Open Access
Add to BibSonomy
Abstract
In the context of quantum information, major efforts have been made to maximize the mutual information by measuring single copies of signal states. In general, one execution of optimal projective measurement extracts all the accessible mutual information. However, in some scenarios, weak measurements are preferred because of kinds of specific requirements, e.g., to distribute secret keys to multi-observers. In this study, we propose a method to construct optimal weak measurements for multi-party quantum communications. Utilizing the method in [Physical Review Letters 120, 160501 (2018)] to classify the mutual information, the theoretical study shows that by successively performing this optimal weak measurement, all accessible information can be obtained by multiple observers. This conclusion is experimentally verified by a cascaded measurement apparatus that can perform six successive weak measurements on heralded single photons. The experimental results clearly indicate that almost all accessible mutual information is extracted by this sequence of optimal weak measurements; meanwhile, none of the information is destroyed or residual. Thus, this optimal weak measurement is an efficient and reliable tool for performing quantum communication tasks. The consistence between the experimental and theoretical results verifies that the classifying method in [Phys. Rev. Lett. 120, 160501 (2018)] can be applied to characterize realistic quantum measurements.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum measurements are designed to extract information from various types of quantum states. Specifically, in the context of quantum communications, classical information is encoded into quantum states and is then transmitted between separated nodes [1,2]. In principle, one can repeat measurements on identically prepared copies of a quantum state to maximize the mutual information, which is defined as the reduction of the entropy (uncertainty) about the initial signal states by performing a measurement. However, in practical quantum communication protocols, the requirements on efficiency and security usually allow only one-shot measurements and the observer has to make judgements about the state based on a certain outcome [3]. The identification of approaches used to make the best use of every single copy of quantum states and design an optimal measurement [4–7] to attain the largest amount of information is essential for these quantum communication tasks.
Currently, almost all the quantum communication protocols adopt the binary encoding, in which two orthogonal or nonorthogonal states are encoded as 0 and 1. The observer tries to discriminate what state he receives with a priori knowledge of the binary states, although the non-orthogonal states cannot be fully discriminated in the sense of their certain labeling after a measurement. Thus, the decoding process can be viewed as a state discrimination task [8–12], in which the observer tries to perform an optimal projective measurement (OPM) to attain the so-called accessible mutual information define [13]. The commonly used OPMs in practical quantum communications are projective von Neumann measurements given by rank-one projections, which are widely used in quantum state tomography [14], quantum teleportation [15,16] or quantum key distribution [17]. Different from the OPM which attains the maximal mutual information over all possible POVMs, a weak measurement (WM) extracts partial information. Meanwhile, some information may be destroyed or residual in the postmeasurement state. This residual information not only allows a possibility to recover the postmeasurement state [18,19], but also the information gain by other observers [20–22]. Therefore, a quantum measurement can be fully characterized by dividing the total accessible mutual information into three components: extracted, residual and destroyed [23].
Although OPM is the most widely used tool in various quantum information processing approaches, the post-measurement states are solely determined by the measurement operators and have no residual information for further applications. In contrast, by using WM, it is possible to extract information from a qubit by multiple observers [24]. Reference [24] proposes an unambiguous state discrimination strategy and each POVM (positive operator-valued measure) has three elements, namely, two error-free measurements and a failure measurement. As a result, some of the photons are discarded due to the fail measurement, and the contained information is lost. Therefore, how to formulate WM to extract all the accessible mutual information is an essential ingredient for quantum communications. In this study, we target to the problem to distinguish two non-orthogonal signal states by multiple observers with WM. Unfortunately, normal weak measurement may destroy some information; therefore, even there is no photon loss in the measurements, multiple observers cannot obtain the total accessible mutual information by performing WMs. In this sense, we propose the method to construct the optimal weak measurement (OWM), whereby the optimal means the observers can attain the total accessible mutual information with none information destroyed or residual by successively performing these OWMs. The effect of OWM is characterized using the manner described in Ref. [23], and we show that it can approach the power of OPM after several executions. We further experimentally demonstrate our method to discriminate two nonorthogonal polarized photon states. Our results indicate that we can extract approximately all the accessible mutual information without an attempt to perform OPM.
2. Theory
2.1 Fraction of mutual information
We consider a quantum communication task between Alice and other observers, e.g., Bob, Charlie, etc. Alice encodes the classical information into an ensemble quantum states $\Omega$, which consists of (orthogonal or nonorthogonal) binary signal states $\rho _{1}$ and $\rho _{2}$ with prior probability distributions $p_{1}$ and $p_{2}$. Alice sends it to Bob, who performs a measurement and then sends the post-measurement states to next observer, and so it goes on. With the pre-knowledge of the binary states and fixed prior probabilities, the observers need to identify which state Alice prepares. In principle, each observer can perform any generalized measurement $\Pi$={$\Pi _{j}:j=1,2,\ldots ,M,\sum _{j=1}^{M}\Pi _{j}=\mathbb{1}$} to decode the information from his measurement outcomes, and the mutual information he extracts can be calculated as [23]
$H(\Omega )$ and $H(\Omega |\Pi _{j})$ respectively denote the entropy of the binary states, and the conditional entropy that describes the remaining ignorance about the signal states given the measurement outcome $\Pi _{j}$ with a probability $P_{\Pi _{j}}$.The accessible information about the binary states is defined as the maximal mutual information attainable over all possible POVMs, which can be written as [23]
In principle, when Bob’s measurement is not optimal, any other observer can perform a subsequent measurement on the postmeasurement state to proceed further. The residual information can be defined as the accessible information in the ensemble of postmeasurement states $\Omega ^{(j)}$, denoted as $I_{acc}(\Omega ^{(j)})$, which can only be attained with a subsequent optimal measurement. Thus, the final ignorance about the signal states can be calculated as
where $\Omega ^{f}$ denotes the states after performing a generalized measurement $\Pi$ and a subsequent optimal measurement.The maximal information that can be extracted through these two successive measurements is given by
In general, a quantum measurement destroys some information $I_{des}(\Omega :\Pi )=I_{acc}(\Omega )-I'_{max}(\Omega ,\Pi )$ when it acts on a certain quantum state. Therefore, a quantum measurement can be fully characterized by three variables (i.e., the fractions of extracted information $E$, residual information $R$, and destroyed information $D$), which can be obtained by normalizing $I(\Omega :\Pi )$, $I'_{max}(\Omega ,\Pi )-I(\Omega :\Pi )$ and $I_{des}(\Omega :\Pi )$ with $I_{acc}(\Omega :\Pi )$, respectively [23].2.2 Optimal measurement
For the task of discriminating two signal states, the observer always attempts to perform a measurement to extract the maximal amount of mutual information or attain a minimum average error probability. When the two encoded states are pure, it has been confirmed that these two requirements can be satisfied simultaneously by the so-called optimal measurement [25]. Thus, how to construct the optimal measurement is a crucial question in binary quantum communication schemes. In 1976, Helstrom proposed a general method to construct the optimal measurement to achieve a minimum error probability in binary state discrimination [6], which can be summarized as the following Definition.
Definition 1 We need to discriminate $\rho _{1},\rho _{2}$ with prior probabilities $p_{1},p_{2}$, and $p_{1}+p_{2}=1$. Define $\Lambda =p_{1}\rho _{1}-p_{2}\rho _{2}$, for which the eigenvector is $|\eta _{k}\rangle$ and $\eta _{k}$ is the corresponding eigenvalue. Then, we can construct OPM: $\Pi _{1}=\sum _{p}|\eta _{p}\rangle \langle \eta _{p}|$, $\Pi _{2}=I-\Pi _{1}$, with $\eta _{p}>0$.
Definition 2 With this OPM, a special type of OWM can be constructed as
in which exists $m$ s.t. $c_{1}^{m}\neq c_{2}^{m}$ and $\sum _{m}A^{(m)}=I$ . The corresponding Kraus operator is: $\digamma ^{(m)}=\sum _{l}\sqrt {c_{l}^{m}}\Pi _{l}$ , m labels the outcome of OWM.Here, if $c_{1}^{m}=c_{2}^{m}$ for all values of $m$, it means all the measurement operators are identities, which cannot extract any information.
Our main result can be stated as follows. Theorem 1. If each observer performs this OWM successively on a single copy of the signal state, the total extracted information approaches the accessible information that is acquired by OPM. Quantitatively, for $n$ repetitions of OWM, when $n\rightarrow \infty$, the fractions of mutual information evolves as $E\rightarrow 1$, meanwhile $D\rightarrow 0$ and $R\rightarrow 0$.
To prove Theorem 1, we introduce three Lemmas as following:
Lemma 1. In the pure binary states quantum communication, to discriminate the ensemble of postmeasurement states $\Omega ^{(m)}$ which is conditioned on a particular measurement outcome $A^{m}$ of one repetition of OWM, the OPM is identical to that for the initial binary signal states.
Lemma 2. In the pure binary states quantum communication, the total extracted information by one repetition of OWM and a subsequent OPM is equivalent to the accessible information of the binary signal states, which can be expressed as $I'_{max}=I_{acc}$; therefore, none of the information is destroyed in the measurement and $D=0$.
Lemma 3. In the pure binary states quantum communication, when OWM is executed with a sufficient number of repetitions, the final postmeasurement states of $\rho _{1}$ and $\rho _{2}$ tend to be indistinguishable with OPM, which means that none of the information is residual with $R=0$.
See the appendix A,B and C for proofs of these three lemmas, respectively.
When we have $D=R=0$, we conclude all of the accessible information is extracted by the optimal measurement. In contrast, non-optimal measurements acquire less information than optimal measurement, and the difference between them are necessarily destroyed or remains in the post-measurement states. From this point of view, our successive OWMs achieve a same effect with OPM.
To verify Theorem 1, a theoretical calculation was performed to simulate the performance of OWM in the task to discriminate the binary states, which are expressed as
Fig. 1. Distribution of mutual information as a function of repetitions to perform OWM. Totally two sets of OWM are studied, of which the two parameters in Eq. (6) are selected as $(\xi =0, \eta =0.9)$ for (a),(b),(c) and (d), and as $(\xi =0.2, \eta =0.8)$ for (e),(f),(g) and (h). For each set of OWM, four different groups of binary signal states are calculated with $\alpha$ in Eq. (5) as $0^{\circ }$, $15^{\circ }$, $30^{\circ }$ and $44^{\circ }$ for (a,e), (b,f), (c,g) and (d,h), respectively. The prior probabilities for these binary signal states are set as $p_{1}=p_{2}=0.5$.
Fig. 2. Distribution of mutual information as a function of repetition to perform general WM on the binary signal states with $\alpha =15^{\circ }$ in Eq. (5) and the prior probabilities as $p_{1}=p_{2}=0.5$. The two parameters in Eq. (6) are selected to be $(\xi =0, \eta =0.9)$ for (a, b, c and d), and $(\xi =0.2, \eta =0.8)$ for (e, f, g and h). The misalignment angle $\theta$ between the basis of OPM and performed WM are selected to be $10^{\circ }$, $30^{\circ }$, $45^{\circ }$ and $70^{\circ }$ for (a,e), (b,f), (c,g) and (d,h), respectively.
We further study the performance of WMs which are not made up of the basis of OPM, by setting the binary state in Eq. (5) to be $\alpha =15^{\circ }$. Totally eight sets of WM are studied by varying the values of $\xi$ and $\eta$, and the misalignment angle $\theta$ between the basis of OPM and WM, which is subject to $\langle \omega | \mu \rangle =\cos (\theta )$. As shown in Fig. 2, when the misalignment angle is nonzero, a considerable amount of information is destroyed. Thus, the extracted information cannot attain the total accessible information. In an extreme scenario when the misalignment angle is selected as $45^{\circ }$, the extracted information remains zero for each repetition of WM; eventually, the total accessible information is destroyed, and none of the information remains in the postmeasurement state.
The theoretical study clearly indicates that when WM is constructed on the basis of OPM, the observers can discriminate the pure binary signal states with maximal extracted information by successively performing OWM. Nonetheless, it is important to show that this method is indeed a realistic advantage compared with the alternative methods.
In this experiment, by encoding the binary information into the polarization of single photons, we test the performance of OWM to discriminate the binary states. The encoding in polarization of photons allows both weak measurements and projective measurements, hence we can study both total accessible mutual information and the information distribution in each measurement. The experimental setup is shown in Fig. 3, which can be divided into three parts: binary states preparation, OWM apparatus and OPM apparatus. The 810-nm photon pairs are generated from a Sagnac interferometer by pumping a periodically poled KTP (PPKTP) crystal with a single mode 405-nm laser. Then, one photon is encoded for binary states communication, and the other photon is detected directly as triggers to herald the signal photon. The OWM apparatus consists of six cascaded Mach-Zehnder interferometers (MZIs) each of which is in charge of one repetition of OWM. The symmetry of the MZI is necessary for preserving the coherence of the binary states, and the two exits of each MZI correspond to the two measurement outcomes. A complete setup for this scheme requires exponentially increasing number of MZIs; and thus, totally $2^{n}-1$ MZIs are needed and $2^{n}$ detectors are required for $n$ sequential OWMs. Therefore, we perform a proof-of-principle experiment with the six cascaded MZIs. With this setup, each of the $n^{6}=64$ outcomes is detected separated in time, and the corresponding measurement probability is calculated as the ratio of corresponding photon count to the total count of the 64 outcomes.
Fig. 3. Experimental setup for discriminating binary states with successive weak measurements. On Alice’s side, the heralded single photon is initially generated by pumping a periodically poled KTP (PPKTP) crystal in a polarized Sagnac interferometer and then triggered by its paring photon. After blocking the pump beam by a band pass filter (BPF), the state encoder, which consists of Polarizer, half wave-plate 1(HWP1) and quarter wave-plate 1 (QWP1), encodes the binary information to the heralded single photons. Then, observers can implement either OPM or OWM on these photons to decode the binary information. OWM can be performed successively with six cascaded Mach-Zehnder interferometers, each of which represents one observer and consists of two beam displacers (BDs) and two HWPs mounted in programmable motorized rotation stage (PMRS). By reflecting the photons with a flipping mirror before QWP2, OPM can be carried out by a polarization analyzer (PA) including HWP2, QWP2 and a polarization beam splitter (PBS). Finally, these photons are detected by single photon detectors (SPDs) linked to a coincidence unit (CU). FC - fiber collimator, SMF - single mode fiber.
For each repetition of OWM, the measurement parameters $\xi$ and $\eta$ are determined by the angle of the HWPs in the corresponding MZI. In order to verify our theory, this apparatus should be capable of implementing not only OWM, but also performing OPM with the polarization analyzer (PA) to exploit the remaining information in the postmeasurement states, or extract the total amount of accessible information $I_{acc}(\Omega :\Pi )$ to estimate the fractions of $E$, $R$ and $D$ in each repetition of OWM. By setting the angles of the two HWPs on the two arms as $\vartheta _1=\arcsin (\sqrt {\xi }/2)$ ($\pi /4-\arcsin (\sqrt {\xi }/2)$) and $\vartheta _2=\arcsin (\sqrt {\eta }/2)$ ($\pi /4-\arcsin (\sqrt {\xi }/2)$), each observer can perform $A^2$ ($A^1$) with single MZI. Therefore, for $n$ MZIs, there are $2^{n}$ measurement outcomes, which can be measured one by one with the SPD at the top-right corner of Fig. 3; therefore, the measured probability does not suffer from the varying detection efficiency. The probability of a certain outcome is calculated as the ratio of recorded photon counting to the total photon counting of $2^{n}$ outcomes, and then $E$ can be estimated through Eq. (1). The residual information $R$ can be determined by performing OPM on the post-measurement states with the PA. The destroyed information $D$ is determined by subtracting $E$ and $R$ from $I_{acc}(\Omega :\Pi )$. By setting all the HWPs in MZIs to be $0^\circ$, $I_{acc}(\Omega :\Pi )$ can be estimated by performing OPM on initial binary signal states with the PA.
Two settings of OWM in the form of Eq. (6) are experimentally studied in this work, both of which are constructed with the basis of OPM. The experimental results of $E$, $R$ and $D$ for each repetition are shown in Fig. 4. The extracted information rapidly increases and approaches the total accessible information within six repetitions. For Fig. 4(a), the proportions of total accessible information acquired in each execution of OWM are 0.59, 0.18, 0.11, 0.05, 0.03 and 0.015, which are summarised as 0.975. For Fig. (b), the proportions of total accessible information acquired in each execution of OWM are 0.64, 0.27, 0.07, 0.015, 0.003 and 0.0004, which are summarised as 0.9984. Meanwhile, the residual information decreases to zero, and none of the information is destroyed during this process. These results are consistent with the theoretical prediction, and indicate that OWM can efficiently discriminate binary signal states without an attempt to implement OPM.
Fig. 4. Experimental results for discriminating binary signal states by six successive observers. Two settings of OWM ((a) $\xi =0.1,\eta =0.9$ and (b) $\xi =0,\eta =0.8$) are implemented to discriminate the binary signal states in the form of Eq. (5) with $\alpha =15^\circ$ and $p_{1}=p_{2}=0.5$. For each setting, the distribution of mutual information is plotted for six successive repetitions of OWM as shown by separated dots, and the solid lines connect the corresponding theoretical values of these dots.
3. Discussion and conclusion
Our OWM is apparently similar with non-destructive measurement, since both of them aims to preserve information in the post-measurement state; hence following observers can extract remained information. However, they are intrinsically different. In each non-destructive measurement, there is necessarily a detection on an auxiliary system. In each repetition of our optimal weak measurement, the photon polarization is coupled to the two exits (path modes) of the interferometer, and then another coupling is executed without detection since more path modes can be introduced. Eventually, the photons are detected and destructed on the detector. Although we perform six repetitions of weak measurements, we actually make one detection, which is after the six steps of photon evolution. A similar proposal can be found in Ref. [26], in which the detectors are placed in a public area separated from all the observers. Once a certain detector clicks, all the observers can acquire the outcome in their respective measurement.
By characterizing the measurement with the extracted, residual and destroyed fractions of mutual information, we show that one can discriminate two pure signal states with WM. Specially, when WM is constructed with the basis of OPM, the total accessible information can be distributed to multiple observers by successively implementing this OWM, since OWM avoids destroying information. Furthermore, by changing the strength of a certain OWM, the attainable mutual information for each observer can be adjusted accordingly. These advantages make OWM a flexible tool in quantum communication protocols using binary states.
Appendix A: proof of lemma 1
Lemma 1: In the binary states quantum communication, to discriminate the ensemble of postmeasurement states $\Omega ^{(m)}$ which is conditioned on a particular measurement outcome $A^{(m)}$ of one repetition of OWM, the OPM is identical to that for the initial binary signal states.
Proof. In order to prove Lemma 1, we construct operator $\Lambda '$ to be similar to $\Lambda$ but with the postmeasurement states $\rho '_{1}$ and $\rho '_{2}$, and prove that the eigenvectors and the sign of the corresponding eigenvalues of $\Lambda '$ are the same as those of $\Lambda$. Thus, OPM is unaltered to discriminate the states pre- and post- OWM.
From Definition 1, we obtain
Taking $\rho _{1}=|\psi _{1}\rangle \langle \psi _{1}|$ and $\rho _{2}=|\psi _{2}\rangle \langle \psi _{2}|$ into Eq. (7), we haveAfter using the generalized Karus operator ${K^{(m)}}=\sum _{q} \sqrt {c^m_{q}}*| \eta _{q} \rangle \langle \eta _{q} |$ on the binary signal states, the postmeasurement states will be in the following form:
We can construct operator $\Lambda '=p_{1}\rho '_{1}-p_{2}\rho '_{2}$ to be similar to $\Lambda$ with these two postmeasurement states, written as
By combining Eqs. (10) and 12, we obtain
On the basis of Eq. (13), we can conclude that $\Lambda$ and $\Lambda '$ have the same eigenvectors and the sign of the eigenvalues. Therefore, the OPM, which is composed of these eigenvectors as defined in Definition 1, does not change for the pre- and post- OWM binary states.Appendix B: proof of lemma 2
Lemma. 2: In the binary states quantum communication, the total extracted information by one repetition of OWM and a subsequent OPM is equivalent to the accessible information of the binary signal states, which can be expressed as $I'_{max}=I_{acc}$.
According to the definitions in Eqs. (2) and (3), $I'_{max}$ and $I_{acc}$ can be calculated as
where $p_{i}$ is the prior probability of the ensemble of signal states, and $l,m$ denote the $l_{th}$ and $m_{th}$ outcome of OPM and OWM, respectively. From Lemma. 1, OPM for the postmeasurement states after one repetition of OWM is still $\Pi _{l}$. Therefore, $p(i,m,l)$ denotes the probability of having the outcomes of the first OWM and subsequent OPM as $A^{m}$ and $\Pi _{l}$, respectively, when the signal state is $\rho _{i}$. It can be calculated as $p(m,l)$ denotes the marginal of $p(i,m,l)$ over the label $i$ of the input states and can be calculated as $p(i,l)$ denotes the probability of having the outcome as $\Pi _{l}$ when the signal state is $\rho _{i}$ and can be calculated as $p(l)$ denotes the marginal of $p(i,l)$ over the label $i$ of the input states and can be calculated as Considering ${\digamma ^{(m)}}=\sum _{l}\sqrt {c_{l}^{m}}\Pi _{l}$ and $\sum _{m}c_{l}^{m}=1$, we have By combining these results with Eqs. (14) and 15, we obtain $I'_{max}=I_{acc}$, which means that $D=0$ with one repetition of OWM. On applying this corollary to the postmeasurement states, we conclude that none of the information is destroyed by successive OWMs.Appendix C: proof of lemma 3
Lemma. 3: In the binary states quantum communication, if Bob repeats OWM with a large number of repetitions, the final postmeasurement states of $\rho _{1}$ and $\rho _{2}$ tend to be indistinguishable with OPM. This means none of the information is residual.
With $|\eta _{k}\rangle$ as bases, the density matrix of the binary signal states can be written as
Consider $\digamma ^{(m_{1})}=\sqrt {c_{1}^{m_{1}}}\Pi _{1}+\sqrt {c_{2}^{m_{1}}}\Pi _{2}$, for the first repetition of OWM, the probability of having the outcome as $A^{m}$ can be calculated as
Similarly, after $n$ repetitions of OWM, the probability of all possible outcomes can be calculated as
The expression of $p(m)$ contains two items, each of which can be regarded as the probability of a binomial distribution. For the first item, the probabilities of the two mutually exclusive results are $c_{1}^{1}$ and $c_{1}^{2}$, respectively. For the second item, these two probabilities are $c_{2}^{1}$ and $c_{2}^{2}$.
As a result, for both items in Eq. (24), each implementation of OWM can be regarded as a binomial sampling. According to the law of large numbers, when $n$ is sufficiently large, which means $n\rightarrow \infty$, the frequency of two outcomes in each item approaches the expectation value with probability 1. Specifically, the frequencies of $c_{1}^{1}$ and $c_{1}^{2}$ in the first item are $n\times c_{1}^{1}$ and $n\times c_{1}^{2}$, and the frequencies of occurrence of $c_{2}^{1}$ and $c_{2}^{2}$ in the second item are $n\times c_{2}^{1}$ and $n\times c_{2}^{2}$. The probabilities of other different frequencies approach 0. In other words, for $n$ repetitions of OWM, only when the frequencies of $A^{1}$ and $A^{2}$ are $n\times c_{1}^{1}$ and $n\times c_{1}^{2}$, the first item in Eq. (24) is nonzero. In this case, the final postmeasurement states of the binary signal states become
Similarly, when the frequencies of $A^{1}$ and $A^{2}$ are $n\ast c_{2}^{1}$ and $n\ast c_{2}^{2}$, the second item in Eq. (24) is nonzero and the final postmeasurement states of the binary signal states become Please note that the above derivation requires $c_{1}^{1}\neq c_{2}^{1}$ $c_{1}^{2}\neq c_{2}^{2}$; otherwise, the measurement operators are identities with which $\rho _{i}$ is unaltered and none information is acquired. These results also require that $A^{1}$ and $A^{2}$ are diagonal matrixes that satisfy commutativity, which can be obtained from Definition 1. For both cases, the two signal states degenerate to an identical postmeasurement state, which cannot be discriminated with an optimal measurement, and the residual information is 0.For OWM with more than two outcomes, each repetition of OWM can be regarded as a random sampling on a multinomial distribution; and thus, we will have a similar result with two-outcome scenario.
Funding
China Postdoctoral Science Foundation (2017M612073); National Postdoctoral Program for Innovative Talents (BX201600146); National Natural Science Foundation of China (11474267, 11774335, 61322506, 61327901); Science Foundation of the Chinese Academy of Sciences (ZDRW-XH-2019-1); Chinese Academy of Sciences Key Research Program of Frontier Sciences (QYZDY-SSWSLH003); Fundamental Research Funds for the Central Universities (WK2030020019, WK2470000026); National Key Research and Development Program of China (2016YFA0302700, 2017YFA0304100); Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-SLH003); Anhui Initiative in Quantum Information Technologies (AHY020100, AHY060300).
Acknowledgments
Authors thank CAS Key Laboratory of Quantum Information, University of Science and Technology.
Disclosures
The authors declare no conflicts of interest.
References
1. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001). [CrossRef]
2. M. A. Nielsen and I. L. Chuang, Quantum information theory (Cambridge University, 2000).
3. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
4. S. Massar and S. Popescu, “Optimal extraction of information from finite quantum ensembles,” in Asymptotic Theory Of Quantum Statistical Inference: Selected Papers, M. Hayashi, ed. (World Scientific, 2005).
5. M. Berta, F. G. Brand ao, C. Majenz, and M. M. Wilde, “Conditional decoupling of quantum information,” Phys. Rev. Lett. 121(4), 040504 (2018). [CrossRef]
6. C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys. 1(2), 231–252 (1969). [CrossRef]
7. R. Han, G. Leuchs, and J. A. Bergou, “Helstrom measurement: A nondestructive implementation,” Phys. Rev. A 101(3), 032103 (2020). [CrossRef]
8. V. A. Vilnrotter, “Quantum receiver for distinguishing between binary coherent-state signals with partitioned-interval detection and constant-intensity local lasers,” NASA IPN Progress Rep. 42, 189 (2012).
9. C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Discrimination of binary coherent states using a homodyne detector and a photon number resolving detector,” Phys. Rev. A 81(6), 062338 (2010). [CrossRef]
10. C. R. Müller, M. Usuga, C. Wittmann, M. Takeoka, C. Marquardt, U. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14(8), 083009 (2012). [CrossRef]
11. K. Tsujino, D. Fukuda, G. Fujii, S. Inoue, M. Fujiwara, M. Takeoka, and M. Sasaki, “Quantum receiver beyond the standard quantum limit of coherent optical communication,” Phys. Rev. Lett. 106(25), 250503 (2011). [CrossRef]
12. M. M. Wilde, Quantum information theory (Cambridge University, 2013).
13. R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007). [CrossRef]
14. D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” in Asymptotic Theory of Quantum Statistical Inference: Selected Papers, M. Hayashi, ed. (World Scientific, 2005).
15. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993). [CrossRef]
16. S. L. Braunstein, G. M. D’Ariano, G. J. Milburn, and M. F. Sacchi, “Universal teleportation with a twist,” Phys. Rev. Lett. 84(15), 3486–3489 (2000). [CrossRef]
17. A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991). [CrossRef]
18. Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak measurement,” Phys. Rev. Lett. 109(15), 150402 (2012). [CrossRef]
19. G. Chen, Y. Zou, X.-Y. Xu, J.-S. Tang, Y.-L. Li, J.-S. Xu, Y.-J. Han, C.-F. Li, G.-C. Guo, H.-Q. Ni, Y. Yu, M.-F. Li, G.-W. Zha, and Z.-C. Niu, “Experimental test of the state estimation-reversal tradeoff relation in general quantum measurements,” Phys. Rev. X 4(2), 021043 (2014). [CrossRef]
20. B.-G. Englert, “Fringe visibility and which-way information: An inequality,” Phys. Rev. Lett. 77(11), 2154–2157 (1996). [CrossRef]
21. F. Sciarrino, M. Ricci, F. De Martini, R. Filip, and L. Mišta Jr, “Realization of a minimal disturbance quantum measurement,” Phys. Rev. Lett. 96(2), 020408 (2006). [CrossRef]
22. T. Ralph, S. Bartlett, J. O’Brien, G. Pryde, and H. Wiseman, “Quantum nondemolition measurements for quantum information,” Phys. Rev. A 73(1), 012113 (2006). [CrossRef]
23. R. Han, G. Leuchs, and M. Grassl, “Residual and destroyed accessible information after measurements,” Phys. Rev. Lett. 120(16), 160501 (2018). [CrossRef]
24. J. Bergou, E. Feldman, and M. Hillery, “Extracting information from a qubit by multiple observers: Toward a theory of sequential state discrimination,” Phys. Rev. Lett. 111(10), 100501 (2013). [CrossRef]
25. L. B. Levitin, “Optimal quantum measurements for two pure, mixed states,” in Quantum Communications and Measurement, Belavkin, V.P., Hirota, Osamu, Hudson and R.L., ed. (Springer, 1995).
26. M. Solis-Prosser, P. González, J. Fuenzalida, S. Gómez, G. Xavier, A. Delgado, and G. Lima, “Experimental multiparty sequential state discrimination,” Phys. Rev. A 94(4), 042309 (2016). [CrossRef]
