Abstract
1. INTRODUCTION
Quantum coherence [1] arising from the superposition principle is the key factor deviating quantum physics from classical when describing particle systems such as photons [2], which may have a classical wave theory, and matters in classical physics such as atoms [3] or ions [4]. Coherence leads to interference already known in classical optics, and also a purely quantum phenomenon in bipartite and multipartite systems, quantum entanglement [5]. There have been several measures to quantify coherence, and they should fulfill the publicly accepted four conditions of nonnegativity, monotonicity, strong monotonicity, and convexity [1]. One example of coherence measures is the norm of coherence introduced in Ref. [6]. Note that coherence is dependent on the reference basis we choose. A pure state has zero coherence if it is one of the basis states, but it may have coherence in other bases.
Coherence and entanglement have many similar aspects [7]. For instance, both can be treated as physical resources [1,6,8], and we often need to preserve coherence against decoherence noises as well as entanglement in theories and experiments. The amount of entanglement has been quantified in many ways, such as concurrence [5,9]. Entanglement dynamics in open systems has been studied. Konrad
Compared with entanglement, the theoretical and experimental results on coherence dynamics in open systems are limited. We need some general law to determine its evolution equation to help us design the effective coherence preservation schemes. Theoretically, Hu
Sign up for Photonics Research TOC. Get the latest issue of Photonics Research delivered right to you!Sign up now
2. THEORY
An arbitrary pure state in -dimensional Hilbert space can be expressed as . When all , the state is known as the maximally coherent state (MCS) [6] , which should be one of the most coherent states in . The quantum operation is described by a set of Kraus operators satisfying . After the operation, state becomes . If each Kraus operator is diagonal in the reference basis , the operation is known as a genuinely IO (GIO), which has the property that an incoherent state is untouched after acting on it, i.e., for any incoherent state , a GIO can preserve it [1,25]. Moreover, all Kraus operators of GIO are diagonal in the reference basis [25]. Therefore, the relations between GIO and IO, SIO, maximally IO (MIO) are [25]. In Refs. [26–29], the -concurrence, which is the geometric mean of the Schmidt coefficients for pure states, has been proposed for fully entangled states (i.e., with nonzero Schmidt coefficients) in systems. It can describe the entanglement factorization law in bipartite high-dimensional systems [16]. Similarly, one can quantify the full coherence of a -dimensional quantum state by defining the -coherence as
This is the coherence factorization law: in -dimensional Hilbert space, if a GIO acts on an arbitrary state , pure or mixed, and produces the final state , then the coherence measure of the final state is the product of the coherence of the initial state and the MCS after the operation , that is,
We can see that the coherence of the final state is defined by two factors, one of which is solely determined by the initial state, and the other is related only to the operation. The operation term is reflected by the coherence of a state, which is similar to Choi–Jamiołkowski isomorphism, also known as channel-state duality [30]. An illustration of the principle is shown in Fig. 1. The proof of the law uses the stronger theorem
Figure 1.Illustration of the coherence factorization law under genuinely incoherent operation (GIO)
Similarly,
3. QUBIT EXPERIMENT
The coherence of qubit is irrelevant to the diagonal elements of the density matrix, so quantum state tomography [31] at the and base is enough to yield this value:
The experimental setup of qubit is shown in Fig. 2(a). An attenuated pulsed 808 nm light from a laser source is collimated into a single-mode fiber and prepared at the horizontal polarization by a half-wave plate (HWP) and a polarizing beam splitter (PBS) after emitting from the collimator. Then an HWP with its fast axis at angle and a beam displacer (BD) produce the initial state , while the two paths have different polarizations. The HWP at angle acting on both paths changes the amplitudes at and polarizations. A quarter-wave plate (QWP) at 0° is inserted at path 2 to realize the phase factor in . To compensate for the optical path difference, another QWP is inserted at path 1 right after the first BD. Then a thick lithium niobate () crystal introduces an optical path difference between the and components larger than the coherence length of the light source from the birefringence effect, and thus destroys the coherence between the two components [32]. To eliminate the polarization difference, an HWP at 22.5° converts and polarizations to diagonal () and anti-diagonal (), respectively. Two HWPs and the second BD select the component and merge the two paths together, converting the path information into polarization while discarding half the photons. Before merging, two thin glass plates are inserted at the two paths to compensate for the residual phase difference. Then the photons pass a tomography device consisting of a QWP, an HWP, and a PBS, before being coupled into another single-mode fiber, sent to a single-photon detector (SPD), and counted by a computer. Dark counts are subtracted to increase the signal-to-noise ratio.
Figure 2.(a) Setup of the qubit coherence factorization law verification experiment. The photons are prepared as the initial state using a half-wave plate (HWP) with its fast axis at angle
We measured with different and values using Eq. (6). Then we removed the optical elements for the operation process from the 90° HWP to and directly performed tomography on the initial states with different values. The relation between of the initial state and the theoretical curve is plotted in Fig. 3(a). In Figs. 3(b)–3(e), we plot of the final states in red dots, the product on the right-hand side of Eq. (2) in blue dots, and the theoretical curves with different values. The product values are close to the directly measured values, verifying the factorization law.
Figure 3.Experimental results for qubit experiment. (a) Measured coherence
4. QUTRIT EXPERIMENT
We use three path DOFs to study the qutrit scenario, where the initial state is , and we design a phase damping operation between two of the paths and the other one. The corresponding Kraus operators are and . Designing a true phase damping operation on all the paths is more complex, as the polarization DOF as an auxiliary is two-dimensional. When , the coherence is preserved, while it is completely destroyed when . The experimental setup is similar to the qubit experiment except for an additional path, a different wave plate setup to realize the operation, and a different tomography method. The essential parts are shown in Fig. 2(b). and are the angles of the wave plates controlling the initial state and the operation, respectively. In the tomography process, wave plates, two BDs, and a PBS are needed to project the quantum state to 15 eigenstates of eight Gell-Mann matrices [31,33,34]. The density matrix is calculated from the averages of these Hermitian operators. See Appendix A for more details. Then we can obtain the coherence via . However, if some of the off-diagonal elements should be zero while others are nonzero, a small value from the experimental errors will cause the measured to be significantly larger than zero. For example, in an experiment, if the moduli of the three measured off-diagonal elements are 0.01, 0.2, and 0.2, we have . So, when the theoretical , the experimental value will be inaccurate unless most of the off-diagonal elements are zero.
We choose (MCS) and 7.5° (another initial state) and take different values to perform state tomography. To show the impact of the phase damping operation on the moduli of density matrix elements, we present values from our experiment in Fig. 4(a) when the initial state is MCS and , 15°, 30°, and 45°. Off-diagonal elements and decay as increases, while and the diagonal elements are roughly unchanged. The coherence values of the initial states are taken when . The measured values of final states, the products (not applicable for MCS), and the theoretical curves are plotted in Fig. 4(b). Coherence decays with the increase in . When , coherence should become zero, but the experimental value is inaccurate as we have stated before, and the error calculated from the deviation of Poisson distribution at the angle is larger than others.
Figure 4.(a) Moduli of density matrix elements
The law of Eq. (2) still holds for mixed initial states, which are statistical mixtures of different pure states. Weighted averages of count data can be used to simulate the mixed state scenario. We let the initial state be (), where is the state when , and use the weighted averages of count data from and as the new count data to simulate the mixed state scenario. Under six types of operations where , 7.5°, 15°, 22.5°, 30°, and 37.5°, corresponding to the six solid dots in Fig. 4(b), the calculated values (light red curves) and the product values (light blue curves) are shown in Fig. 5 when takes values from zero to one. The two curves are close to each other, verifying the factorization law with mixed input states.
Figure 5.
5. DISCUSSION AND CONCLUSION
For qudits with higher dimensions, we can still use the path DOF, but the optical setup is more vulnerable to errors from the misalignment of optical devices, making the measured values deviate from theoretical ones. Nevertheless, we performed a four-dimensional experiment, projected the states after operation for qudit state tomography [31], and found the equation still agrees well for a certain operation and initial state: , .
There are other types of operations that satisfy the law. One example is that all the Kraus operators are multiplied by the same permutation matrix on the left, changing the order of the reference bases while keeping the coherence value. For example, a qutrit operation described by is related to the Kraus operators of a GIO by
But it does not hold when the permutation matrices for each are different. Also, some quantum operations cannot be described in the form above, but they are special to satisfy it as well. One example is the qubit amplitude decay channel
In summary, we have presented and proved a factorization law for qudits under GIO, in that using the -coherence measure, the coherence of the state after the operation can be factorized into the product of an initial state term and an operation term. To verify the law, we used an optical setup to test qubit and qutrit cases using a given set of initial states and operations. Our work provides an indirect method to measure the final coherence of quantum states after a specific kind of evolution, and would play an important role in the simplification of coherence measurements, as well as the discovery of other laws about quantum coherence. For example, there are other types of coherence metrics, such as the convex-roof norm (see Ref. [7] for more details) whose calculation method is uncertain yet. We can define as the minimum statistical average of all possible pure state combinations of :
APPENDIX A: QUTRIT STATE TOMOGRAPHY
The Gell-Mann matrices are used in qutrit state tomography just as Pauli matrices in the qubit case. For four-dimensional qudits, the matrices are the direct sums of two Pauli matrices. The original matrix () can be found in Refs. [
We define as the probability to be projected to state . According to the eigenvalues and the corresponding eigenvectors, the average value of each Gell-Mann matrix is
Calculating and is unnecessary if we need only its off-diagonal elements.
References
[1] T. Baumgratz, M. Cramer, M. B. Plenio. Colloquium: quantum coherence as a resource. Rev. Mod. Phys., 89, 041003(2017).
[2] C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.-M. Raimond, S. Haroche. Progressive field-state collapse and quantum non-demolition photon counting. Nature, 448, 889-893(2007).
[3] Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel. An atom-sorting machine. Nature, 442, 151(2006).
[4] J. Benhelm, G. Kirchmair, C. F. Roos, R. Blatt. Towards fault-tolerant quantum computing with trapped ions. Nat. Phys., 4, 463-466(2008).
[5] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki. Quantum entanglement. Rev. Mod. Phys., 81, 865-942(2009).
[6] T. Baumgratz, M. Cramer, M. B. Plenio. Quantifying coherence. Phys. Rev. Lett., 113, 140401(2014).
[7] X. Qi, T. Gao, F. Yan. Measuring coherence with entanglement concurrence. J. Phys. A, 50, 285301(2017).
[8] E. Chitambar, G. Gour. Quantum resource theories. Rev. Mod. Phys., 91, 025001(2019).
[9] W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 80, 2245-2248(1998).
[10] T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, A. Buchleitner. Evolution equation for quantum entanglement. Nat. Phys., 4, 99-102(2008).
[11] O. J. Farías, C. L. Latune, S. P. Walborn, L. Davidovich, P. H. S. Ribeiro. Determining the dynamics of entanglement. Science, 324, 1414-1417(2009).
[12] T. Yu, J. H. Eberly. Phonon decoherence of quantum entanglement: robust and fragile states. Phys. Rev. B, 66, 193306(2002).
[13] F. Mintert, A. R. R. Carvalho, M. Kuś, A. Buchleitner. Measures and dynamics of entangled states. Phys. Rep., 415, 207-259(2002).
[14] A. R. R. Carvalho, M. Busse, O. Brodier, C. Viviescas, A. Buchleitner. Optimal dynamical characterization of entanglement. Phys. Rev. Lett., 98, 190501(2007).
[15] J.-S. Xu, C.-F. Li, X.-Y. Xu, C.-H. Shi, X.-B. Zou, G.-C. Guo. Experimental characterization of entanglement dynamics in noisy channels. Phys. Rev. Lett., 103, 240502(2009).
[16] M. Tiersch, F. de Melo, A. Buchleitner. Entanglement evolution in finite dimensions. Phys. Rev. Lett., 101, 170502(2008).
[17] A. R. R. Carvalho, F. Mintert, A. Buchleitner. Decoherence and multipartite entanglement. Phys. Rev. Lett., 93, 230501(2004).
[18] G. Gour. Evolution and symmetry of multipartite entanglement. Phys. Rev. Lett., 105, 190504(2010).
[19] M.-L. Hu, H. Fan. Evolution equation for quantum coherence. Sci. Rep., 6, 29260(2016).
[20] Y. Yao, X. Xiao, L. Ge, C. P. Sun. Quantum coherence in multipartite systems. Phys. Rev. A, 92, 022112(2015).
[21] Z. Xi, Y. Li, H. Fan. Quantum coherence and correlations in quantum system. Sci. Rep., 5, 10922(2015).
[22] J. Ma, B. Yadin, D. Girolami, V. Vedral, M. Gu. Converting coherence to quantum correlations. Phys. Rev. Lett., 116, 160407(2016).
[23] S.-J. Xiong, Z. Sun, Q.-P. Su, Z.-J. Xi, L. Yu, J.-S. Jin, J.-M. Liu, F. Nori, C.-P. Yang. Experimental demonstration of one-shot coherence distillation: realizing
[24] K.-D. Wu, T. Theurer, G.-Y. Xiang, C.-F. Li, G.-C. Guo, M. B. Plenio, A. Streltsov. Quantum coherence and state conversion: theory and experiment. NPJ Quantum Inf., 6, 22(2020).
[25] J. I. de Vicente, A. Streltsov. Genuine quantum coherence. J. Phys. A, 50, 045301(2016).
[26] A. Uhlmann. Roofs and convexity. Entropy, 12, 1799-1832(2010).
[27] G. Gour. Family of concurrence monotones and its applications. Phys. Rev. A, 71, 012318(2005).
[28] H. Fan, K. Matsumoto, H. Imai. Quantify entanglement by concurrence hierarchy. J. Phys. A, 36, 4151-4158(2003).
[29] H. Barnum, N. Linden. Monotones and invariants for multi-particle quantum states. J. Phys. A, 34, 6787-6805(2001).
[30] C. Datta, S. Sazim, A. K. Pati, P. Agrawal. Coherence of quantum channels. Ann. Phys., 397, 243-258(2018).
[31] R. T. Thew, K. Nemoto, A. G. White, W. J. Munro. Qudit quantum-state tomography. Phys. Rev. A, 66, 012303(2002).
[32] P. G. Kwiat, A. J. Berglund, J. B. Altepeter, A. G. White. Experimental verification of decoherence-free subspaces. Science, 290, 498-501(2000).
[33] M. Gell-Mann. Symmetries of baryons and mesons. Phys. Rev., 125, 1067-1084(1962).
[34] Z.-H. Liu, K. Sun, J. K. Pachos, M. Yang, Y. Meng, Y.-W. Liao, Q. Li, J.-F. Wang, Z.-Y. Luo, Y.-F. He, D.-Y. Huang, G.-R. Ding, J.-S. Xu, Y.-J. Han, C.-F. Li, G.-C. Guo. Topological contextuality and anyonic statistics of photonic-encoded parafermions. PRX Quantum, 2, 030323(2021).
Set citation alerts for the article
Please enter your email address