week ending 14 JUNE 2013
PHYSICAL REVIEW LETTERS
PRL 110, 240405 (2013)
Matrix Product States for Quantum Metrology Marcin Jarzyna and Rafał Demkowicz-Dobrzan´ski Faculty of Physics, University of Warsaw, ul. Hoz˙a 69, PL-00-681 Warszawa, Poland (Received 5 February 2013; revised manuscript received 23 April 2013; published 14 June 2013) We demonstrate that the optimal states in lossy quantum interferometry may be efficiently simulated using low rank matrix product states. We argue that this should be expected in all realistic quantum metrological protocols with uncorrelated noise and is related to the elusive nature of the Heisenberg precision scaling in the asymptotic limit of a large number of probes. DOI: 10.1103/PhysRevLett.110.240405
PACS numbers: 03.65.Ta, 02.70.c, 06.20.Dk, 42.50.St
Over recent years, advancements in quantum engineering have pushed nonclassical concepts such as entanglement and squeezing, previously regarded as largely academic topics, close to practical applications. Quantum features of light and atoms helped to improve the performance of measuring devices that operate in the regime where the precision is limited by the fundamental laws of physics [1]. One of the most spectacular examples of practical applications of quantum metrology can be found in gravitational wave detectors [2], where the original idea [3] of employing squeezed states of light to improve the sensitivity of an interferometer has found its full-scale realization [4,5]. No less impressive are experiments with trapped entangled ions demonstrating spectroscopic resolution enhancement crucial for the operation of atomic clocks [6–8]. When standard sources of laser light are being used, any interferometric experiment may be fully described by treating each photon individually and claiming that each photon interferes only with itself. Sensing a phase delay between the two arms of the interferometer via intensity measurements may be regarded as many independent repetitions of single photon interferometric experiments. N independent experiments result in data that allow pffiffiffiffithe parameter to be estimated with error scaling as 1= N — the so-called standard quantum limit or the shot noise limit. If, however, an experiment cannot be split into N-independent processes, as is, e.g., the case with the N probing photons being entangled, the above reasoning is invalid, and one can in principle achieve the 1=N estimation precision—the Heisenberg scaling [9–12]—with the help of, e.g., the N00N states [13]. Still, in all realistic experimental setups, decoherence typically makes the relevant quantum features, such as squeezing or entanglement, die out very quickly [14,15]. Recently, it has been rigorously shown for optical interferometry with loss [16,17], as well as for more general decoherence models [18,19], that if decoherence acts independently on each of the probes, one can get at best pffiffiffiffi c= N asymptotic scaling of error—precision that is better than the classical one only by a constant factor c which depends on the type of decoherence and its strength. 0031-9007=13=110(24)=240405(5)
One can therefore appreciate the Heisenberg-like decrease in uncertainty only in the regime of small N, where the precise meaning of ‘‘small’’ depends on the decoherence strength [18], and in typical cases is of the order of 10 photons or atoms. This indicates that in the limit of a large number of probes, almost optimal performance can be achieved by dividing the probes into independent groups where only the probes from a given group are entangled among each other. Clearly, the size ofpthe ffiffiffiffi group that is needed to approach the fundamental c= N bound up to a given accuracy will depend on the strength of decoherence. Nevertheless, irrespective of how small the decoherence strength is, for N large enough, the size of the group will saturate at some point, and therefore, asymptotically the optimal state may be regarded as only locally correlated. A natural class of states efficiently representing locally correlated states are the matrix product states (MPSs) [20–24], which have proved to be highly successful in simulating low-energy states of complex spin systems. Until now no attempt has been made, however, to employ MPSs for quantum metrology purposes. Establishing this connection is the essence of the present Letter. A basic quantum metrology scheme is depicted in Fig. 1. The N probe input state ji travels through N parallel noisy channels whose action is parametrized by an ^ x is performed on the unknown value . A measurement output density matrix ^ ¼ N ðjihjÞ yielding a result ^ x with probability pðxjÞ ¼ Trð^ x Þ. The estimation procedure is completed by specifying an estimator function ~ ðxÞ. Eventually we are left with the estimated value ~ which in general will be different of the parameter , from . We denote the average uncertainty of estimation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ Þ2 i, where the average is performed by ¼ hð over different measurement results x. The main goal of theoretical quantum metrology is to find strategies that minimize . For this purpose, one has to find the optimal estimator, measurement, and input state. This in general is a difficult task. To simplify the problem, one may resort to the quantum Cramer-Rao inequality [25–28]
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FIG. 1. Quantum metrology with matrix product states. The N parallel quantum channels act on the input state ji inscribing parameter value as well as causing local decoherence on each ^ x on the output state ^ allows us of the probes. Measurement ~ to make an estimate ðxÞ based on the measurement result x. The input state is a matrix product state obtained by the action of maps A½k k on pairs of virtual D-dimensional systems prepared in such a way that the two adjacent systems corresponding to different neighboring particles are in a maximally entangled state j’D i.
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; kFð^ Þ
Fð^ Þ ¼ Trð^ L^ 2 Þ
(1)
that bounds the precision of any unbiased estimation strategy based on k-independent repetitions of an experiment. Fð^ Þ is the quantum Fisher information (QFI) written in terms of L^ , the so-called symmetric logarithmic derivative defined implicitly as 2ðd^ =dÞ ¼ L^ ^ þ ^ L^ . For pure states, the formula for QFI simplifies to Fðj iÞ ¼ 4ðh_ j_ i jh_ j ij2 Þ, where j_ i ¼ ðdj i=dÞ. The bound is known to be saturable in the asymptotic limit of k ! 1 in the sense that there exist a measurement and an estimator that yields equality in Eq. (1). The main benefit of using QFI is that since it does not depend on the measurement or the estimator, the only remaining optimization problem is the maximization of Fð^ Þ over the input states. Since the optimal states in the regime of a large number of probes N (not k) may be regarded as consisting of independent groups, the Cramer-Rao bound may be saturated even for k ¼ 1, provided N is large enough [27,29]. This makes the QFI an even more appealing quantity than in the decoherence-free case where some controversies arise on the practical use of the strategies based on the optimization of the QFI [30,31]. The maximization of QFI over the most general input states for large N may still be challenging though, and even if it was successful, it might not provide insight into the structure of the optimal states. This is the place where MPSs are useful. A general MPS of N qubits is defined as jiMPS
1 ¼ pffiffiffiffiffiffiffi N
1 X 1 ...N ¼0
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PHYSICAL REVIEW LETTERS
PRL 110, 240405 (2013)
TrðA½1 1
. . . A½N Þj1 N
. . . N i;
(2)
where A½k k are square complex matrices of dimension D D, D is called the bond dimension, and N is the
normalization factor. In operational terms, a MPS is generated by assuming that each qubit is substituted by a pair of D-dimensional virtual systems. Adjacent systems corresponding to different neighboring particles prepared pffiffiffiffiare P in maximally entangled states j’D i ¼ 1= D D ¼1 j; i PD ½k (Fig. 1) and maps Ak ¼ ;¼1 Ak ;; jk ih; j are applied to the pair of virtual systems corresponding to the kth particle [23]. Such a description of state is very efficient, provided the bond dimension D increases slowly with N. In a most favorable case when D may be assumed to be bounded, D < Dmax , the number of coefficients needed to specify an N qubit state in the asymptotic regime of large N will scale as ND2max (linear in N), as opposed to the standard 2N scaling. It should be noted, however, that in many quantum metrological models, in particular, the ones based on the QFI, the search for the optimal input probe states may be restricted to symmetric (bosonic) states [14,18,32,33]. Even though the description of a symmetric N qubit pure state is efficient and requires only N þ 1 parameters, the use of MPSs may still offer a significant advantage as the symmetric MPS description involves matrices A which are identical for different particles: A½k ¼ A and commute under the trace—TrðA1 . . . AN Þ does not depend on the order of matrices. Provided D is asymptotically bounded or grows slowly with N, one can still benefit significantly from the use of MPSs in the large N regime. In order to demonstrate the power of the MPS approach, we apply it to the most thoroughly analyzed and relevant model in quantum metrology—the lossy interferometer. We will not specify the nature of the physical systems (atoms, photons) but will rather refer to abstract twolevelprobes, with orthogonal states j0i, j1i. The parameter to be estimated is the relative phase delay that a probe experiences being in the j1i vs j0i state. The decoherence mechanism amounts to a loss of probes where each of the probes is lost independently of the others with probability 1 . As such, this is an example of a general scheme depicted in Fig. 1. Since the distinguishability of probes offers no advantage for phase estimation [33], we move to the symmetric state P description where the general N probe state reads ji ¼ N n¼0 n jn;N ni, and jn;N ni represents the n and N n probes in states j0i and j1i, respectively. The output state ^ can be written explicitly as ^ ¼
N Nl X0 X l0 ¼0 l1 ¼0
pl0 l1 jl0 l1 ihl0 l1 j;
(3)
where X1 1 Nl jl0 l1 i ¼ pffiffiffiffiffiffiffiffiffi ein nl0 l1 jn l0 ; N n l1 i; pl0 l1 n¼l0 n ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n nl n n Nn n ð1 Þl ; (4) Bl ¼ l0 l1 ðÞ ¼ Bl0 Bl1 ; l
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FIG. 2. Log-log plots of the phase estimation precision pffiffiffiffi with losses ( ¼ 0:9) as a function of the number of probes N optimized over input states in (a) the QFI approach where ¼ 1= F with F given by Eq. (5) and (b) the Ramsey interferometry with given by Eq. (7). The solid gray curves ordered from top pffiffiffiffiffiffiffi ffi to bottom correspond to matrix product states of increasing bond dimension D: D ¼ 1 (lightest gray, no correlations: ¼ 1= N ), D ¼ 2, D ¼ 3, D ¼ 4, D ¼ 5 (darkest gray). The black solid curves correspond to the optimal states obtained with brute force optimization, with extrapolation for higher N (dashed) obtained using scaling formulas of Ref. [16] (a)ffi and optimization over the spin-squeezed states [40] (b). The dashed-dotted lines represent the asymptotic bound ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 =N [16–19]. The upper-right insets depict the bond dimension D needed to reach the optimal precision with at most 1% discrepancy: ¼ 0:9 (solid), ¼ 0:3 (dashed). Lower-left insets present normalized absolute values of elements of the optimal diagonal MPS matrix A for bond dimension D ¼ 4 and ¼ 0:9.
and pl0 l1 is a normalization factor which can be interpreted as a probability to lose l0 , l1 probes in states j0i and j1i, respectively. As the output state ^ is mixed, QFI is not explicitly given in terms of the input state parameters, as it requires involved calculation of the symmetric logarithmic derivative or other equivalent quantities [18,34,35]. In general, due to convexity, QFI for a mixed state is smaller than a weighted sum of QFIs for pure states into which a mixed state may be decomposed. Nevertheless, it was shown in Ref. [36] that assuming additional knowledge of how many photons were lost being in the state j0i and how many being in the state j1i does not improve the estimation precision appreciably. Hence, one can approximate the QFI with a weighted sum of the QFI’s all pure states entering the mixture ^ : ~ ^ Þ ¼ Fð^ Þ & Fð
N Nl X0 X l0 ¼0 l1 ¼0
pl0 ;l1 Fðjl0 ;l1 iÞ:
(5)
This approximation simplifies the calculations significantly, since in our case Fðjl0 l1 iÞ ¼ 4ðhl0 l1 jn^ 2 jl0 l1 i
^ l0 l1 ij2 Þ with n^ being the excitation number jhl0 l1 jnj ^ N ni ¼ njn; N ni. Direct optimization operator njn; of Eq. (5) over the input state parameters n involves N þ 1 variables. This approach was taken in Refs. [33,36]. Here we consider the class of symmetric MPSs parametrized with two diagonal (assuring commutativity) D D matrices A0 , A1 . These MPSs are parametrized with 2D complex numbers, instead of N þ 1, and read explicitly
jiMPS
sffiffiffiffiffiffiffiffiffi N 1 X N TrðAn0 ANn ¼ pffiffiffiffiffiffiffi Þjn; N ni: 1 n N n¼0
(6)
Thanks to the simple form of Eq (5), it is possible to compute F~ directly on A matrices, and there is no need to go back to the less efficient standard description as would be the case with Eq. (1). Figure 2(a) illustrates the precision obtained using MPSs for the case of relatively small losses ¼ 0:9. As one can see, the MPS approximation is excellent. In particular, the upper-right inset shows that already D ¼ 5 is sufficient to obtain less than 1% discrepancy for N 500. We have confirmed this observation for different and observed that for higher losses (lower ), lower D are required to obtain a given level of approximation for a particular N—an effect that should be much more spectacular for larger N reflecting the fact that stronger decoherence diminishes the role of quantum correlations. Moreover, we have observed that optimal matrices A0 , A1 have the same diagonal values which are ordered complementarily—the highest in A0 is paired with lowest one in A1 , etc. N is higher the closer the diagonal values approach each other, as can be seen from the lower-left inset on Fig. 2. This confirms the intuition that with increasing N, the optimal states are becoming less distinct from the product state—all diagonal values of A equal. The peculiarity of phase estimation is that in the decoherence-free case, optimal QFI is achieved for the N00N state which, even though it has nonlocal correlations, it is an example of a MPS with D ¼ 2. This makes the MPS capable of approximating the optimal states very well even for low loss and small N [N & 1=ð1 Þ]—an
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PHYSICAL REVIEW LETTERS
ability that in general will not hold for other estimation problems. Taking now a more operational approach, not based on the QFI, one may consider a concrete measurement scheme with a particular observable O^ being measured. A simple error-propagation formula for yields ¼ ^ ^ O=jdh Oi=dj. In the Ramsey spectroscopy setup [37], or equivalently in the Mach-Zehnder interferometer with a photon number difference measurement, one effectively measures a component of the total angular momentum operator J^ of N spins 1=2—if a qubit j0i, j1i is treated as a spin-1=2 particle [11]. If the phase-dependent rotation U^ ¼ eiJ^z is being sensed by the measurement of the J^x observable, the explicit formula for estimation uncertainty at the optimal operation point ¼ 0 calculated for ^ from Eq. (3) reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N 2 J^ : ¼ ^ 2x þ hJ y i 4hJ^y i2
(7)
The search for the optimal state amounts to minimizing the above quantity. Since it depends only on first and second ^ it is simple to implement numerically using moments of J, MPSs. The results are presented in Fig. 2(b). It is clear that MPSs are capable of capturing the essential feature of the optimal states—the squeezing of the J^x —with relatively low bond dimensions D. Moreover, the upper-right inset indicates that the required bond dimension D is reduced much more significantly with increasing decoherence strength than in the QFI approach. The lower-left inset confirms again that the structure of the optimal states gets closer to the product state structure with increasing N. We have also applied the MPS approach to Ramsey spectroscopy with other decoherence models including independent dephasing, depolarization, and spontaneous emission and have obtained completely analogous results. In summary, we have shown that MPSs are very well suited for achieving the optimal performance in realistic quantum metrological setups and may reduce the numerical effort while searching for the optimal estimation strategies. Even though we have based our presentation on a single model of lossy phase estimation, we anticipate these conclusions to be valid in all metrological setups where decoherence makes the asymptotic Heisenberg scaling unachievable—the intuitive argument being that no largescale strong correlations are needed to reach the optimal performance. An intriguing open question remains: is it possible, as it is in many-body physics problems, to obtain an exponential reduction in numerical complexity thanks to the use of MPSs? This is not possible when the optimal states are known to be symmetric, as in the lossy phase estimation. In problems, however, where distinguishability of probes is essential as, e.g., in Bayesian multiparameter estimation [38,39], MPSs might demonstrate their full
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potential when the impact of decoherence is taken into account. We thank Janek Kołodyn´ski and Konrad Banaszek for fruitful discussions and support. This research was supported by the European Commission under the Integrating Project Q-ESSENCE, Polish NCBiR under the ERA-NET CHIST-ERA project QUASAR, and the Foundation for Polish Science under the TEAM Programme.
[1] V. Giovanetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011). [2] B. P. Abbott et al., Rep. Prog. Phys. 72, 076901 (2009). [3] C. M. Caves, Phys. Rev. D 23, 1693 (1981). [4] J. Abadie et al. (LIGO Scientific Collaboration), Nat. Phys. 7, 962 (2011). [5] H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Gra¨f, K. Danzmann, and R. Schnabel, Classical Quantum Gravity 27, 084027 (2010). [6] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996). [7] P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, Science 309, 749 (2005). [8] C. Ospelkaus, U. Warring, Y. Colombe, K. R. Brown, J. M. Amini, D. Leibfried, and D. J. Wineland, Nature (London) 476, 181 (2011). [9] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004). [10] D. W. Berry and H. M. Wiseman, Phys. Rev. Lett. 85, 5098 (2000). [11] H. Lee, P. Kok, and J. P. Dowling, J. Mod. Opt. 49, 2325 (2002). [12] M. Zwierz, C. A. Perez-Delgado, and P. Kok, Phys. Rev. Lett. 105, 180402 (2010). [13] J. P. Dowling, Phys. Rev. A 57, 4736 (1998). [14] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. Ekert, M. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865 (1997). [15] A. Shaji and C. M. Caves, Phys. Rev. A 76, 032111 (2007). [16] S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, Phys. Rev. A 83, 021804(R) (2011). [17] J. Kolodynski and R. Demkowicz-Dobrzanski, Phys. Rev. A 82, 053804 (2010). [18] B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011). [19] R. Demkowicz-Dobrzan´ski, J. Kolodyn´ski, and M. Guta, Nat. Commun. 3, 1063 (2012). [20] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Commun. Math. Phys. 115, 477 (1988). [21] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992). [22] F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006). [23] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Inf. Comput. 7, 401 (2007). [24] F. Verstraete, J. I. Cirac, and V. Murg, Adv. Phys. 57, 143 (2008).
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[25] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994). [26] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). [27] O. E. Barndorff-Nielsen, R. D. Gill, and P. E. Jupp, J. R. Stat. Soc. Ser. B 65, 775 (2003). [28] M. G. A. Paris, Int. J. Quantum. Inform. 07, 125 (2009). [29] J. Kahn and M. Guta, Commun. Math. Phys. 289, 597 (2009). [30] P. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Phys. Rev. Lett. 104, 103602 (2010). [31] V. Giovannetti and L. Maccone, Phys. Rev. Lett. 108, 210404 (2012). [32] P. Kolenderski and R. Demkowicz-Dobrzan´ski, Phys. Rev. A 78, 052333 (2008).
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[33] R. Demkowicz-Dobrzan´ski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. A 80, 013825 (2009). [34] G. To´th and D. Petz, Phys. Rev. A 87, 032324 (2013). [35] A. Fujiwara and H. Imai, J. Phys. A 41, 255304 (2008). [36] U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Phys. Rev. Lett. 102, 040403 (2009). [37] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Phys. Rev. A 46, R6797 (1992). [38] G. Chiribella, G. M. D’Ariano, P. Perinotti, and M. F. Sacchi, Phys. Rev. Lett. 93, 180503 (2004). [39] E. Bagan, M. Baig, A. Brey, R. Munoz-Tapia, and R. Tarrach, Phys. Rev. A 63, 052309 (2001). [40] J. Ma, X. Wang, C. P. Sun, and F. Nori, Phys. Rep. 509, 89 (2011).
240405-5
PHYSICAL REVIEW LETTERS
PRL 110, 240405 (2013)
Matrix Product States for Quantum Metrology Marcin Jarzyna and Rafał Demkowicz-Dobrzan´ski Faculty of Physics, University of Warsaw, ul. Hoz˙a 69, PL-00-681 Warszawa, Poland (Received 5 February 2013; revised manuscript received 23 April 2013; published 14 June 2013) We demonstrate that the optimal states in lossy quantum interferometry may be efficiently simulated using low rank matrix product states. We argue that this should be expected in all realistic quantum metrological protocols with uncorrelated noise and is related to the elusive nature of the Heisenberg precision scaling in the asymptotic limit of a large number of probes. DOI: 10.1103/PhysRevLett.110.240405
PACS numbers: 03.65.Ta, 02.70.c, 06.20.Dk, 42.50.St
Over recent years, advancements in quantum engineering have pushed nonclassical concepts such as entanglement and squeezing, previously regarded as largely academic topics, close to practical applications. Quantum features of light and atoms helped to improve the performance of measuring devices that operate in the regime where the precision is limited by the fundamental laws of physics [1]. One of the most spectacular examples of practical applications of quantum metrology can be found in gravitational wave detectors [2], where the original idea [3] of employing squeezed states of light to improve the sensitivity of an interferometer has found its full-scale realization [4,5]. No less impressive are experiments with trapped entangled ions demonstrating spectroscopic resolution enhancement crucial for the operation of atomic clocks [6–8]. When standard sources of laser light are being used, any interferometric experiment may be fully described by treating each photon individually and claiming that each photon interferes only with itself. Sensing a phase delay between the two arms of the interferometer via intensity measurements may be regarded as many independent repetitions of single photon interferometric experiments. N independent experiments result in data that allow pffiffiffiffithe parameter to be estimated with error scaling as 1= N — the so-called standard quantum limit or the shot noise limit. If, however, an experiment cannot be split into N-independent processes, as is, e.g., the case with the N probing photons being entangled, the above reasoning is invalid, and one can in principle achieve the 1=N estimation precision—the Heisenberg scaling [9–12]—with the help of, e.g., the N00N states [13]. Still, in all realistic experimental setups, decoherence typically makes the relevant quantum features, such as squeezing or entanglement, die out very quickly [14,15]. Recently, it has been rigorously shown for optical interferometry with loss [16,17], as well as for more general decoherence models [18,19], that if decoherence acts independently on each of the probes, one can get at best pffiffiffiffi c= N asymptotic scaling of error—precision that is better than the classical one only by a constant factor c which depends on the type of decoherence and its strength. 0031-9007=13=110(24)=240405(5)
One can therefore appreciate the Heisenberg-like decrease in uncertainty only in the regime of small N, where the precise meaning of ‘‘small’’ depends on the decoherence strength [18], and in typical cases is of the order of 10 photons or atoms. This indicates that in the limit of a large number of probes, almost optimal performance can be achieved by dividing the probes into independent groups where only the probes from a given group are entangled among each other. Clearly, the size ofpthe ffiffiffiffi group that is needed to approach the fundamental c= N bound up to a given accuracy will depend on the strength of decoherence. Nevertheless, irrespective of how small the decoherence strength is, for N large enough, the size of the group will saturate at some point, and therefore, asymptotically the optimal state may be regarded as only locally correlated. A natural class of states efficiently representing locally correlated states are the matrix product states (MPSs) [20–24], which have proved to be highly successful in simulating low-energy states of complex spin systems. Until now no attempt has been made, however, to employ MPSs for quantum metrology purposes. Establishing this connection is the essence of the present Letter. A basic quantum metrology scheme is depicted in Fig. 1. The N probe input state ji travels through N parallel noisy channels whose action is parametrized by an ^ x is performed on the unknown value . A measurement output density matrix ^ ¼ N ðjihjÞ yielding a result ^ x with probability pðxjÞ ¼ Trð^ x Þ. The estimation procedure is completed by specifying an estimator function ~ ðxÞ. Eventually we are left with the estimated value ~ which in general will be different of the parameter , from . We denote the average uncertainty of estimation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ Þ2 i, where the average is performed by ¼ hð over different measurement results x. The main goal of theoretical quantum metrology is to find strategies that minimize . For this purpose, one has to find the optimal estimator, measurement, and input state. This in general is a difficult task. To simplify the problem, one may resort to the quantum Cramer-Rao inequality [25–28]
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FIG. 1. Quantum metrology with matrix product states. The N parallel quantum channels act on the input state ji inscribing parameter value as well as causing local decoherence on each ^ x on the output state ^ allows us of the probes. Measurement ~ to make an estimate ðxÞ based on the measurement result x. The input state is a matrix product state obtained by the action of maps A½k k on pairs of virtual D-dimensional systems prepared in such a way that the two adjacent systems corresponding to different neighboring particles are in a maximally entangled state j’D i.
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; kFð^ Þ
Fð^ Þ ¼ Trð^ L^ 2 Þ
(1)
that bounds the precision of any unbiased estimation strategy based on k-independent repetitions of an experiment. Fð^ Þ is the quantum Fisher information (QFI) written in terms of L^ , the so-called symmetric logarithmic derivative defined implicitly as 2ðd^ =dÞ ¼ L^ ^ þ ^ L^ . For pure states, the formula for QFI simplifies to Fðj iÞ ¼ 4ðh_ j_ i jh_ j ij2 Þ, where j_ i ¼ ðdj i=dÞ. The bound is known to be saturable in the asymptotic limit of k ! 1 in the sense that there exist a measurement and an estimator that yields equality in Eq. (1). The main benefit of using QFI is that since it does not depend on the measurement or the estimator, the only remaining optimization problem is the maximization of Fð^ Þ over the input states. Since the optimal states in the regime of a large number of probes N (not k) may be regarded as consisting of independent groups, the Cramer-Rao bound may be saturated even for k ¼ 1, provided N is large enough [27,29]. This makes the QFI an even more appealing quantity than in the decoherence-free case where some controversies arise on the practical use of the strategies based on the optimization of the QFI [30,31]. The maximization of QFI over the most general input states for large N may still be challenging though, and even if it was successful, it might not provide insight into the structure of the optimal states. This is the place where MPSs are useful. A general MPS of N qubits is defined as jiMPS
1 ¼ pffiffiffiffiffiffiffi N
1 X 1 ...N ¼0
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PRL 110, 240405 (2013)
TrðA½1 1
. . . A½N Þj1 N
. . . N i;
(2)
where A½k k are square complex matrices of dimension D D, D is called the bond dimension, and N is the
normalization factor. In operational terms, a MPS is generated by assuming that each qubit is substituted by a pair of D-dimensional virtual systems. Adjacent systems corresponding to different neighboring particles prepared pffiffiffiffiare P in maximally entangled states j’D i ¼ 1= D D ¼1 j; i PD ½k (Fig. 1) and maps Ak ¼ ;¼1 Ak ;; jk ih; j are applied to the pair of virtual systems corresponding to the kth particle [23]. Such a description of state is very efficient, provided the bond dimension D increases slowly with N. In a most favorable case when D may be assumed to be bounded, D < Dmax , the number of coefficients needed to specify an N qubit state in the asymptotic regime of large N will scale as ND2max (linear in N), as opposed to the standard 2N scaling. It should be noted, however, that in many quantum metrological models, in particular, the ones based on the QFI, the search for the optimal input probe states may be restricted to symmetric (bosonic) states [14,18,32,33]. Even though the description of a symmetric N qubit pure state is efficient and requires only N þ 1 parameters, the use of MPSs may still offer a significant advantage as the symmetric MPS description involves matrices A which are identical for different particles: A½k ¼ A and commute under the trace—TrðA1 . . . AN Þ does not depend on the order of matrices. Provided D is asymptotically bounded or grows slowly with N, one can still benefit significantly from the use of MPSs in the large N regime. In order to demonstrate the power of the MPS approach, we apply it to the most thoroughly analyzed and relevant model in quantum metrology—the lossy interferometer. We will not specify the nature of the physical systems (atoms, photons) but will rather refer to abstract twolevelprobes, with orthogonal states j0i, j1i. The parameter to be estimated is the relative phase delay that a probe experiences being in the j1i vs j0i state. The decoherence mechanism amounts to a loss of probes where each of the probes is lost independently of the others with probability 1 . As such, this is an example of a general scheme depicted in Fig. 1. Since the distinguishability of probes offers no advantage for phase estimation [33], we move to the symmetric state P description where the general N probe state reads ji ¼ N n¼0 n jn;N ni, and jn;N ni represents the n and N n probes in states j0i and j1i, respectively. The output state ^ can be written explicitly as ^ ¼
N Nl X0 X l0 ¼0 l1 ¼0
pl0 l1 jl0 l1 ihl0 l1 j;
(3)
where X1 1 Nl jl0 l1 i ¼ pffiffiffiffiffiffiffiffiffi ein nl0 l1 jn l0 ; N n l1 i; pl0 l1 n¼l0 n ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n nl n n Nn n ð1 Þl ; (4) Bl ¼ l0 l1 ðÞ ¼ Bl0 Bl1 ; l
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FIG. 2. Log-log plots of the phase estimation precision pffiffiffiffi with losses ( ¼ 0:9) as a function of the number of probes N optimized over input states in (a) the QFI approach where ¼ 1= F with F given by Eq. (5) and (b) the Ramsey interferometry with given by Eq. (7). The solid gray curves ordered from top pffiffiffiffiffiffiffi ffi to bottom correspond to matrix product states of increasing bond dimension D: D ¼ 1 (lightest gray, no correlations: ¼ 1= N ), D ¼ 2, D ¼ 3, D ¼ 4, D ¼ 5 (darkest gray). The black solid curves correspond to the optimal states obtained with brute force optimization, with extrapolation for higher N (dashed) obtained using scaling formulas of Ref. [16] (a)ffi and optimization over the spin-squeezed states [40] (b). The dashed-dotted lines represent the asymptotic bound ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 =N [16–19]. The upper-right insets depict the bond dimension D needed to reach the optimal precision with at most 1% discrepancy: ¼ 0:9 (solid), ¼ 0:3 (dashed). Lower-left insets present normalized absolute values of elements of the optimal diagonal MPS matrix A for bond dimension D ¼ 4 and ¼ 0:9.
and pl0 l1 is a normalization factor which can be interpreted as a probability to lose l0 , l1 probes in states j0i and j1i, respectively. As the output state ^ is mixed, QFI is not explicitly given in terms of the input state parameters, as it requires involved calculation of the symmetric logarithmic derivative or other equivalent quantities [18,34,35]. In general, due to convexity, QFI for a mixed state is smaller than a weighted sum of QFIs for pure states into which a mixed state may be decomposed. Nevertheless, it was shown in Ref. [36] that assuming additional knowledge of how many photons were lost being in the state j0i and how many being in the state j1i does not improve the estimation precision appreciably. Hence, one can approximate the QFI with a weighted sum of the QFI’s all pure states entering the mixture ^ : ~ ^ Þ ¼ Fð^ Þ & Fð
N Nl X0 X l0 ¼0 l1 ¼0
pl0 ;l1 Fðjl0 ;l1 iÞ:
(5)
This approximation simplifies the calculations significantly, since in our case Fðjl0 l1 iÞ ¼ 4ðhl0 l1 jn^ 2 jl0 l1 i
^ l0 l1 ij2 Þ with n^ being the excitation number jhl0 l1 jnj ^ N ni ¼ njn; N ni. Direct optimization operator njn; of Eq. (5) over the input state parameters n involves N þ 1 variables. This approach was taken in Refs. [33,36]. Here we consider the class of symmetric MPSs parametrized with two diagonal (assuring commutativity) D D matrices A0 , A1 . These MPSs are parametrized with 2D complex numbers, instead of N þ 1, and read explicitly
jiMPS
sffiffiffiffiffiffiffiffiffi N 1 X N TrðAn0 ANn ¼ pffiffiffiffiffiffiffi Þjn; N ni: 1 n N n¼0
(6)
Thanks to the simple form of Eq (5), it is possible to compute F~ directly on A matrices, and there is no need to go back to the less efficient standard description as would be the case with Eq. (1). Figure 2(a) illustrates the precision obtained using MPSs for the case of relatively small losses ¼ 0:9. As one can see, the MPS approximation is excellent. In particular, the upper-right inset shows that already D ¼ 5 is sufficient to obtain less than 1% discrepancy for N 500. We have confirmed this observation for different and observed that for higher losses (lower ), lower D are required to obtain a given level of approximation for a particular N—an effect that should be much more spectacular for larger N reflecting the fact that stronger decoherence diminishes the role of quantum correlations. Moreover, we have observed that optimal matrices A0 , A1 have the same diagonal values which are ordered complementarily—the highest in A0 is paired with lowest one in A1 , etc. N is higher the closer the diagonal values approach each other, as can be seen from the lower-left inset on Fig. 2. This confirms the intuition that with increasing N, the optimal states are becoming less distinct from the product state—all diagonal values of A equal. The peculiarity of phase estimation is that in the decoherence-free case, optimal QFI is achieved for the N00N state which, even though it has nonlocal correlations, it is an example of a MPS with D ¼ 2. This makes the MPS capable of approximating the optimal states very well even for low loss and small N [N & 1=ð1 Þ]—an
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ability that in general will not hold for other estimation problems. Taking now a more operational approach, not based on the QFI, one may consider a concrete measurement scheme with a particular observable O^ being measured. A simple error-propagation formula for yields ¼ ^ ^ O=jdh Oi=dj. In the Ramsey spectroscopy setup [37], or equivalently in the Mach-Zehnder interferometer with a photon number difference measurement, one effectively measures a component of the total angular momentum operator J^ of N spins 1=2—if a qubit j0i, j1i is treated as a spin-1=2 particle [11]. If the phase-dependent rotation U^ ¼ eiJ^z is being sensed by the measurement of the J^x observable, the explicit formula for estimation uncertainty at the optimal operation point ¼ 0 calculated for ^ from Eq. (3) reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N 2 J^ : ¼ ^ 2x þ hJ y i 4hJ^y i2
(7)
The search for the optimal state amounts to minimizing the above quantity. Since it depends only on first and second ^ it is simple to implement numerically using moments of J, MPSs. The results are presented in Fig. 2(b). It is clear that MPSs are capable of capturing the essential feature of the optimal states—the squeezing of the J^x —with relatively low bond dimensions D. Moreover, the upper-right inset indicates that the required bond dimension D is reduced much more significantly with increasing decoherence strength than in the QFI approach. The lower-left inset confirms again that the structure of the optimal states gets closer to the product state structure with increasing N. We have also applied the MPS approach to Ramsey spectroscopy with other decoherence models including independent dephasing, depolarization, and spontaneous emission and have obtained completely analogous results. In summary, we have shown that MPSs are very well suited for achieving the optimal performance in realistic quantum metrological setups and may reduce the numerical effort while searching for the optimal estimation strategies. Even though we have based our presentation on a single model of lossy phase estimation, we anticipate these conclusions to be valid in all metrological setups where decoherence makes the asymptotic Heisenberg scaling unachievable—the intuitive argument being that no largescale strong correlations are needed to reach the optimal performance. An intriguing open question remains: is it possible, as it is in many-body physics problems, to obtain an exponential reduction in numerical complexity thanks to the use of MPSs? This is not possible when the optimal states are known to be symmetric, as in the lossy phase estimation. In problems, however, where distinguishability of probes is essential as, e.g., in Bayesian multiparameter estimation [38,39], MPSs might demonstrate their full
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potential when the impact of decoherence is taken into account. We thank Janek Kołodyn´ski and Konrad Banaszek for fruitful discussions and support. This research was supported by the European Commission under the Integrating Project Q-ESSENCE, Polish NCBiR under the ERA-NET CHIST-ERA project QUASAR, and the Foundation for Polish Science under the TEAM Programme.
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