Limits on sequential sharing of nonlocal advantage of quantum coherence

The characteristics of a quantum state without any classical analog are fundamental and key issue of quantum physics Nielsen . Formally, one can introduce different forms of nonlocal correlations to characterize these intriguing characteristics, including Bell nonlocality confirming the nonexistence of the local hidden variable model Bell1 ; Bell2 , Einstein-Podolsky-Rosen (EPR) steering confirming the nonexistence of the local hidden state model steer1 ; steer2 , quantum entanglement originating from the superposition principle of states QE , and quantum discord which is rooted in the noncommutativity of operators QD . These nonlocal correlations are crucial physical resources for quantum communication and computation tasks which outperform their classical counterparts.

For a given quantum state, when one assumes no-signaling among its parties, the monogamy relation imposes constraints on the number of observers who can share the quantum correlations in this state monoe ; monon ; monos ; monoc ; monod . But if the no-signaling condition is partially relaxed, e.g., a single Bob holds half of an entangled pair, while multiple Alices (say, Alice₁, Alice₂, etc.) hold the other half of that pair and perform weak measurements sequentially and independently on their half, then the prior measurement of Alice₁ implicitly signals to Alice₂ by her choice of measurement setting, likewise, Alice₂ signals to Alice₃, and so on. Thereby the monogamy constraints might be relaxed to allow sequential sharing of quantum correlations. In this context, a double violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality with equal sharpness of measurements has been theoretically predicted shareBT1 ; shareBT2 ; shareBT3 ; shareBT4 and experimentally verified shareBE1 ; shareBE2 ; shareBE3 . Further studies showed that an unbounded number of CHSH violations can be achieved using weak measurements with unequal sharpness shareBT6 ; shareBT7 ; shareBT8 . The weak measurement scenario has also been extended to investigate sequential sharing of tripartite Bell nonlocality shareBT9 , EPR steering shareST1 ; shareST2 ; shareST3 ; shareST4 , and bipartite entanglement shareET1 . Then an open question to ask is whether other forms of quantum correlations could be sequentially shared, especially for those beyond the two-qubit case or stronger than Bell nonlocality.

As a fundamental property in quantum theory, quantum coherence is not only an embodiment of the superposition principle of states, but is also intimately related to quantum correlations among the constituents of a system Ficek . In particular, following the resource theory of entanglement QE and quantum discord QD , Baumgratz et al. coher introduced a resource theoretic framework for quantifying coherence, within which the fascinating properties and potential applications of coherence have been investigated in a number of contexts Plenio ; Hu ; Wu ; ad1 ; ad2 ; ad3 . The resource theory of coherence can also be used to interpret those already known quantum correlations coen1 ; coen2 ; coen3 ; coqd1 ; coqd2 ; coqd3 ; coqd4 and introduce other quantifiers of correlations Hu , among which is the nonlocal advantage of quantum coherence (NAQC) naqc1 ; naqc2 . It is captured by violations of the coherence steering inequalities, which is similar to the Bell nonlocality captured by violation of the CHSH inequality CHSH . Specifically, for a $(d\times d)$-dimensional state $\rho_{AB}$ with $d$ being a power of a prime (hereafter, we call it the two-qudit state), when the steered coherence on $B$ after local measurements on $A$ exceeds a threshold, we say that there is NAQC in the sense that such a coherence is unattainable for any product state. As for its hierarchy with other quantum correlations, the set of states with NAQC forms a subset of entangled states naqc1 ; naqc2 and for the $d=2$ case, it is also a subset of Bell nonlocal states naqc3 .

In this work, we investigate how many Alices could sequentially demonstrate NAQC with a single Bob. We first analyze the information-disturbance trade-off of the $d$-dimensional unsharp measurements. Then in the context of unsharp measurements, we show that different from the sequential sharing of Bell nonlocality shareBT1 ; shareBT2 ; shareBT3 ; shareBT4 ; shareBE1 ; shareBE2 ; shareBE3 ; shareBT6 ; shareBT7 ; shareBT8 ; shareBT9 , EPR steering shareST1 ; shareST2 ; shareST3 ; shareST4 , and entanglement shareET1 , at most one Alice can demonstrate the NAQC with Bob. In particular, such a limit exists even when one considers the weak measurements with optimal pointer states. These results may enrich our comprehension on the interplay between NAQC and measurements on high-dimensional systems.

In 2014, Baumgratz et al. coher introduced a resource theoretic framework for quantifying coherence. Within this framework, the incoherent states are defined as those described by the diagonal density operators, and for a given state described by the density operator $\rho$, the amount of coherence could be quantified by its minimal distance to the set $\mathcal{I}$ of incoherent states in the same Hilbert space. By fixing the basis $\{|i\rangle\}$ which can be recognized as the normalized eigenbasis of a Hermitian operator $\mathcal{O}$, Baumgratz et al. coher further identified two coherence measures, that is, the $l_{1}$ norm and relative entropy of coherence, which are given by

where the subscripts $l_{1}$ and $re$ indicate the metrics of the two coherence measures, while $S(\rho)$ and $S(\rho_{\mathrm{diag}})$ are the von Neumann entropies of $\rho$ and $\rho_{\mathrm{diag}}=\sum_{i}\langle i|\rho|i\rangle|i\rangle\langle i|$, respectively.

Starting from the above coherence measures, one can then introduce the NAQC which captures the nonlocal property of a bipartite state. Specifically, the NAQC characterizes the ability of one party to steer the coherence of the other one when they share a two-qudit state $\rho_{AB}$. To illustrate such a nonlocal characteristics, we suppose qudit $A$ ($B$) belongs to Alice (Bob) and denote by $\{A^{v}\}$ the set of $d+1$ mutually unbiased observables. Alice measures randomly one of the observables on qudit $A$ and informs Bob of her choice $A^{v}$ and outcome $a$. Then the conditional state of qudit $B$ will be given by

where $\Pi^{v}=\{\Pi^{v}_{a}\}$ denotes the measurement operator of Alice, $\mathds{1}$ is the identity operator, and $p_{\Pi^{v}_{a}}=\mathrm{Tr}[(\Pi^{v}_{a}\otimes\mathds{1})\rho_{AB}]$ is the probability of Alice’s measurement outcome $a$.

After Alice’s local measurements, Bob can measure coherence of the conditional states on $B$ in different reference bases. In the first framework, Bob chooses with equal probability $1/d$ one of the eigenbasis of $\{A^{u}\}_{u\neq v}$, then one can obtain the average steered coherence (ASC) $N_{AB}^{\alpha}(\rho_{AB})$ ($\alpha=l_{1}$ or $re$) and the criterion for capturing NAQC in $\rho_{AB}$ is given by naqc1 ; naqc2

where the critical value $N_{c}^{\alpha}$ is obtained by first summing the single-qudit coherence over the $d+1$ mutually unbiased bases and then maximizing it over all the single-qudit states naqc2 . In the second framework, Bob chooses the eigenbasis of $A^{\beta_{v}}$ after Alice’s measurement $\Pi^{v}$, with $\beta=\{\beta_{v}\}_{v=0}^{d}$ being a permutation of the set $\{0,1,\ldots,d\}$ with elements $\beta_{v}$. Then the criterion for capturing NAQC in $\rho_{AB}$ becomes naqc2

where the maximum is taken over the $(d+1)!$ (the factorial of $d+1$) possible permutations of $\{0,1,\ldots,d\}$.

For the special $d=2$ case, the sharing of NAQC captured by the criterion of Eq. (3) has been studied sharenaqc ; sharenaqce . However, Eq. (3) is less efficient than Eq. (4) in capturing NAQC naqc2 , hence we will focus on the latter when discussing sharing of NAQC by sequential observers.

In the framework of von Neumann-type measurement von , the measurement process on a $d$-dimensional state $\rho_{0}$ implies interaction of the system with the apparatus which induces the map: $\mathcal{E}(\rho_{0}\otimes|\phi\rangle\langle\phi|)=\sum_{ij}\Pi_{i}\rho_{0}\Pi_{j}\otimes|\phi_{i}\rangle\langle\phi_{j}|$, where $\{\Pi_{i}\}$ denotes the measurement operators, $|\phi\rangle$ is the initial pointer state of the apparatus, and $|\phi_{i}\rangle$ is the postmeasurement state of the pointer associated with the outcome $i$. By tracing out the pointer states one can obtain the nonselective postmeasurement state as $\rho=\sum_{ij}\Pi_{i}\rho_{0}\Pi_{j}\langle\phi_{i}|\phi_{j}\rangle$, where $\langle\phi_{i}|\phi_{j}\rangle$ may be different for different $i\neq j$. Without loss of generality, here we consider $\langle\phi_{i}|\phi_{j}\rangle\equiv F$ ($\forall i\neq j$) for simplicity, then $\rho$ can be reformulated as

where $F\in[0,1]$ is the quality factor of the measurement, with $F=0$ corresponding to the usual projective (strong) measurement. $F$ measures the extent to which the system state remains undisturbed after the measurement and depends on the pointer of the apparatus by its definition shareBT1 .

One point to be stressed here is that the reduced disturbance of a weak measurement will induce reduced information gain. To quantify such a quantity (i.e., the information gain or precision of the measurements), one needs to choose a complete orthogonal set of states $\{|\varphi_{i}\rangle\}$ as reading states because the set $\{|\phi_{i}\rangle\}$ is non-orthogonal. As a result, the probability of getting the outcome $i$ and the associated (unnormalized) postmeasurement state $\rho_{i}=\langle\varphi_{i}|\mathcal{E}(\rho_{0}\otimes|\phi\rangle\langle\phi|)|\varphi_{i}\rangle$ are given by

where $|\langle\varphi_{i}|\phi_{i}\rangle|^{2}$ ($|\langle\varphi_{i}|\phi_{j\neq i}\rangle|^{2}$) is the probability of obtaining the correct (wrong) outcome. When the measurement is unbiased, i.e., $|\langle\varphi_{i}|\phi_{i}\rangle|^{2}$ is independent of $i$ and $|\langle\varphi_{i}|\phi_{j\neq i}\rangle|^{2}$ is independent of $j\neq i$, Eq. (6) can be reformulated as

where $G=1-d|\langle\varphi_{i}|\phi_{j\neq i}\rangle|^{2}$ is the precision of the measurement which quantifies the information gain from the measured system and we have denoted by $\mathcal{F}=[(1+d_{1}G)(1-G)]^{1/2}$, where $d_{1}=d-1$. $G$ also depends on the pointer states of the measuring apparatus. Usually, one has the trade-off $F^{2}+G^{2}\leqslant 1$, and for $F^{2}+G^{2}=1$, the trade-off is said to be optimal in the sense that the measurement yields the highest precision for a given quality factor shareBT1 .

In this paper, we follow the framework of Refs. shareBT1 ; shareBT2 ; shareST2 and consider the $d$-dimensional unsharp measurements represented by the set of effect operators

where $\{E^{v}_{a}\}$ represents the measurement settings with $d$ possible outcomes per setting, $0<\lambda\leqslant 1$ represents the sharpness parameter, and $\Pi^{v}_{a}=|\phi^{v}_{a}\rangle\langle\phi^{v}_{a}|$ is the projector. In the following, we restrict ourselves to $\Pi^{v}_{a}$ ($v=0,1,\ldots,d$) constructed by the $d+1$ mutually unbiased bases MUB1 ; MUB2 :

where $\delta_{an}$ is the Delta function and $i$ represents the imaginary unit. One can note that the unsharp measurement operator $E^{v}_{a}$ corresponds to a linear combination of the projector $\Pi^{v}_{a}$ with the white noise. It satisfies the relation $\sum_{a}E^{v}_{a}=\mathds{1}$ ($\forall v$) and belongs to the class of positive-operator-valued measurements. In contrast to the conventional strong measurement which enables an extraction of the maximum information and destroys completely the system to be measured, for the unsharp measurements the system is weakly coupled to the probe and thus provides less information about the system while producing less disturbance weak1 ; weak2 . Hence, the postmeasurement states retain some original properties of the measured system which might be observed by the subsequent observers. Moreover, $E^{v}_{a}$ reduces to the projective (strong) measurements when $\lambda=1$, and for such a special case, the basis comprising $\Pi^{v}_{a}$ is an essential ingredient for introducing the flag additivity condition which is equivalent to the strong monotonicity and convexity of a coherence measure new1 ; new2 .

Note that for $d=2$, the effect operators can also be written as $E_{\pm}=(\mathds{1}\pm\lambda\hat{n}\cdot\vec{\sigma})/2$, where $\hat{n}$ is a unit vector in $\mathbb{R}^{3}$ and $\vec{\sigma}$ is a vector composed of the three Pauli operators.

From Eq. (8) one can obtain that for any initial state $\rho_{0}$, the nonselective postmeasurement state is given by Luders

and the probability of getting the outcome $a$ is given by

where

Then by comparing Eqs. (5) and (7) with Eqs. (10) and (11), one can see that the quality factor and precision of the unsharp measurements (8) are respectively given by

Hence unsharpening the measurements with a parameter $\lambda$ enables the control of the trade-off between disturbance and information gain. For $d=2$, one always has the optimal trade-off $F^{2}+G^{2}=1$. But for the prime $d\geqslant 3$, $F^{2}+G^{2}\leqslant 1$, and the equality holds only for $\lambda=1$, which corresponds to $F=0$ and $G=1$, namely, the case of a projective (strong) measurement shareBT1 .

In Fig. 1 we give a plot of the trade-off between the quality factor $F$ and precision $G$ for the unsharp measurements of Eq. (8) with the first ten primes, and for comparison, we also show the trade-off for the Gauss pointer considered usually and the simple square pointer shareBT1 . For $d=2$, as mentioned before, it saturates the optimal trade-off constraint $F^{2}+G^{2}=1$, i.e., for any $F$, there exists optimal measurement pointer that achieves the maximum $G$ shareBT1 . For the prime $d\geqslant 3$, as shown in Fig. 1, although the corresponding trade-off is not optimal, it is still better than that given by the Gauss pointer for any $F$ (if $d=3$) or when $F$ is smaller than a threshold (if $d\geqslant 5$), and with an increase in $d$, it approaches gradually the trade-off $F+G=1$ given by the square pointer shareBT1 .

To address the question for sequential sharing of the NAQC, we consider a scenario in which multiple Alices (say, Alice₁, Alice₂, etc.) have access to half of an entangled qudit pair and a spatially separated single Bob has access to the other half, and they agree on the measurement settings $\{E^{v}\}$ in prior shareBT1 . First, Alice₁ and Bob share the state $\rho_{A_{1}B}$ and Alice₁ proceeds by choosing randomly one of $\{E^{v}\}$ and performs the unsharp measurements on qudit $A_{1}$. She then passes the measured qudit (we rename it as qudit $A_{2}$) on to Alice₂ who measures again and passes it on to Alice₃, and so on until the last Alice. During the whole process, every Alice is assumed to be ignorant of the measurement settings chosen by the former Alices, that is, communications among them are forbidden and each Alice chooses independently and randomly one of the measurement setting. Our aim is to determine the maximum number of Alices whose statistics of measurements can demonstrate NAQC with a spatially separated single Bob.