Quantum complementarity from a measurement-based perspective

Quantum mechanics imposes fundamental limits on an observer’s information gain in complementary measurements. In the light of Bohr’s complementarity principle [1], quantum systems possess mutually exclusive properties that are equally real, and a measurement to reveal one property would inevitably preclude all the complementary ones. Characterizing this subtle relationship between measurement strategy and information gain is significant for the sophisticated manipulation of quantum measurements in various tasks, from demonstrating genuine nonclassical features of quantum objects to general quantum information processing.

Wootters and Zurek [2] proposed the first quantitative statement of complementarity relation by taking an information-theoretical perspective into the competitive tradeoff between the wave-like and particle-like behaviors of single photons. This kind of wave-particle duality relations (WPDRs) are currently expressed in a concise inequality form [3, 4, 5, 6] for photons within the Mach-Zehnder interferometer (MZI; see Fig. 3). For example, Jaeger et al [4] and Englert [6] obtained the duality relation $\mathcal{V}^{2}+\mathcal{D}^{2}\leq 1$ between fringe visibility (wave property) $\mathcal{V}$ and path distinguishability (particle property) $\mathcal{D}$. It is thus obvious that better which-way information implies less wave information, and vice versa.

Heisenberg’s uncertainty principle [7] is another fundamental concept in quantum mechanics which captures similar underlying physics of complementarity. It states that outcomes of specific measurements, e.g., position and momentum of a single particle, cannot be predicted with certainty simultaneously. Modern formulations of the uncertainty principle typically use entropic uncertainty measures [8, 9] due to their operational significance and the widespread applications [10] of entropic uncertainty relations (EURs), e.g., in the security analysis of quantum protocols [11, 12, 13].

The connections and contrasts between uncertainty and complementarity have been intensively debated [14, 15, 16, 17, 18, 19, 20]. It has been wondered whether novel complementarity relations can be derived directly from the well-studied and already-proven EURs. Particularly, Coles et al [20] proved that several WPDRs in the two-way interferometer can be equivalently reformulated as EURs for complementary observables. Thus two fundamental concepts of quantum mechanics are unified in this simple case.

Nevertheless, entropy is a natural measure of lack of information regarding only observation-independent properties and becomes conceptually inadequate [21] for quantum properties which are contextual and do not exist prior to measurements [22, 23]. To avoid this dilemma, Brukner and Zeilinger proposed an operationally invariant information measure of quantum systems [24]. It is naturally aligned with the concept of complementarity as being elegantly defined as the sum of individual measures of information gain over a complete set of mutually unbiased bases (CMUBs) [25, 26, 27, 28]—complementary bases—independent of particular choices of CMUBs and invariant under unitary time evolution. These intriguing properties inspired a series of insightful investigations [29, 30, 31, 32, 33, 34, 35, 36], including quantum state estimation [29, 30] and uncertainty relations for MUBs [34, 35].

In this paper, we adopt the operationally invariant information measure [24] and develop a general framework for characterizing quantum complementarity beyond WPDRs, in terms of basic limits on one’s ability to gain information about quantum systems through complementary measurement setups, i.e., information exclusion relations (IERs). We emphasize that when considering generalized measurements, identifying certainty of outcome statistics with information gain or visibility of physical property faces conceptual challenge—an outcome predictable with 100% certainty not necessarily reflects the complete information of the measured system. In contrast to IERs [37, 38, 39, 40, 41] that utilize entropic mutual information or deriving complementarity relations from EURs [20], our theory applies to generalized measurements and well captures the complete information of quantum systems as conserved quantities comprised of complementary pieces, highlighting the interplay between different pieces of information and their complementary nature.

This paper is structured as follows. In Sec. II, we introduce some preliminary notations. In Sec. III, we propose a measure of information gain in individual measurements while formalizing the concept of complementary information. In Sec. IV, we proceed to establish IERs which restrict one’s weighted sum of information gains over multiple measurements, with and without quantum memory respectively. In Sec. V, we show how our IERs lead to tight WPDRs. In Sec. VI, we discuss practical applications of our IERs. Finally, we briefly conclude this work in Sec. VII.

On a $d$-dimensional Hilbert space $\mathcal{H}_{d}$, each generalized measurement, i.e., positive-operator-valued measure (POVM), is a collection of positive semi-definite operators (called effects) $\mathcal{M}=\{M_{i}\}$ that sum up to the identity operator: $M_{i}\geq 0$ and $\sum_{i}M_{i}=\mathbbm{1}_{d}$. In particular, the measurement of a nondegenerate observable is described by rank-1 projectors onto its eigenvectors, i.e., rank-1 projective measurement. When a quantum state $\rho$ is measured, the outcome probabilities are given by Born’s rule, $p_{i}=tr(M_{i}\rho)$.

The Choi-Jamiołkowski isomorphism [42] allows us to represent each operator $O$ on $\mathcal{H}_{d}$ as a vector $\ket{O}$ of the product space $\mathcal{H}_{d}^{\otimes 2}$:

where $\ket{\psi_{d}}=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}\ket{i}\otimes\ket{i}^{*}$ is the maximally entangled isotropic state and $tr_{2}(\cdot)$ denotes the partial trace over the second space. A useful property of Eq. (1) that will be exploited is that $\braket{O_{1}|O_{2}}=tr(O_{1}^{+}O_{2})$ holds for any two operators $O_{1}$ and $O_{2}$ on $\mathcal{H}_{d}$.

In the light of Kochen-Specker’s theorem [22] (see also Ref. [23]), it is impossible to assign a definite noncontextual value to every quantum observable. During a measurement, all that an observer has is the probabilistic occurrence of one outcome (labeled contextual value 1), which simultaneously negates the occurrence of other outcomes (labeled contextual values 0). The information content of quantum systems is thus reflected in the statistics of these contextual binary strings.

Consider an experimental setup to perform the measurement $\mathcal{M}=\{M_{i}\}$ on individual copies of a quantum state that is unknown to the experimenter. Each time the $i$th outcome occurs, the experimenter gets a squared deviation $(1-tr(M_{i})/d)^{2}$ from the expectation $tr(M_{i})/d$ for the completely mixed state—least information state—or gets $(0-tr(M_{i})/d)^{2}$ otherwise. After repeating the experiments large enough $N$ times, the total squared deviation is $D_{i}^{2}=N[p_{i}(1-tr(M_{i})/d)^{2}+(1-p_{i})(tr(M_{i})/d)^{2}]$, which consists of two contributions $D_{i}^{2}=\Delta_{i}^{2}+B_{i}^{2}$. Wherein $\Delta_{i}^{2}=N[p_{i}(1-p_{i})]$ is the total uncertainty (variance), which determines the width $2\Delta_{i}/N$ of the confidence interval $[p_{i}-\frac{1}{N}\Delta_{i},p_{i}+\frac{1}{N}\Delta_{i}]$ for estimating the outcome probabilities $\{p_{i}\}$.

What truly discriminates the measured state from the completely mixed state, on the other hand, is the total squared bias $B_{i}^{2}=N(p_{i}-tr(M_{i})/d)^{2}$. We suggest the measure of information gain on the state $\rho$ in each individual trial of the measurement $\mathcal{M}=\{M_{i}\}$ to be the sum of mean squared bias over all outcomes

In the above, we leverage the isomorphism (1) to define the view operator of a measurement $\mathcal{M}$ as

where $\tilde{M}_{i}=M_{i}-\frac{1}{d}tr(M_{i})\mathbbm{1}_{d}$ is traceless, or equivalently, $\ket{\tilde{M}_{i}}=\ket{M_{i}}-|\psi_{d}\rangle\langle\psi_{d}|M_{i}\rangle$ is orthogonal to $\ket{\psi_{d}}$. View operators are positive semi-definite, $\hat{G}\geq 0$ on the $(d^{2}-1)$-dimensional subspace $\mathcal{H}_{\perp\psi_{d}}$ of $\mathcal{H}_{d}^{\otimes 2}$ orthogonal to $\ket{\psi_{d}}$, and vanish for trivial POVMs whose effects are all proportional to the identity, $M_{i}=\frac{1}{d}tr(M_{i})\mathbbm{1}_{d}$.

Now we are able to formalize our idea of complementary information. Let $\tilde{\rho}=\rho-\mathbbm{1}_{d}/d$, observe that the outcome probabilities of a measurement $\mathcal{M}$ on the state $\rho$ are encoded in the expansion coefficients of the vector $\ket{\tilde{\rho}_{\mathcal{M}}}=\hat{G}(\mathcal{M})\cdot\ket{\rho}=\hat{G}(\mathcal{M})\cdot\ket{\tilde{\rho}}=\sum_{i}(p_{i}-\text{tr}(M_{i})/d)\ket{\tilde{M}_{i}}$ under the basis $\{\ket{\tilde{M}_{i}}\}$. The vector $\ket{\tilde{\rho}_{\mathcal{M}}}$ encodes the complete information of $\rho$ if $\ket{\tilde{\rho}}$ lies in the subspace of $\mathcal{H}_{\perp\psi_{d}}$ on which the view operator $\hat{G}(\mathcal{M})$ is invertible, whereas if $\ket{\tilde{\rho}}$ is orthogonal to that space, $\ket{\tilde{\rho}_{\mathcal{M}}}$ vanishes and $\mathcal{M}$ cannot be employed to distinguish $\rho$ from the completely mixed state (see Fig. 1 for an illustration of the geometric relations between the above vectors). In the sense above, two nontrivial measurements $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ satisfying

are complementary since, if the complete information of $\rho$ is accessible through $\mathcal{M}_{1}$, then no information gain is accessible through $\mathcal{M}_{2}$, and vice versa. We prove in Appendix. A that measurements in mutually unbiased bases (MUBs) [25, 26, 27, 28] are complementary.

It is worth mentioning that the combined view operator $\hat{G}=\sum_{\theta}\hat{G}(\mathcal{M}_{\theta})$ associated with a set of POVMs $\mathscr{M}=\{M_{\theta}\}$ on $\mathcal{H}_{d}$ can be positive definite (invertible) on $\mathcal{H}_{\perp\psi_{d}}$. In this case, no POVM can be complementary to all POVMs of $\mathscr{M}$ simultaneously. This means that $\mathscr{M}$ is informationally-complete and $\hat{G}$ offers a complete view to all $d$-dimensional quantum states. Utilizing the isomorphism (1), arbitrary unknown state $\rho$ can then be reconstructed from the vector $\hat{G}\ket{\tilde{\rho}}=\ket{\tilde{\rho}_{\mathscr{M}}}$ encoding the outcome statistics as follows

For further readings on the topic of state estimation, we recommend Refs. [43, 44].

Interestingly, the combined view operator associated with CMUBs of $\mathcal{H}_{d}$, i.e., $d+1$ mutually unbiased bases (MUBs) [25, 26, 27, 28], is simply the identity operator $\mathbbm{1}_{\perp\psi_{d}}=\mathbbm{1}_{d}\otimes\mathbbm{1}_{d}-|\psi_{d}\rangle\langle\psi_{d}|$ on $\mathcal{H}_{\perp\psi_{d}}$ (see Appendix. A). Thus the operationally invariant measure [24] of complete information content contained in quantum states can be restateted in our language as

This measure naturally coincides with Bohr’s idea [1] that only the totality of complementary properties together exhausts the complete information of objects.

To formulate quantum complementarity into information exclusion relations, next we focus on the measurement scenarios where distinct measurements on individual copies of a quantum system are selected with biased (non-uniform) probabilities.

We argue that the tight WPDRs are particular examples of the IERs (7) and (10) for complementary observables. To see this, let us consider two complementary setups of the Mach-Zehnder interferometer depicted in Fig. 3: (i) the second beam splitter is removed to gain the path information of single photons inside the interferometer (let $\sigma^{\rm p}$ denote the associated path observable with binary outcomes “$+1$” and “$-1$”, corresponding to clicks in detectors $D_{0}$ and $D_{1}$ respectively ); (ii) BS₂ is inserted in and the phase shift $\phi$ is adjustable to reveal wave properties of photons (let $\sigma_{\phi}^{\rm w}$ denote the associated wave observable with binary outcomes “$\pm 1$”). It takes some calculation (see Appendix. C) to see that Eq. (7) leads to the equality

where $\braket{\sigma}=tr(\sigma\rho)$ denotes the average of observable $\sigma$, and $G(\sigma^{\rm p})_{\rho}=\frac{1}{2}\braket{\sigma^{\rm p}}^{2}$ and $G(\sigma_{\phi}^{\rm w})_{\rho}=\frac{1}{2}\braket{\sigma_{\phi}^{\rm w}}^{2}$ are the respective information gains (2) for measuring the path and wave observables in the qubit $\rho$.

Observe that the information gain regarding an individual wave observable oscillates as the phase shift $\phi$ varies, Eq. (11) essentially depicts an interference pattern of the wave information. To make it clearer, let $\vec{\phi}$ and $\vec{\phi^{\prime}}$ be two real unit vectors at an angle of $\phi^{\prime}-\phi$. Eq. (11) can then be restated as

It is interesting to note that the average values of wave observables behave like the “amplitudes of wave information” and interfere with each other, see Fig. 4a. Notably, the average interference intensity $\mathcal{I}=I_{\rm com}(\rho)-G(\sigma^{\rm p})_{\rho}$ on the r.h.s. of Eq. (12) disappears if the photon exhibits particle property only—the complete information content of $\rho$ is accessible through measuring the path observable or, formally, $I_{\rm com}(\rho)=G(\sigma^{\rm p})_{\rho}$. In this view, $\mathcal{I}=G(\sigma_{\phi}^{\rm w})_{\rho}+G(\sigma_{\phi+\pi/2}^{\rm w})_{\rho}$ (see the case $\phi^{\prime}-\phi=\pm\frac{\pi}{2}$ in the preceding equation) emerges as a measure of wave property which can be determined by measuring two complementary wave observables.

Conventionally, the wave property is frequently quantified by the fringe visibility [3, 4, 5, 6]

where $p_{\phi}^{i}$ is the probability that the $i$th detector clicks when the observable $\sigma^{\rm w}_{\phi}$ is measured. We remark here that the average interference intensity is precisely half of the fringe visibility squared, i.e., $\mathcal{V}=\max_{\phi}|\braket{\sigma^{\rm w}_{\phi}}|=\sqrt{\mathcal{I}/2}$ (see also Fig. 4b for an illustration). Combined with the squared path distinguishability $\mathcal{D}^{2}=\braket{\sigma^{\rm p}}^{2}=2G(\sigma^{\rm p})_{\rho}$, we then arrive at the WPDR $\mathcal{V}^{2}+\mathcal{D}^{2}=2tr(\rho^{2})-1$ [60]. We therefore see that the WPDR originates from the IER $\big{[}G(\sigma_{\phi}^{\rm w})_{\rho}+G(\sigma_{\phi+\pi/2}^{\rm w})_{\rho}\big{]}+G(\sigma^{\rm p})_{\rho}=I_{\rm com}(\rho)$ for three complementary observables, including the path observable and two wave observables with phase difference satisfying $\phi^{\prime}-\phi=\pm\pi/2$.

Theorem 1 applies also to the quantum delayed-choice experiment [61], where complementary properties of photons are measured in a single experimental setup. As shown in Fig. 5, the presence of BS₂ is controlled by an ancilla qubit, the value of which determines whether to reveal the wave property or particle property. In this case, Eq. (7) limits an observer’s weighted average information gain about three complementary observables, with (unnormalized) weights $w_{1}=\cos^{2}\beta$ for the path observable and $w_{2}=w_{3}=\sin^{2}\beta$ for the wave observables. Therefore, the true nature of complementarity does not prohibit the observation of complementary properties in a single measurement setup, but necessarily restricts one’s simultaneous information gain about them.

Another interesting issue concerns the WPDRs when an observer has side information about single photons in the Mach-Zehnder interferometer, but without direct access to them. Let us consider two photons in the bipartite state $\rho_{AB}$. As a measure of information about photon A conditioned on photon B, we turn to the complement of the conditional linearized entropy (8) below

This is non-negative and reduces to the complete information of the reduced state $\rho_{A}$, $I(A|B)=I_{\rm com}(\rho_{A})$ when $\rho_{AB}=\rho_{A}\otimes\rho_{B}$ is a product state.

We derive in Appendix. C the following generalization of Eq. (12),

Here, $\vec{\rho}_{\phi B}=tr_{A}\big{[}(\rho_{B}^{-1/4}\rho_{AB}\rho_{B}^{-1/4})(\sigma_{\phi}^{\rm w}\otimes\mathbbm{1}_{B})\big{]}\ \vec{\phi}$ is the “amplitude of conditional information” which connects to the conditional information $I(\sigma_{\phi}^{\rm w}|B)$ through its squared modulus $tr(\vec{\rho}_{\phi B}^{\ 2})=2I(\sigma_{\phi}^{\rm w}|B)$.

Equation (15) manifests the interference pattern of conditional information amplitude, with the r.h.s. of it being the interference intensity. Combining the average intensity (wave property) $I(A|B)-I(\sigma^{\rm p}|B)=I(\sigma_{\phi}^{\rm w}|B)+I(\sigma^{\rm w}_{\phi+\pi/2}|B)$ with the conditional which-way information (particle property) $I(\sigma^{\rm p}|B)$, we then obtain the WPDR $\big{[}I(\sigma_{\phi}^{\rm w}|B)+I(\sigma_{\phi+\pi/2}^{\rm w}|B)\big{]}+I(\sigma^{\rm p}|B)=I(A|B)$. Again, we see that a tight WPDR saturated by all bipartite systems with dimension $d_{A}=2$ arises from an IER for three complementary observables, wherein two complementary wave observables constitute the complete description of wave property.

Our theory for characterizing information complementarity from a measurement-based perspective enables us to analyze the behaviors of quantum systems through their manifestations in versatile measurement setups, without delving into the exhaustive calculations with quantum state parameters. As two examples, we explore the implications of our IERs (7, 10) for entanglement detection and EURs respectively.

In summary, we have developed a general approach to formulate the complementarity principle quantitatively in terms of basic limits on one’s ability to gain information on quantum systems under versatile measurement setups, with and without memory respectively. Our framework sheds new light on the interpretation of wave-particle duality for single photons in the two-way interferometry experiments from an information-theoretical perspective. Extending this interpretation to multi-path interferometers presents an intriguing avenue for future investigation. Moreover, our IERs have direct applications in certifying genuine quantum features of physical systems, such as entanglement detection based on local measurement outcomes. An extended form of EURs can also be derived from our IERs, which could offer practical advantages in various quantum information processing.