB}$ relative to that measurement minus the residual coherence of the measured marginal. The paper posed two open questions: (i) a necessary and sufficient condition for $\delta^{A|B}(R,\rho^{AB})=0$ is unknown — compatibility $[\rho^{AB},R]=0$ is only sufficient, and the question is nontrivial exactly when $R$ is degenerate, since the nondegenerate case was already solved by Yadin et al. (Phys. Rev. X 6, 041028 (2016)); and (ii) the physical meaning, operational interpretation, and applications of the normalized POVM-based coherence functional $\tilde C_G(\rho)=S(\rho\|\sum_n M_n^{1/2}\rho M_n^{1/2})$ are undetermined. Neither question has been resolved in the subsequent literature, and the two remain open, jointly aimed at completing the operational interpretation of measurement-rooted coherence quantifiers.

Background

The resource theory of quantum coherence treats states diagonal in a preferred orthonormal basis as free (incoherent) and quantifies the coherence of a state $\rho$ by its distance to that set; the standard measure is the relative entropy of coherence $C_{\rm re}(\rho)=S(\rho\|\Delta(\rho))$, where $S(\rho\|\sigma)=\mathrm{Tr}\,\rho(\log\rho-\log\sigma)$ is the quantum relative entropy and $\Delta(\rho)=\sum_n |n\rangle\langle n|\rho|n\rangle\langle n|$ is dephasing, i.e. a projective measurement in the preferred basis with outcomes discarded. A positive-operator-valued measure (POVM) is a collection of positive operators $\{M_n\}$ with $\sum_n M_n=\mathbb 1$; a projective measurement of an observable $R=\sum_n r_n P_n$ uses orthogonal spectral projectors $\{P_n\}$, and $R$ is called degenerate when some eigenspace $\mathrm{ran}(P_n)$ has dimension greater than one.

Quantum discord (Ollivier–Zurek; Henderson–Vedral) quantifies the part of the correlations in a bipartite state $\rho^{AB}$ — measured by the quantum mutual information $I(\rho^{AB})=S(\rho^A)+S(\rho^B)-S(\rho^{AB})$, with $S(\rho)=-\mathrm{Tr}\,\rho\log\rho$ the von Neumann entropy — that is destroyed by measuring one subsystem: discord is the gap between $I(\rho^{AB})$ and the best classical correlation achievable after a measurement on $B$.

Yao–Dong–Xiao–Li–Sun (Phys. Rev. A 96, 052322 (2017)) connected these two programs through the measurement process itself. For a fixed observable $R$ on $B$ with spectral projectors $\{P_n\}$, the local Lüders operation is

$$L_B(\rho)=\sum_n(\mathbb 1\otimes P_n)\,\rho\,(\mathbb 1\otimes P_n),$$

the state update that dephases between (but not within) the eigenspaces of $R$, and the Lüders-type discord is

$$\delta^{A|B}(R,\rho^{AB})=I(\rho^{AB})-I(L_B(\rho^{AB})).$$

The paper proves the identity $\delta^{A|B}=C_{\rm re}^{A|B}(R,\rho^{AB})-C_{\rm re}(R,\rho^B)$: the discord equals the local coherence of $\rho^{AB}$ relative to the Lüders measurement of $R$ minus the residual coherence left in the measured marginal. When $R$ is nondegenerate, $L_B$ is full dephasing and the zero-discord states were already characterized by Yadin–Ma–Girolami–Gu–Vedral (Phys. Rev. X 6, 041028 (2016)): $\delta^{A|B}=0$ exactly on mixtures $\rho^{AB}=\sum_\alpha p_\alpha\rho_A^\alpha\otimes\rho_B^\alpha$ whose $B$-components are perfectly distinguishable in the eigenbasis of $R$. When $R$ is degenerate the Lüders update is strictly weaker — degenerate Lüders post-measurement states can even remain entangled (Luo–Fu 2013) — and the paper states that the necessary and sufficient condition for $\delta^{A|B}=0$ is open, with compatibility $[\rho^{AB},R]=0$ known to be sufficient only.

For general POVMs, the paper introduces (following Coles–Metodiev–Lütkenhaus, who needed such quantities in numerical quantum key distribution) the functional $C_G(\rho)=S(\rho\|\sum_n M_n\rho M_n)$ together with its normalized Lüders analogue

$$\tilde C_G(\rho)=S\left(\rho\,\Big\|\,\sum_n M_n^{1/2}\,\rho\,M_n^{1/2}\right),$$

where the second argument is now always a density operator, since the generalized Lüders map $\Lambda_M(\rho)=\sum_n M_n^{1/2}\rho M_n^{1/2}$ is trace-preserving, $\mathrm{Tr}\,\Lambda_M(\rho)=\sum_n\mathrm{Tr}(\rho M_n)=\mathrm{Tr}\,\rho=1$ (it is also unital, $\Lambda_M(\mathbb 1)=\mathbb 1$). The paper establishes the nonnegativity $\tilde C_G\ge 0$ but leaves the physical meaning and applications of $\tilde C_G$ undetermined.

Problem Statement

Characterize, necessary and sufficiently, the pairs $(R, \rho^{AB})$ of a (generally degenerate) observable $R$ on subsystem $B$ and a bipartite state for which the Lüders-type quantum discord vanishes,

$$\delta^{A|B}(R,\rho^{AB}) = I(\rho^{AB}) - I\big(L_B(\rho^{AB})\big) = 0,$$

where $\{P_n\}$ are the spectral projectors of $R$, $L_B(\rho)=\sum_n(\mathbb 1\otimes P_n)\rho(\mathbb 1\otimes P_n)$ is the local Lüders operation, and $I$ is the quantum mutual information; and determine the physical meaning — an operational interpretation and concrete applications — of the normalized POVM coherence functional

$$\tilde C_G(\rho) = S\left(\rho\,\Big\|\,\sum_n M_n^{1/2}\rho M_n^{1/2}\right)$$

for a general POVM $\{M_n\}$, where $S(\cdot\|\cdot)$ is the quantum relative entropy.

The two questions share one scientific objective: completing the operational interpretation of measurement-rooted coherence quantifiers, namely a state-independent account of when a general Lüders or POVM measurement leaves all correlations of the measured system undisturbed, and of what the residual coherence such a measurement necessarily leaves behind quantifies. Consistent with the source, both questions are open-ended, and a complete resolution addresses both parts — a state-independent characterization of when $\delta^{A|B}=0$ and an operational interpretation of $\tilde C_G$ meeting the acceptance standard stated below; a rigorous solution of either part alone is an accepted partial result. For the characterization, the nondegenerate case is already settled by the Yadin–Ma–Girolami–Gu–Vedral result (zero discord exactly on product mixtures whose $B$-components are perfectly distinguishable in the eigenbasis of $R$), and Lüders-invariant states (e.g. product mixtures whose $B$-components are supported inside single degenerate eigenspaces of $R$) always have $\delta^{A|B}=0$; the open content therefore concerns genuinely degenerate $R$. For the functional, a complete resolution identifies at least one operational task or operational quantity for which $\tilde C_G$ is the exact figure of merit (or for which it admits an explicitly stated tight bound), together with at least one concrete application; note that $\tilde C_G$ reduces automatically to the relative entropy of coherence when $\{M_n\}$ is projective, since $P_n^{1/2}=P_n$.

References

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