Quantum correlations and complementarity of vectorial light fields

TL;DR

This paper investigates quantum correlations in vector light fields, revealing a fundamental triality identity that links wave-particle duality and establishes a general quantum complementarity relation across multiple slits and states.

Contribution

It introduces a novel framework connecting field coherence matrices with photon density matrices, transforming duality inequalities into equalities and uncovering fundamental quantum interference physics.

Findings

Derived a direct link between field coherence and photon density matrices.

Revealed a hidden information-theoretic contribution to wave-particle duality.

Established a universal quantum complementarity relation for vector-light fields.

Abstract

We explore quantum correlations of general vector-light fields in multislit interference and show that the $n$th-order field-coherence matrix is directly linked with the reduced $n$-photon density matrix. The connection is utilized to examine photon wave-particle duality in the double-slit configuration, revealing that there is a hidden information-theoretic contribution that complements the standard inequality associated with such duality by transforming it into a strict equality, a triality identity. We also establish a general quantum complementarity relation among the field correlations and the particle correlations which holds for any number of slits, correlation orders, and vector-light states. The framework that we advance hence uncovers fundamental physics about quantum interference.

Equations

$$|\Psi ⟩=C_1|20⟩+C_2|02⟩+C_3|11⟩,$$

$$|\Psi ⟩=C_1|xx⟩+C_2|yy⟩+C_3\frac{|xy⟩+|yx⟩}{\sqrt{2}},$$

$${\rho }_{i_1{i}_2,j_1{j}_2}^{(2)}=⟨\Psi |{\hat{a}}_{j_1}^{†}{\hat{a}}_{j_2}^{†}{\hat{a}}_{i_1}{\hat{a}}_{i_2}|\Psi ⟩,$$

$${\varrho }_{i_1,j_1}^{(2;1)}:=\sum _{k\in \{x,y\}}{\rho }_{i_{1}k,j_{1}k}^{(2)}\equiv ⟨\Psi |{\hat{a}}_{j_1}^{†}{\hat{a}}_{i_1}|\Psi ⟩,$$

$${\mathbf{G}}^{(n)}(x_1,\dots ,x_{2n})$$

$$\otimes {\hat{\mathbf{E}}}^{(+)}(x_{n+1})\otimes \cdots \otimes {\hat{\mathbf{E}}}^{(+)}(x_{2n})].$$

$${\hat{\mathbf{E}}}^{(+)}(x_m)=E_0\sum _{s=1}^2{\hat{a}}_{ms}{\mathbf{e}}_{ms}{e}^{i(\mathbf{k}\cdot {\mathbf{r}}_m-\omega t_m)},$$

$${\mathbf{G}}^{(n)}(x_{m_1},\dots ,x_{m_{2n}})=I_0^n\sum _{s_1,\dots ,s_{2n}}{W}^{(n)}({\kappa }_1,\dots ,{\kappa }_{2n})$$

$$\underset{\mu =1}{\overset{n}{⨂}}{\mathbf{e}}_{{\kappa }_{\mu }}^{†}{e}^{-i(\mathbf{k}\cdot {\mathbf{r}}_{m_{\mu }}-\omega t_{m_{\mu }})}\underset{\nu =n+1}{\overset{2n}{⨂}}{\mathbf{e}}_{{\kappa }_{\nu }}{e}^{i(\mathbf{k}\cdot {\mathbf{r}}_{m_{\nu }}-\omega t_{m_{\nu }})},$$

$$W^{(n)}({\kappa }_1,\dots ,{\kappa }_{2n})=\mathrm{tr}(\hat{\rho }{\hat{a}}_{{\kappa }_1}^{†}\cdots {\hat{a}}_{{\kappa }_n}^{†}{\hat{a}}_{{\kappa }_{n+1}}\cdots {\hat{a}}_{{\kappa }_{2n}})$$

$$N$$

$${\rho }_{i_1\dots i_N,j_1\dots j_N}^{(N)}:=N!⟨i_1,\dots ,i_N|{\hat{\rho }}^{(N)}|j_1,\dots ,j_N⟩,$$

$${\varrho }_{i_1\dots i_M,j_1\dots j_M}^{(N;M)}:=$$

$$\frac{1}{(N-M)!}\sum _{l_{M+1},\dots ,l_N}{\rho }_{i_1\dots i_M{l}_{M+1}\dots l_N,j_1\dots j_M{l}_{M+1}\dots l_N}^{(N)}$$

$$\varrho^{(N;M)}_{i_{1}\ldots i_{M},j_{1}\ldots j_{M}}=\mathrm{tr}\big{[}\hat{\rho}^{(N)}\hat{a}_{j_{1}}^{\dagger}\cdots\hat{a}_{j_{M}}^{\dagger}\hat{a}_{i_{1}}\cdots\hat{a}_{i_{M}}\big{]}.$$

$${\varrho }_{i_1\dots i_M,j_1\dots j_M}^{(M)}=\sum _{N=0}^{\infty }{p}_N{\varrho }_{i_1\dots i_M,j_1\dots j_M}^{(N;M)},$$

$$W^{(n)}({\kappa }_1,\dots ,{\kappa }_{2n})={\varrho }_{{\kappa }_{n+1}\dots {\kappa }_{2n},{\kappa }_1\dots {\kappa }_n}^{(n)}.$$

$$\hat{\boldsymbol{\mathcal{E}}}=\big{[}\hat{E}_{1}^{(+)}\!(x_{1}),\hat{E}_{2}^{(+)}\!(x_{1}),\ldots,\hat{E}_{1}^{(+)}\!(x_{L}),\hat{E}_{2}^{(+)}\!(x_{L})\big{]}\!,$$

$$\boldsymbol{\mathcal{G}}^{(1)}=\mathrm{tr}\big{(}\hat{\rho}\,\hat{\boldsymbol{\mathcal{E}}}^{\dagger}\!\otimes\hat{\boldsymbol{\mathcal{E}}}\big{)},$$

$${\mathcal{G}}^{(1)}=\left[\begin{array}{ccc}{\mathbf{G}}^{(1)}(x_1,x_1)& \cdots & {\mathbf{G}}^{(1)}(x_1,x_L)\\ ⋮& \ddots & ⋮\\ {\mathbf{G}}^{(1)}(x_L,x_1)& \cdots & {\mathbf{G}}^{(1)}(x_L,x_L)\end{array}\right].$$

$$F^{(1)}=\frac{\mathcal{K}}{\mathcal{K}-1}\Big{[}\big{\|}\tilde{\boldsymbol{\mathcal{G}}}^{(1)}\big{\|}_{\mathrm{F}}^{2}-\frac{1}{\mathcal{K}}\Big{]};\quad\tilde{\boldsymbol{\mathcal{G}}}^{(1)}=\frac{\boldsymbol{\mathcal{G}}^{(1)}}{\mathrm{tr}\boldsymbol{\mathcal{G}}^{(1)}},$$

$$S^{(1)}=\frac{\mathcal{K}}{\mathcal{K}-1}\big{\{}1-\mathrm{tr}\big{[}\tilde{\varrho}^{(1)}\big{]}^{2}\big{\}};\quad\tilde{\varrho}^{(1)}=\frac{\varrho^{(1)}}{\mathrm{tr}\varrho^{(1)}},$$

$$\big{\|}\tilde{\boldsymbol{\mathcal{G}}}^{(1)}\big{\|}_{\mathrm{F}}^{2}=\mathrm{tr}\big{[}\tilde{\varrho}^{(1)}\big{]}^{2},$$

$$F^{(1)}+S^{(1)}=1.$$

$$x_1$$

$$D(x_1,x_2)=\frac{|{\bar{N}}_1-{\bar{N}}_2|}{{\bar{N}}_1+{\bar{N}}_2},$$

$$V(x)=\frac{2({\bar{N}}_1{\bar{N}}_2{)}^{1/2}}{{\bar{N}}_1+{\bar{N}}_2}{g}^{(1)}(x_1,x_2),$$

$$g^{(1)}(x_1,x_2)=\frac{\Vert {\mathbf{G}}^{(1)}(x_1,x_2){\Vert }_{\mathrm{F}}}{[\mathrm{tr}{\mathbf{G}}^{(1)}(x_1,x_1)\mathrm{tr}{\mathbf{G}}^{(1)}(x_2,x_2){]}^{1/2}}.$$

$$F^{(1)}=D^2(x_1,x_2)+V^2(x)\le 1,$$

$$D^2(x_1,x_2)+V^2(x)+S^{(1)}=1,$$

$$D^2(x_1,x_2)+V^2(x)$$

$$D^2(x_1,x_2)+S^{(1)}$$

$$V^2(x)+S^{(1)}$$

$$\boldsymbol{\mathcal{G}}^{(n)}=\mathrm{tr}\big{[}\hat{\rho}\,(\hat{\boldsymbol{\mathcal{E}}}^{\dagger})^{\otimes n}\otimes(\hat{\boldsymbol{\mathcal{E}}})^{\otimes n}\big{]},$$

$$F^{(n)}=\frac{\mathcal{K}^{n}}{\mathcal{K}^{n}-1}\Big{[}\big{\|}\tilde{\boldsymbol{\mathcal{G}}}^{(n)}\big{\|}_{\mathrm{F}}^{2}-\frac{1}{\mathcal{K}^{n}}\Big{]};\ \ \tilde{\boldsymbol{\mathcal{G}}}^{(n)}=\frac{\boldsymbol{\mathcal{G}}^{(n)}}{\mathrm{tr}\boldsymbol{\mathcal{G}}^{(n)}},$$

$$S^{(n)}=\frac{\mathcal{K}^{n}}{\mathcal{K}^{n}-1}\big{\{}1-\mathrm{tr}\big{[}\tilde{\varrho}^{(n)}\big{]}^{2}\big{\}};\quad\tilde{\varrho}^{(n)}=\frac{\varrho^{(n)}}{\mathrm{tr}\varrho^{(n)}},$$

$$\big{\|}\tilde{\boldsymbol{\mathcal{G}}}^{(n)}\big{\|}_{\mathrm{F}}^{2}=\mathrm{tr}\big{[}\tilde{\varrho}^{(n)}\big{]}^{2},$$

$$F^{(n)}+S^{(n)}=1.$$

Full text

Introduction.—The seminal quantum theory of optical coherence Glauber63 ; Mandelbook , dealing with field correlations of light, is ubiquitous in the physical sciences; it is widely exploited in quantum optics Fortsch13 ; Tenne19 , atomic physics Hodgman11 ; Perrin12 , optomechanics Cohen15 ; Ockeloen18 , quantum simulation Barrett13 , quantum electronics Brange15 , and cosmology Giovannini11 , among other research areas. Recently quantum coherence of genuine vector-light fields was examined in double-slit interference, revealing a new fundamental aspect of photon wave-particle duality Norrman17 . The rapid progress in quantum information science has at the same time led to an ever-growing interest towards nonclassical correlations that may prevail in multipartite quantum compositions of diverse physical nature Horodecki09 ; Chiara18 . These correlations lie at the heart of foundational quantum physics, with applications in quantum teleportation, quantum cryptography, and quantum computation.

Quantum correlations of indistinguishable systems, in particular, have attracted broad interest lately Eckert02 ; Tichy11 . Such correlations may occur among modes Huang94 ; Zanardi01 ; Zanardi04 , usually met in operational quantum science, or between particles Paskauskas01 ; Li01 ; Ghirardi04 , mainly considered in many-body physics. The latter are regarded more fundamental as they are a prerequisite for the former Wiseman03 . In atomic physics, where correlation functions render schemes for joint probability detection of atoms Bach04 , Glauber’s contribution Glauber63 ; Glauber99 can be viewed as a unified framework that subsumes field and particle correlations KRZ . In quantum electrodynamics, however, where photons have no well-defined position operator IBB09 , particle-like information-theoretic content of the electromagnetic field is customarily omitted.

In this Letter, we investigate the relationship between field correlations and particle correlations of true vectorial light of any quantum state in multislit interference. We show that the $n$th-order field correlations are directly connected to the particle correlations among $n$ photons. This relationship is especially employed to explore quantum complementarity in the celebrated double-slit setup, resulting in the discovery of a tight equality which may be interpreted as describing photon wave-particle triality. In the general case with any number of slits, we derive a fundamental complementarity relation between the $n$th-order field correlations and the $n$th-order particle correlations, stating that there is a strict trade-off between these two types of correlations for all quantum states of light. Our work may be viewed as the most general framework regarding quantum complementarity of field-particle correlations in vector-light interference, providing deeper insights into foundational quantum-interference physics.

Modes and particles.—Let us first discuss the notions of modes and particles in the description of quantum light. As an example we consider a pure, unit-trace normalized, two-photon state $\hat{\rho}^{(2)}=\left|\Psi\right\rangle\left\langle\Psi\right|$ involving two orthogonal modes ($x$ and $y$).

In the second-quantized (mode) representation,

[TABLE]

Nevertheless, the two representations contain very distinct information about the correlations in the system. The reduced one-mode state associated with Eq. (1a) and given by $\mathrm{diag}\left(|C_{1}|^{2},|C_{2}|^{2},|C_{3}|^{2}\right)$ describes entanglement between the two modes. However, the reduced one-photon state corresponding to Eq. (1b), with matrix elements

[TABLE]

characterizes quantum correlations between the two photons, which owing to the bosonic indistinguishability cannot be operationally classified as pure entanglement Li01 . While unitary operations achievable by linear optics may alter the mode correlations Enk03 ; JETP , the primordial particle (photon) correlations Wiseman03 remain invariant.

Quantum correlations in vector-light interference.—The complete information about the $n$th-order quantum-field correlations of vectorial light footnote3 , at $2n$ space-time points $x_{1}=(\mathbf{r}_{1},t_{1}),\ldots,x_{2n}=(\mathbf{r}_{2n},t_{2n})$, is encoded in the $n$th-order correlation (or coherence) matrix Glauber63 ; Mandelbook

[TABLE]

Here $\hat{\mathbf{E}}^{(+)}(x)$ and $\hat{\mathbf{E}}^{(-)}(x)$ are the positive and negative frequency parts of the electric-field operator, while $\hat{\rho}$ is the density operator of the quantum state.

Let us consider a vector-light field in $L$-slit interference. The electric-field operator at slit $m\in\{1,\ldots,L\}$ reads

[TABLE]

where $E_{0}$ is a constant, $\omega$ is the angular frequency, and $\mathbf{k}$ is the wave vector. The polarization vectors $\mathbf{e}_{ms}$ fulfill the transversality condition $\mathbf{k}\cdot\mathbf{e}_{ms}=0$ and the ortonormality condition $\mathbf{e}_{ms}^{\ast}\cdot\mathbf{e}_{ms^{\prime}}=\delta_{ss^{\prime}}$, while the annihilation operators $\hat{a}_{ms}$ satisfy the commutation relations $[\hat{a}_{ms},\hat{a}_{m^{\prime}s^{\prime}}^{\dagger}]=\delta_{mm^{\prime}}\delta_{ss^{\prime}}$ as well as $[\hat{a}_{ms},\hat{a}_{m^{\prime}s^{\prime}}]=0=[\hat{a}_{ms}^{\dagger},\hat{a}_{m^{\prime}s^{\prime}}^{\dagger}]$. After inserting Eq. (4) into Eq. (Quantum correlations and complementarity of vectorial light fields), and using the abbreviations $\kappa_{1}=m_{1}s_{1},\ldots,\kappa_{2n}=m_{2n}s_{2n}$, the $n$th-order correlation matrix for the slits $m_{1},\ldots,m_{2n}\in\{1,\ldots,L\}$ becomes

[TABLE]

We next expand the state density operator in Eq. (5b) as $\hat{\rho}=\sum_{N=0}^{\infty}p_{N}\hat{\rho}^{(N)}$, in which $\hat{\rho}^{(N)}$ are $N$-photon states and $p_{N}$ are probability distributions. The average photon number of the whole state $\hat{\rho}$ is thereby $\bar{N}=\sum_{N=0}^{\infty}p_{N}N$. In atomic physics, various superselection rules exclusively admit density operators of such “block-diagonal” form, as superposition states of different number of particles (“off-diagonal” terms) are prohibited within first quantization. Because Eq. (5b) preserves the total number of photons, we may consider multiphoton states of the above form without loss of generality.

[TABLE]

Combining Eqs. (5b), (6c), and (6d) now yields

[TABLE]

Equation (7) establishes a fundamental relation between the field correlations and the particle correlations, stating that the $n$th-order cross-spectral density and thereby the $n$th-order correlation (coherence) matrix in Eq. (5) are fully specified by the reduced $n$-photon density matrix. We emphasize that even if first-quantized reduced density matrices are ubiquitous in atomic physics, they are rarely met in quantum optics.

First-order correlations and complementarity.—Let us investigate the implications of Eq. (7) in the context of first-order correlations. For this we introduce the vector

[TABLE]

where $\hat{E}_{s}^{(+)}(x_{m})$ is the $s\in\{1,2\}$ polarized part of the field operator in slit $m\in\{1,\ldots,L\}$, as given in Eq. (4). We then construct the first-order field density matrix

[TABLE]

[TABLE]

Analogously to polarimetric purity Gil07 ; Gil18 , we define the first-order degree of field purity $F^{(1)}$ according to

[TABLE]

with $L\leq\mathcal{K}\leq 2L$ being the number of modes considered (e.g., $\mathcal{K}=L$ if the field is fully polarized in all openings) and where the subscript F stands for the Frobenius norm. Because $\mathrm{tr}\mathbf{G}^{(1)}(x_{m},x_{m})=I_{0}\bar{N}_{m}$, with $\bar{N}_{m}$ being the average photon number in slit $m$, we have $\mathrm{tr}\boldsymbol{\mathcal{G}}^{(1)}=I_{0}\bar{N}$. The degree of purity obeys $0\leq F^{(1)}\leq 1$ and is a measure of all the first-order field correlations within the system. The maximum $F^{(1)}=1$ is saturated when the field is completely polarized and first-order coherent at all slits. In this case the correlation matrices in Eq. (9b) factorize in the respective space-time variables Glauber63 ; Mandelbook and merely one eigenvalue of the field density matrix $\boldsymbol{\mathcal{G}}^{(1)}$ is nonzero. Likewise, the minimum $F^{(1)}=0$ takes place if the field is unpolarized and first-order incoherent for all openings, with additionally $\bar{N}_{1}=\ldots=\bar{N}_{L}$, and in such scenario all eigenvalues are equal.

The first-order particle correlations in the system are described by the one-photon reduced density matrix $\varrho^{(1)}$. To quantify the amount of such particle correlations, we utilize the linear entropy of the trace-normalized $\varrho^{(1)}$,

[TABLE]

with $\mathrm{tr}\varrho^{(1)}=\bar{N}$ and the prefactor ensuring $0\leq S^{(1)}\leq 1$. The upper bound $S^{(1)}=1$ stands for maximal first-order particle correlations in the full $\bar{N}$-photon state $\hat{\rho}$ and is only met when $\tilde{\varrho}^{(1)}$ is maximally mixed. The lower bound $S^{(1)}=0$ corresponds to total lack of first-order particle correlations and is solely encountered if $\tilde{\varrho}^{(1)}$ is pure.

From Eqs. (5), (7), and (9b) we learn that

[TABLE]

and on further combining Eqs. (10a)–(10c) we end up with the first main result of this Letter:

[TABLE]

Equation (11) forms a fundamental quantum complementarity relation between the field correlations represented by $F^{(1)}$ and the particle correlations represented by $S^{(1)}$. It shows that these two quantum correlation species are mutually exclusive, i.e., a variation of $F^{(1)}$ or $S^{(1)}$ alters its complementary partner such that the totality of first-order correlations within the system remains unchanged. Equation (11) is very general as it sets no restrictions on the quantum state or the number of slits; it covers any mixed state in multislit vector-light interference.

Wave-particle triality.—The arguably most recognized manifestion of complementarity is wave-particle duality, restricting the coexistence of “which-path information” and intensity-fringe visibility of quantum objects Greenberger88 ; Jaeger95 ; Englert96 . Photons, however, can exhibit interference not merely in terms of intensity fringes but also via polarization-state fringes, a unique characteristic of vector-light fields with no correspondence in scalar-light interference footnote3 ; Setala06a ; Leppanen14 . Implications of such polarization modulation in quantum-light complementarity have been recently studied in the celebrated double-slit configuration, leading to previously unexplored, fundamental physical findings about photon wave-particle duality Norrman17 ; Norrman18 .

[TABLE]

In the observation plane $\mathcal{B}$, the photon number and the polarization-state variations of the vector-light field are described by the total visibility Norrman17

[TABLE]

including the degree of vector-light coherence

[TABLE]

The degree of coherence satisfies $0\leq g^{(1)}(x_{1},x_{2})\leq 1$ and thus also the total visibility is bounded as $0\leq V(x)\leq 1$. Equation (12b) is the vector-light generalization of the usual visibility relation met in the scalar framework that characterizes the variations of all four Stokes parameters at $\mathcal{B}$ Norrman17 . Therefore, the total visibility does not generally coincide with the intensity visibility but it encompasses also the polarization modulation. For instance, a photon in an even and orthogonally polarized superposition at the slits leads to $V(x)=1$, with maximal polarization-state fringes (due to first-order coherence), although in this case the intensity-fringe visibility is zero Norrman17 .

[TABLE]

From Eqs. (5), (9b), (10a), and (12a)–(12c) we now obtain that in this double-slit vector-light scenario

[TABLE]

which is the recently reported complementarity relation for vectorial quantum light Norrman17 . Such complementarity, or wave-particle duality, is thereby already an inherent part and an important physical manifestation of $F^{(1)}$. The upper limit in Eq. (13a) is always saturated when the photon field is first-order coherent (in the vector sense), viz., $g(x_{1},x_{2})=1$. This holds for any pure single-photon vector-light state, yet not for pure quantum states in general Norrman17 , and highlights very essential physics about wave-particle duality in vector-light interference: for vectorial light the photon path predictability couples not to the intensity visibility (as for coherent scalar light in the classical domain Eberly17 ) but instead to the total visibility which accounts also for the polarization modulation.

If the field is not first-order coherent, i.e., $g(x_{1},x_{2})<1$, then $D^{2}(x_{1},x_{2})+V^{2}(x)<1$ and thus there is no longer a strict complementarity between the path predictability and total visibility. Remarkably, however, on combining Eqs. (11) and (13a) we discover that

[TABLE]

stating that the particle correlations complement the lack of field correlations (and vice versa) such that the whole system is governed by a strict equality, a triality identity. It should not be confused with triality relations involving the concurrence and intensity visibility Jakob10 ; Qian18 , since $S^{(1)}$ quantifies correlations among indistinguishable particles and $V(x)$ characterizes also the polarization modulation. Yet, as $S^{(1)}$ is specified by the reduced one-photon state, Eq. (13b) is in line with the standpoint that wave-particle duality/triality of light is a single-photon feature Qian18 ; Qian19 . Equation (13b) thus unveils a fundamental physical facet concerning vector-light quantum complementarity in the double-slit setup, and constitutes the second main result of this Letter.

In fact, Eq. (13b) encompasses three specific dualities:

[TABLE]

The first case, Eq. (13c), is the usual scenario with first-order coherent fields and no first-order particle correlations; all pure single-photon states fall into this category. The second case, Eq. (13d), is encountered for fields being first-order incoherent and thus displaying no intensity or polarization-state variations in the observation plane; yet first-order correlations among particles are possible. The last case, Eq. (13e), is met for zero path predictability and in this scenario both first-order field correlations as well as first-order particle correlations can be present. For other cases, $0<g^{(1)}(x_{1},x_{2})<1$ and $\bar{N}_{1}\neq\bar{N}_{2}$, the system is governed by the triality relation (13b).

Higher-order correlations and complementarity.—As a final point we investigate quantum complementarity with higher-order correlations. To this end we use $\hat{\boldsymbol{\mathcal{E}}}$ in Eq. (8) and introduce the general $n$th-order field density matrix

[TABLE]

which is Hermitian, non-negative definite, and covers all possible $n$th-order intra-field and inter-field correlations. The $n$th-order degree of field purity $F^{(n)}$ is defined as

[TABLE]

The information on the $n$th-order particle correlations is encoded in the reduced $n$-photon density matrix $\varrho^{(n)}$. Similarly to Eq. (10b), we quantify the amount of these correlations in the whole state $\hat{\rho}$ via the linear entropy

[TABLE]

with the normalizations ensuring $0\leq S^{(n)}\leq 1$ for all $n$. The bound $S^{(n)}=1$ [$S^{(n)}=0$] is only met when $\tilde{\varrho}^{(n)}$ is maximally mixed (pure), representing maximal (full lack of) $n$th-order particle correlations in the system.

Equations (5), (7), (8), and (14) now yield

[TABLE]

which together with Eqs. (15a) and (15b) eventually leads to the third and last major discovery of this Letter:

[TABLE]

Equation (16) states that there is a strict trade-off among field correlations and particle correlations for any $n$, such that the total sum of quantum correlations is invariant. The first-order relation (11) and triality identity (13b) follow from this fundamental complementarity relation. We may thus view Eq. (16) as the most general statement of field-particle duality as regards quantum correlations in vector-light interference; it covers any number of slits, any correlation orders, and any states of light.

Conclusions.—In summary, we have studied quantum correlations in general multislit vector-light interference and established a fundamental complementarity relation between the field correlations and particle correlations that holds for any correlation orders and quantum states. It dictates that these two quantum correlation species are not interchangeable but mutually exclusive and reflects the intrinsic field-particle nature of light at a deep level. For this, we derived a link which connects the $n$th-order field-coherence matrix and the reduced $n$-photon state. The relation was also used to study photon wave-particle duality in the double-slit setup, showing that the particle correlations complement the vector-light duality inequality by transforming it into an equality, a triality identity. Our work thus provides a quantum information-theoretic foundation of optical coherence for vector-light fields, uncovers fundamental aspects concerning field-particle complementarity, and identifies directions towards future research involving quantum interference.

Acknowledgments.—We thank A. T. Friberg, A. Z. Khoury, S. Franke-Arnold, F. De Zela, K. Rzążewski, K. Pawłowski, and R. Seiringer for fruitful discussions and correspondence. A. Norrman acknowledges the Swedish Cultural Foundation in Finland for financial support.