Coherence as uncertainty

Coherence of a quantum state (with respect to a fixed incoherent basis) is usually quantified by distancelike quantities, among which a particularly convenient and intuitive quantifier of coherence is based on the Hilbert-Schmidt distance between the quantum state and its dephased counterpart. This quantifier has a simple structure and many nice properties, although it is not a coherence monotone in the conventional resource-theoretic framework of coherence. Here we reveal its information-theoretic significance by showing that it coincides with uncertainty, as quantified by the variance of the state in the incoherent basis. The key point here is to regard the state as an observable, and to regard the incoherent basis as an ensemble of states rather than as measurement operators. Our result provides a natural decomposition of coherence into components along the incoherent basis. Furthermore, in terms of the Tsallis 2-entropy, which is also a measure of uncertainty, we provide two alternative interpretations of coherence: as increase of uncertainty caused by decoherence and as the conditional Tsallis 2-entropy in the context of purification. An intrinsic relation between the maximal coherence and the Brukner-Zeilinger invariant information is also established. These identifications of coherence with increase of uncertainty lead us to interpret coherence as a manifestation of quantum uncertainty, which may have implications for both quantum foundations and applications.

Figure 1

Coherence and uncertainty, as two basic quantum manifestations, can be identified in certain contexts. Here coherence is quantified by $C(\rho ,\mathrm{\Pi })=\left|\right|\rho -{\mathrm{\Pi }\left(\rho \right)\left|\right|}^2,$ which is the Hilbert-Schmidt distance between the quantum state $\rho$ and the decohered state $\mathrm{\Pi }\left(\rho \right)={\sum }_i{\mathrm{\Pi }}_i\rho {\mathrm{\Pi }}_i$, with $\mathrm{\Pi }=\{{\mathrm{\Pi }}_i=|i\rangle \langle i|:i=1,2,\cdots ,d\}$ being the von Neumann measurement induced by the incoherent basis $\left\{\right|i\rangle :i=1,2,\cdots ,d\}.$ Uncertainty is quantified by $U(\rho ,\mathrm{\Pi })={\sum }_i{V}_{{\mathrm{\Pi }}_i}\left(\rho \right),$ which is the variance of the quantum state $\rho$ (regarded as an observable) in the von Neumann measurement $\mathrm{\Pi }$. See Sec. 3 for details. It is remarkable that $C(\rho ,\mathrm{\Pi })=U(\rho ,\mathrm{\Pi }),$ which may be symbolized as “Coherence $\sim$ Uncertainty.”

Figure 2

Comparison between $V_{\rho }\left(K\right)$ (variance of an observable) and $V_K\left(\rho \right)$ (variance of a state) for non-negative operators $K$ and $\rho$: The conventional variance $V_{\rho }\left(K\right)$ is usually interpreted as the uncertainty of the observable $K$ (in the state $\rho$). In contrast, the quantity $V_K\left(\rho \right),$ which arises from interchanging $\rho$ and $K$ in $V_{\rho }\left(K\right),$ is the variance of the state $\rho$ (regarded as an observable) in the observable $K$ (regarded as an unnormalized state). It should be emphasized that $V_{\rho }\left(K\right)$ (variance of an observable) is concave in $\rho$, while $V_K\left(\rho \right)$ (variance of a state) is convex in $\rho$.

Figure 3

The uncertainty of a state is classified into three categories: total, quantum, and classical. The total uncertainty of $\rho ,$ as quantified by the variance $V(\rho ,\mathrm{\Pi })$ of the von Neumann measurement $\mathrm{\Pi }$ in the state $\rho ,$ is decomposed into the sum of the quantum part $U(\rho ,\mathrm{\Pi })$ and the classical part $S_2\left(\rho \right),$ which may be interpreted as quantum uncertainty and classical uncertainty, respectively.

Figure 4

Coherence $C(\rho ,\mathrm{\Pi })$ as conditional Tsallis 2-entropy $S_2\left({\tilde{\rho }}^{ab}\right|{\tilde{\rho }}^b)$: The state $\rho ={\rho }^a$ is purified to ${\rho }^{ab}=|{\mathrm{\Psi }}^{ab}\rangle \langle {\mathrm{\Psi }}^{ab}|$ with the help of the ancillary system $b$, and the joint state is decohered to ${\tilde{\rho }}^{ab}=({\mathrm{\Pi }}^a\otimes {\mathbf{1}}^b)\left({\rho }^{ab}\right)={\sum }_i({\mathrm{\Pi }}_i^a\otimes {\mathbf{1}}^b)\rho ({\mathrm{\Pi }}_i^a\otimes {\mathbf{1}}^b)$ via the local von Neumann measurement $\mathrm{\Pi }={\mathrm{\Pi }}^a=\{{\mathrm{\Pi }}_i^a:i=1,2,\cdots ,d\}$ on system $a.$ The coherence of $\rho$ with respect to $\mathrm{\Pi }$ turns out to be the conditional Tsallis 2-entropy of ${\tilde{\rho }}^{ab}$ conditional on the ancillary system $b$, i.e., $C(\rho ,\mathrm{\Pi })=S_2\left({\tilde{\rho }}^{ab}\right|{\tilde{\rho }}^b).$