On the power of weak measurements in separating quantum states. (English) Zbl 1317.81017
Summary: We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when and how a set of consecutive weak measurements converges to a strong measurement. Second, we show that for a small set of consecutive weak measurements, long before their convergence, one can separate close states without causing their collapse. We thus demonstrate a tradeoff between the success probability and the bias of the original vector towards collapse. Next, we use post-selection within the two-state vector formalism and present the non-linear expansion of the expectation value of the measurement device’s pointer to distinguish between two predetermined close vectors.
MSC:
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 81P50 | Quantum state estimation, approximate cloning |
References:
| [1] | Aharonov, Y., Albert, D., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351-1354 (1988) · doi:10.1103/PhysRevLett.60.1351 |
| [2] | Hosten, O., Kwiat, P.: Observation of the spin Hall effect of light via weak measurements. Science 319, 787-790 (2008) · doi:10.1126/science.1152697 |
| [3] | Starling, D.J., Dixon, P.B., Jordan, A.N., Howell, J.C.: Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values. Phys. Rev. A 80, 041803 (2009) · doi:10.1103/PhysRevA.80.041803 |
| [4] | Xu, X.Y., Kedem, Y., Sun, K., Vaidman, L., Li, C.F., Guo, G.C.: Phase estimation with weak measurement using a white light source. Phys. Rev. Lett. 111, 033604 (2013) · doi:10.1103/PhysRevLett.111.033604 |
| [5] | Jordan, A.N., Tollaksen, J., Troupe, J.E., Dressel, J., Aharonov, Y.: Heisenberg scaling with weak measurement: A quantum state discrimination point of view (2014). arXiv:1409.3488 · Zbl 1317.81013 |
| [6] | Aharonov, Y., Botero, A., Popescu, S., Reznik, B., Tollaksen, J.: Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values. Phys. Lett. A 301, 130-138 (2002) · Zbl 0997.81011 · doi:10.1016/S0375-9601(02)00986-6 |
| [7] | Tollaksen, J., Aharonov, Y., Casher, A., Kaufherr, T., Nussinov, S.: Quantum interference experiments, modular variables and weak measurements. New J. Phys. 12, 013023 (2010) · doi:10.1088/1367-2630/12/1/013023 |
| [8] | Aharonov, Y., Popescu, S., Rohrlich, D., Skrzypczyk, P.: Quantum cheshire cats. New J. Phys. 15, 113015 (2013) · Zbl 1451.81018 · doi:10.1088/1367-2630/15/11/113015 |
| [9] | Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87, 052104 (2013) · doi:10.1103/PhysRevA.87.052104 |
| [10] | Aharonov, Y., Cohen, E., Ben-Moshe, S.: Unusual interactions of pre- and post-selected particles. EPJ Web Conf. 70, 00053 (2014) · doi:10.1051/epjconf/20147000053 |
| [11] | Aharonov, Y.; Vaidman, L.; Muga, JG (ed.); etal., The two-state vector formalism: an updated review, 399-447 (2007), Berlin, Heidelberg · doi:10.1007/978-3-540-73473-4_13 |
| [12] | Aharonov, Y., Rohrlich, D.: Quantum Paradoxes: Quantum Theory for the Perplexed. Wiley, Weinheim (2005) · Zbl 1081.81001 · doi:10.1002/9783527619115 |
| [13] | Tamir, B., Cohen, E.: Introduction to weak measurements and weak values. Quanta 2, 7-17 (2013) · Zbl 1351.81030 · doi:10.12743/quanta.v2i1.14 |
| [14] | Aharonov, Y., Cohen, E., Elitzur, A.C.: Foundations and applications of weak quantum measurements. Phys. Rev. A 89, 052105 (2014) · doi:10.1103/PhysRevA.89.052105 |
| [15] | Chefles, A.: Quantum state discrimination. Contemp. Phys. 41, 401-424 (2000) · doi:10.1080/00107510010002599 |
| [16] | Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic press, New York (1976) · Zbl 1332.81011 |
| [17] | Osaki, M., Ban, M., Hirota, O.: Derivation and physical interpretation of the optimum detection operators for coherent-state signals. Phys. Rev. A 54, 1691 (1996) · doi:10.1103/PhysRevA.54.1691 |
| [18] | Ivanovic, I.D.: How to differentiate between non-orthogonal states. Phys. Lett. A 123, 257-259 (1987) · doi:10.1016/0375-9601(87)90222-2 |
| [19] | Bagan, E., Munoz-Tapia, R., Olivares-Renteria, G.A., Bergou, J.A.: Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes. Phys. Rev. A. 86, 030303 (2012) · doi:10.1103/PhysRevA.86.040303 |
| [20] | Herzog, U.: Optimal state discrimination with a fixed rate of inconclusive results. Analytic solution and relation to state discrimination with a fixed error rate. Phys. Rev. A. 86, 032314 (2012) · doi:10.1103/PhysRevA.86.032314 |
| [21] | Zilberberg, O., Romito, A., Starling, D.J., Howland, G.A., Broadbent, C.J., Howell, J.C., Gefen, Y.: Null values and quantum state discrimination. Phys. Rev. Lett. 110, 170405 (2013) · doi:10.1103/PhysRevLett.110.170405 |
| [22] | Chiribella, G., Yang, Y., Yao, A.C.: Quantum replication at the Heisenberg limit. Nat. Commun. 4, 2915 (2013) · doi:10.1038/ncomms3915 |
| [23] | Qiao, C., Wu, S., Chen, Z.B.: Unambiguous discrimination of extremely similar states by a weak measurement (2013). arXiv:1302.5986 |
| [24] | Murch, K.W., Weber, S.J., Macklin, C., Siddiqi, I.: Observing single quantum trajectories of a superconducting quantum bit. Nature 502, 211-214 (2013) · doi:10.1038/nature12539 |
| [25] | Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307-327 (2003) · doi:10.1080/00107151031000110776 |
| [26] | Childs, A.: Quantum information processing in continuous time, Ph.D. Thesis, M.I.T (2004) |
| [27] | Koike, T., Tanaka, S.: Limits on amplification by Aharonov-Albert-Vaidman weak measurement. Phys. Rev. A 84, 062106 (2011) · doi:10.1103/PhysRevA.84.062106 |
| [28] | Jozsa, R.: Complex weak values in quantum measurement. Phys. Rev. A 76, 044103 (2007) · Zbl 1146.76539 · doi:10.1103/PhysRevA.76.044103 |
| [29] | \[|\cos (g\hat{X})-i \langle \hat{A}\rangle_w \sin (g\hat{X})|^2= |\cos (g\hat{X})+b \sin (g\hat{X})|^2= \cos^2(g\hat{X})+b^2\sin^2(g\hat{X})+b\sin (2g\hat{X})= (1-b^2)\cos^2(g\hat{X})+ b^2+b\sin (2g\hat{X})=(1-b^2)( \cos (2g\hat{X})+1)/2 +b^2+b\sin (2g\hat{X})=\frac{(1-b^2)}{2}\cos (2g\hat{X})+\frac{(1+b^2)}{2}+b\sin (2g\hat{X})\]|cos(gX^)-i⟨A^⟩wsin(gX^)|2=|cos(gX^)+bsin(gX^)|2=cos2(gX^)+b2sin2(gX^)+bsin(2gX^)=(1-b2)cos2(gX^)+b2+bsin(2gX^)=(1-b2)(cos(2gX^)+1)/2+b2+bsin(2gX^)=(1-b2)2cos(2gX^)+(1+b2)2+bsin(2gX^) |
| [30] | Use integration by parts on \[\int e^{-\frac{x^2}{2\sigma^2}} \cos (kx) dx\]∫e-x22σ2cos(kx)dx |
| [31] | Note that \[\cos (\eta ) = e^{-2(g\sigma )^2}\] cos(η)=e-2(gσ)2, and \[\sin (\eta )=\sqrt{1-e^{-4(g\sigma )^2}}\] sin(η)=\sqrt{1-e-4(gσ)2}, therefore \[\frac{\sin (\eta )2g \sigma^2}{e^{2(g\sigma )^2}-\cos (\eta )}= \frac{\sqrt{1-e^{-4(g\sigma )^2}} 2g\sigma^2}{e^{2(g\sigma )^2}(1-e^{-4(g\sigma )^2})}\] sin(η)2gσ2e2(gσ)2-cos(η)=\sqrt{1-e-4(gσ)2}2gσ2e2(gσ)2(1-e-4(gσ)2) |
| [32] | Use integration by parts on \[\int e^{-\frac{x^2}{2\sigma^2}}[\cos (2kx)-2kx\sin (2kx)]dx\]∫e-x22σ2[cos(2kx)-2kxsin(2kx)]dx and the previous formula for \[\langle \hat{X}\sin (2g\hat{X})\rangle_{in} \]⟨X^sin(2gX^)⟩in |
| [33] | Troupe, J.E.: Quantum key distribution using sequential weak values. Quantum Stud. Math. Found. 1, 79-96 (2014) · Zbl 1304.81066 · doi:10.1007/s40509-014-0003-9 |
| [34] | von Neumann, J.: Mathematical Foundation of Quantum Mechanics. Princeton University Press, Princeton, NJ, USA (1955) · Zbl 0064.21503 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
