Theory-Independent Measure of Coherence

TL;DR

This paper introduces a theory-independent framework for quantifying coherence, applicable to quantum and non-local models, using sequential measurement statistics to distinguish quantum coherence from other phenomena.

Contribution

It proposes a novel, theory-independent method to measure coherence through measurement outcome deviations, applicable to quantum mechanics and non-local models.

Findings

Introduces two new quantum coherence measures.

Identifies a finite gap in coherence between non-local models and quantum mechanics.

Provides a framework to distinguish quantum coherence from non-locality.

Abstract

We provide a theory independent framework to quantify coherence. In comparison with Bell's theory independent approach to quantum nonlocality, we characterize a general coherence phenomenon with statistics arising from sequential measurements of observables that might not be compatible. By introducing a "decohered" state after the sharp measurement of some preferred observable, we quantify coherence by either the difference of initial "superposed" state from the "decohered" state or the measurement outcome deviations when they subjected to further measurements. Applied to quantum mechanics, the outcome-difference measures yield two novel quantum coherence measures, one of which upper-bounds quantum interference visibility. In the Bell's scenario, we find a finite gap of coherences between a super non-local model and quantum mechanic and therefore our framework can help to single out quantum mechanics beyond non-locality.

Equations

$$C_E(\mathcal{S})=\left|\mathbb{S}(\mathcal{S})-\mathbb{S}(\overline{{\mathcal{S}}_A})\right|.$$

$$C_M=\underset{A^{\prime }}{\sup }\mathrm{Dis}({\mathbf{p}}_{A^{\prime }},{\mathbf{p}}_{A^{\prime }}^A).$$

$$C_M^{\mathrm{K}}$$

$$C_M^{\mathrm{F}}$$

$$C_E(\rho )=S(\rho )-S({\rho }_A)=S(\rho \Vert {\rho }_A),$$

$$C_M^{\mathrm{K}}$$

$$C_M^{\mathrm{F}}$$

$$\mathrm{Tr}\left|\Lambda (\rho )-\Delta (\Lambda (\rho ))\right|\le \mathrm{Tr}\left|\rho -\Delta (\rho )\right|,$$

$$\mathrm{Tr}|\Lambda (\rho )-\Delta (\Lambda (\rho ))|$$

$$\mathrm{Tr}\left|p_1{\rho }_1\oplus p_2{\rho }_2-p_1{\rho }_{1A}\oplus p_2{\rho }_{2A}\right|$$

$$=p_1\mathrm{Tr}|{\rho }_1-{\rho }_{1A}|+p_2\mathrm{Tr}|{\rho }_2-{\rho }_{2A}|.$$

$$V=\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}.$$

$$V$$

$$y=X\cdot Y\oplus x.$$

$$C_M^{\mathrm{K}}({\omega }_{0|Y})$$

$$C_M^{\mathrm{K}}({\omega }_{0|1})+C_M^{\mathrm{K}}({\omega }_{0|0})\ge 1,$$

$$C_M^{\mathrm{K}}({\omega }_{0|1})+C_M^{\mathrm{K}}({\omega }_{0|0})\le \frac{1}{2}({\Delta }_{{\omega }_{0|1}}A+{\Delta }_{{\omega }_{0|0}}A)=0.$$

Full text

Introduction.—Coherence is a fundamentally non-classical feature. It is the resource of many quantum phenomena that provide advantages in practical applications over classical resources 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 . In quantum mechanics (QM), coherence originates from the superposition principle, which states that a superposition of valid states forms a new valid state. Recently, considerable work has been devoted to characterizing and quantifying quantum coherence Cor ; Cd ; CE ; CR ; Cop ; Cro1 ; Cro2 ; rmp ; Bis . These coherence measures were either defined by functions of a density matrix’s off-diagonal entries, or proposed with an analogy with entanglement theory Cor ; rmp . These measures then have been studied on the foundations of quantum resource theory, leading many results rmp .

On the other hand, studying quantum phenomena in a theory-independent manner enables one to find physical ground behind their formulism gs ; Barr ; bel ; gpt ; ic ; wvd ; cl , then yielding a deep understanding of QM. For example, Bell’s inequalities only involve necessary constitutive components while not relying on any specific structures bell , thus appliable to any theory bell ; CHsh ; pr . By the inequalities, it has been found that the no-signalling principle is not enough to specify quantum correlation pr , therefore triggering fruitful research for principles constraining QM correlation ic ; wvd ; cl . It has been found that many features, thought special to QM, can be studied in a theory-independent manner, $e.g.$, teleportation ta ; te , purification pur and entanglement swapping sw . Coherence is another fundamental QM feature, and a theory-independent quantification of it would not only provide intuitive coherence measures but also provide a new viewpoint to specify QM. However, such a framework is missing.

In this Letter, we provide a theory-independent coherence quantification framework. The framework focuses on the features attributed to coherence while not relying on the specific structure of a theory. We introduce classical mixture counterpart (CMC) for a state corresponding to a general physical system (like the local realistic theory in Bell’s inequalities). Coherence is characterized by the difference of “superposed” system from its CMC, and their outcome deviations when subjected to experiment. We then give two kinds of coherence measures, $i.e.$, ensemble-difference (ED) measure and measurement-outcome-difference (MOD) measure. For the MOD measure, we employ Kolmogorov distance and fidelity distance to quantify the outcome difference, denoted as MOD-K and MOD-F, respectively. We then apply the framework to QM, and find that quantum ED measure coincides with the relative entropy measure, while both quantum MOD-K and quantum MOD-F are novel QM coherence measures. As quantum phase-sensitive-interference-visibility (PSIV) is often taken as a coherence measure Bis ; wd1 ; wd2 ; wd3 , we compare quantum MOD-K with PSIV, find that MOD-K captures all the coherence carried by a quantum state and upper-bounds the PSIV. Finally, coherence predicted by Popescu and Rohirlich box model (PR-box) pr is studied, and difference between PR-box coherence and QM coherence is shown.

Superposition and classical mixture.—The key observation from the two-slit interference experiment is that: the photons do not pass through either upper or lower slit independently, but pass through the two slits simultaneously PHY ; wd1 ; Wh , and then coherent superposition of pathes is claimed. Coherence is defined with respect to a preferred observable here denoted by $A$ and not confined to the path. To be more general, $A$ and another observable $A^{\prime}$ (which shall be used) are treated as general discrete valued observables. Note that even through we are talking about observables and measurements, we do not deal with them in QM, and all the mentioned measurements are sharp measurements. For a general underlying theory, we only attach several necessary constitutive properties re ; Von : for any qualified observable, there should be a corresponding accurate measurement referred to as the sharp measurement. For a sharp measurement, we take measurement of observable $A$ for example, each run would yield an outcome $a_{j}$ with probability $p_{j|A}$ then the state of system $\mathcal{S}$ would be prepared in an “eigenstate” of $A$ denoted as $\mathcal{S}_{j|A}$; an immediately sequential measurement of $A$ would repeat the outcome $a_{j}$ sm1 ; sm2 .

After performing sufficient amount circles of measurement $A$ on state ensemble $\mathcal{S}$, the post-measurement ensemble would be transferred to a mixture of “eigenstates”: $\overline{\mathcal{S}_{A}}=\sum_{j}p_{j|A}\mathcal{S}_{j|A}$. $\overline{\mathcal{S}_{A}}$ is defined as the CMC of $\mathcal{S}$ with respect to preferred observable $A$. The CMC is introduced as a free coherence state then taken for the comparation with original state, thus the coherence is characterized.

Ensemble-difference and measurement-outcome-difference coherence measures.—As the sharp measurement is repeatable, the preparation of CMC does not change the statistics of the preferred observable, while destroys the coherence. Our first measure quantifies the change of ensemble state due to coherence destruction. We employ measurement entropy e to identify an state, and it is defined as: $\mathbb{S}(\mathcal{S})=\inf_{A^{\prime}}\{\sum_{j}-p_{j|A^{\prime}}\log p_{j|A^{\prime}}\}$, where $\{p_{j|A^{\prime}}\}$ denotes statistic from measurement of observable $A^{\prime}$. One direct coherence measure with respect to $A$ is defined as,

[TABLE]

ED coherence measure is theory-independent as it does not rely on specific structure.

Our second measure is motivated by quantifying the difference between observable “fringes” from measurements on "photons" with superposed “pathes” and that with mixed “pathes”. We denote by $\{p_{i|A^{\prime}}\}$ and $\{p^{A}_{i|A^{\prime}}\}$ the statistics of measurement $A^{\prime}$ on $\mathcal{S}$ and $\overline{\mathcal{S}_{A}}$, respectively, where $p^{A}_{i|A^{\prime}}=\sum_{j}p_{j|A}p_{i;A^{\prime}|j;A}$ and $p_{i;A^{\prime}|j;A}$ denotes probability of obtaining $i$ when measuring $A^{\prime}$ on ‘eigenstate’ of $A$ with value $j$. The MOD coherence measure reads

[TABLE]

We have many distance measures for probability distribution. In the following, we take the Kolmogorov distance and fidelity distance 1 , with corresponding measures referred to as MOD-K (denoted as $C_{M}^{\rm K}$) and MOD-F (denoted as $C_{M}^{\rm F}$), respectively,

[TABLE]

Here, the fidelity is defined as $\operatorname{F}(\mathbf{p}_{A^{\prime}},\mathbf{p}^{A}_{A^{\prime}})=\sum_{i}\sqrt{p_{i|A^{\prime}}p^{A}_{i|A^{\prime}}}$. From above definitions, it can be seen that these measures are also theory-independent.

Application to QM.—A qualified theory-independent measure should yield a well-defined special measure when applied to a given theory. We now apply our framework to QM and consider a quantum state’s coherence. In QM, a state of system is represented as a density matrix $\mathcal{S}=\rho$, and the sharp measurement as projective measurement sm1 ; sm2 ; Von . Thus CMC of $\rho$ with respect to preferred observable $A$ is given as $\overline{\mathcal{S}_{A}}\equiv\sum_{i_{A}}|i_{A}\rangle\langle i_{A}|\rho|i_{A}\rangle\langle i_{A}|:=\rho_{A}$, where $|i_{A}\rangle\langle i_{A}|$ denotes the $i$-th eigenstate of $A$ re ; Von and $\rho_{A}$ denotes the completely decohered state.

In QM, we have $\mathbb{S}(\rho)=S(\rho):=-\operatorname{Tr}\left(\rho\log\rho\right)$, which is von Neumann entropy 1 ; Von ; e . The ED measure is given as

[TABLE]

where $\operatorname{Tr}\left(\rho_{A}\log\rho_{A}\right)=\operatorname{Tr}\left(\rho\log\rho_{A}\right)$ has been used. $S(\rho\|\rho_{A})$ is the relative entropy between $\rho$ and $\rho_{A}$, and it has been studied as a well-defined coherence measure in Cor ; rmp .

We now show that MOD yields two novel quantum coherence measures. Consider the quantum version of MOD-K, it can be written as

[TABLE]

where $\{\hat{P}_{i|A^{\prime}}\}$ are projective operators of $A^{\prime}$, and $|\rho-\rho_{A}|:=\sqrt{(\rho-\rho_{A})^{{\dagger}}(\rho-\rho_{A})}$. The reason for the third equality is that: the operator $\rho-\rho_{A}$ is hermitian, and its maximum is obtained when bases of $A^{\prime}$ are the eigenvectors of $\rho-\rho_{A}$ 1 . Thus, and the quantum $C_{A}^{\rm K}$ has a simple formula as the trace distance of $\rho$ from $\rho_{A}$.

We have quantum version of MOD-F as

[TABLE]

where $\operatorname{F}_{q}(\rho,\rho_{A})=\operatorname{Tr}\sqrt{\sqrt{\rho}\rho_{A}\sqrt{\rho}}$ is quantum fidelity between the state $\rho$ and $\rho_{A}$. In QM, $\inf_{A^{\prime}}F(\mathbf{p}_{A^{\prime}},\mathbf{p}^{A}_{A^{\prime}})=F_{q}(\rho,\rho_{A})$, which also is a simple formula 1 .

Recently, a general quantum resource measure was proposed as a distance of state $\rho$ from its resource-destroyed state $\mathfrak{D}(\rho)$: $\mathfrak{D}(\rho)=D(\rho,\Gamma(\rho))$ d . Though, our quantum coherence measures are not derived from a quantum resource theory but in a theory independent manner, interestingly, in QM, we find that all $C_{E}$, $C_{M}^{\rm K}$ and $C_{M}^{\rm F}$ formally coincide with $\mathfrak{D}$. Different from the measure $\Gamma(\rho)$, our measures are applicable to a general theory. We shall apply it to PR-box model, while $\mathfrak{D}(\rho)$ is confined to QM.

As quantum $C_{M}^{\rm K}$ and $C_{M}^{\rm F}$ are novel measures, let us now examine them with the requirements proposed in Ref. Cor ; rmp ; T , which has been taken as qualifications for a well-defined quantum coherence measure:

$C(\rho)\geq 0$ for all quantum states, and $C(\rho)=0$ if and only if $\rho$ are incoherent states. 2. 2.

$C(\rho)\geq C(\Lambda(\rho))$ if $\Lambda$ is incoherent operation. 3. 3.

$C(p_{1}\rho_{1}\oplus p_{2}\rho_{2})=p_{1}C(\rho_{1})+p_{2}C(\rho_{2})$ for Bloch diagonal states $\rho$ in the incoherent basis.

As the fidelity shares many similar properties with the trace distance 1 , we only need to examine the quantum MOD-K with the requirements, and the results are applicable to quantum MOD-F. Quantum MOD-K definitely satisfies the first requirement. We shall prove the requirement 2 under dephasing-covariant-incoherent operation. We need to prove

[TABLE]

where $\Delta$ denotes a dephasing map $\Delta(\rho)=\sum_{i}|i\rangle\langle i|\rho|i\rangle\langle i|$, and it maps a state to its CMC. $\Lambda$ denotes a dephasing-covariant-incoherent operation (Chitamber and Gour have made a critical examation on various incoherent operations and showed their containment relationships Cop ) which commutes with $\Delta$ by definition, then $\operatorname{Tr}(|\Lambda(\rho)-\Delta(\Lambda(\rho))|)=\operatorname{Tr}(|\Lambda(\rho)-\Lambda(\Delta(\rho))|)$. We take the trace norm convexity theorem 1 : the trace distance of arbitrary states $\rho_{1}$ and $\rho_{2}$ is contractive under any tracing-preserving quantum operations $\xi$ , $\operatorname{Tr}|\xi(\rho_{1})-\xi(\rho_{2})|\leq\operatorname{Tr}|\rho_{1}-\rho_{2}|$. As $\Lambda\subset\xi$ 1 ; Cor , we have,

[TABLE]

As for the requirement 3, it is obvious that

[TABLE]

Thus quantum MOD-K satisfies all three requirements under dephasing-covariant-incoherent operation. Although the proof under maximal incoherent operation is still missing, quantum MOD-K is physically well-motivated. We shall show that the quantum MOD-K upper-bounds a commonly used interference measure.

In the two-slit interference experiments, PSIV has been used as a quantifier of coherence between two paths wd1 ; wd2 ; wd3 ; wds , which is defined by the sensitivity of measurement outcome to phase shift. In the schemes , an input state $\rho$ is distributed into two ways based on a $\sigma_{z}$–beam splitter, then undergoes a path-dependent phase shift $U^{{\dagger}}_{\phi}\rho U_{\phi}$, where $U_{\phi}=\exp(-\operatorname{i}{\phi}\sigma_{z}/2)$, and merges to a final measurement. The frequencies of outcome $+1$ is $\operatorname{I}$, and the PSIV is commonly defined as 9

[TABLE]

where $\operatorname{I}_{\min}=\inf_{\phi}\operatorname{Tr}\left(U^{{\dagger}}_{\phi}\rho U_{\phi}\hat{P}_{+}\right)$, $\operatorname{I}_{\max}=\sup_{\phi}\operatorname{Tr}\left(U^{{\dagger}}_{\phi}\rho U_{\phi}\hat{P}_{+}\right)$, and $\hat{P}_{+}$ is the projector of the final measurement on the direction $(|0\rangle+|1\rangle)/\sqrt{2}$. By trivial calculus, we find that the visibility is exactly the same as quantum MOD-K with preferred observable $\sigma_{z}$ up to a constant 2, namely $V=2C_{M}^{\rm K}$.

With the above observation, we now explore a general relation between PSIV and quantum MOD-K. Consider a general measurement where state $\rho$ is distributed to $d$ pathes based on a "$A$–beam splitter", and each one undergoes a phase shift before merging to a measurement $A^{\prime}$. Taking all the frequencies into consideration, we generalize Eq. (11) as $V_{d}=\frac{1}{2}\sup_{\Phi,\Phi^{\prime}}\sum_{i}|p_{i|\Phi,A^{\prime}}-p_{i|\Phi^{\prime},A^{\prime}}|$, where $p_{i|\Phi,A^{\prime}}=\operatorname{Tr}(U^{{\dagger}}_{\Phi}\rho U_{\Phi}\hat{P}_{i|A^{\prime}})$, $U_{\Phi}$ denotes phase shift and $\rho_{\Phi}=U^{{\dagger}}_{\Phi}\rho U_{\Phi}$. Quantum MOD-K is found to upper-bound the visibility

[TABLE]

Equation (13) usually does not saturate, because the diagonal terms of $\rho_{\Phi}$ and $\rho_{\Phi^{\prime}}$ are equal, while their anti-diagonal terms cancel each other out. Therefore, the coherence of $\rho$ can not be captured by PSIV, while all are captured by the quantum MOD-K.

Application to PR-box.—Bell’s inequalities triggered fruitful results on quantum correlation sm1 ; sm2 ; MI1 . The inequalities consider two space separated observers, Alice and Bob. They share a correlated system and randomly perform measurements $X,Y\in\{0,1\}$ with outcomes $x,y\in\{0,1\}$. The inequalities expose the discrepancies between general theories. The model is defined as $X\oplus Y=x\cdot y$, then we have

[TABLE]

For each of Bob’s settings and outcomes, he prepares a state $\omega_{y|Y}$ on Alice’s side. By the definition of the model, any measurement on any Alice’s state would yield outcome in a deterministic way. In QM, when measured state is an eigenstate of the desired observable, the measurement would yield outcomes in a deterministic way, and the state would not be disturbed. We now show that PR-box model contradicts with this feature. If PR-box model shares this feature: measured state would be not disturbed if the outcome is yielded deterministic. Taking $\omega_{0|0}$ for example, when measurement is $X=0$, then outcome would be deterministic $x=0$. By the assumption of no disturbance, $x=0$ would be obtained for a sequential measurement of $X=1$. Subjecting $\{X=0,x=0\}$, $\{X=1,x=0\}$ into Eq. (14), one can derive the setting and outcome on Bob’s side as $\{Y=1,y=0\}$. Thus if the assumption is applicable to the model, no-instantaneous messaging principle is violated. To guarantee the principle, there must be coherence.

We now apply MOD-K to the model and consider the coherence of $\omega_{0|Y}$ with respect to observable $X=0$. The state $\omega_{0|Y}$ would give result $x=XY$ deterministic. We have its coherence

[TABLE]

Given the repeatable feature of a sharp measurement, the maximum is obtained when $X^{\prime}=1$ is taken, and $p_{i;1|0;0}$ denotes probability of obtaining $i$ when measuring $X=1$ on the state prepared by the measurement $X=0$ with outcome [math]. As a result, we have

[TABLE]

for PR box while in QM we have

[TABLE]

Here, the inequality is because the following upper bound $C_{M}^{\rm K}=|\langle 0_{A}|\rho|1_{A}\rangle|\leq\sqrt{p_{0|A}p_{1|A}}=\frac{1}{2}\Delta A$ for the coherence in an arbitrary qubit state. Thus if the two quantum states yield the same statistics with the $\omega_{0|1}$ and $\omega_{0|0}$ when subjected to reference measurement, their sum of the coherence must equal to [math]. However, by Eq. (16), the lower bound of the sum is $1$. A finite gap between the amounts of their coherence is shown.

Conclusion.—We have provided a theory-independent coherence-quantification framework. Our framework gives two novel quantum coherence measures when applied to QM, and has been used to study the coherence of PR-box and specify QM from the model based on the amounts of their coherence. As theory-independent measure of non-locality, Bell’s inequalities have triggered fruitful research for principles constraining QM, this coherence quantification framework would pave a novel method to specify QM on the other hand.

Acknowledgement.—This work has been supported by the Chinese Academy of Sciences, the National Natural Science Foundation of China under Grant No. 61125502, and the National Fundamental Research Program under Grant No. 2011CB921300.