Optimal measurements in phase estimation: simple examples

We identify optimal measurement strategies for phase estimation in different scenarios in which the interferometer acts on two-mode symmetric states. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed, this set reduces to merely two possible measurements. For two-qubit symmetric Werner state, we find the optimal measurement and show that estimation from the population imbalance is optimal only if the state is pure. We also determine the optimal measurements for a wide class of symmetric N-qubit Werner-like states. Finally, for a pure symmetric state of N qubits, we find under which conditions the estimation from the full N-body correlation and from the population imbalance is optimal.

[1]  Klaus Blaum,et al.  Radiative cooling of Al−4 and Al−5 in a cryogenic environment , 2012 .

[2]  A. Smerzi,et al.  Phase estimation from atom position measurements , 2010, 1012.3593.

[3]  J. Grond,et al.  Atom interferometry with trapped Bose–Einstein condensates: impact of atom–atom interactions , 2010, 1002.0265.

[4]  Yun Li,et al.  Atom-chip-based generation of entanglement for quantum metrology , 2010, Nature.

[5]  G. Tóth,et al.  Multipartite entanglement and high precision metrology , 2010, 1006.4368.

[6]  R. B. Blakestad,et al.  Creation of a six-atom ‘Schrödinger cat’ state , 2005, Nature.

[7]  Wineland,et al.  Squeezed atomic states and projection noise in spectroscopy. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[8]  F. Piazza,et al.  Sub-shot-noise interferometry from measurements of the one-body density , 2011, 1108.2785.

[9]  Augusto Smerzi,et al.  Fisher information and multiparticle entanglement , 2010, 1006.4366.

[10]  W. Wootters.  Statistical distance and Hilbert space , 1981 .

[11]  V. Vuletić,et al.  States of an ensemble of two-level atoms with reduced quantum uncertainty. , 2008, Physical review letters.

[12]  Holland,et al.  Interferometric detection of optical phase shifts at the Heisenberg limit. , 1993, Physical review letters.

[13]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[14]  Augusto Smerzi,et al.  Quantum theory of phase estimation , 2014, 1411.5164.

[15]  Augusto Smerzi,et al.  Not all pure entangled states are useful for sub-shot-noise interferometry , 2009, 0912.4349.

[16]  Augusto Smerzi,et al.  Ultrasensitive two-mode interferometry with single-mode number squeezing. , 2013, Physical review letters.

[17]  S. Lloyd,et al.  Advances in quantum metrology , 2011, 1102.2318.

[18]  Philipp Treutlein,et al.  Quantum metrology with a scanning probe atom interferometer. , 2013, Physical review letters.

[19]  G. Milburn,et al.  Generalized uncertainty relations: Theory, examples, and Lorentz invariance , 1995, quant-ph/9507004.

[20]  C. F. Roos,et al.  ‘Designer atoms’ for quantum metrology , 2006, Nature.

[21]  Moore,et al.  Spin squeezing and reduced quantum noise in spectroscopy. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[22]  M. Oberthaler,et al.  Squeezing and entanglement in a Bose–Einstein condensate , 2008, Nature.

[23]  H. Proudhon,et al.  Methodology for studying strain inhomogeneities in polycrystalline thin films during in situ thermal loading using coherent x-ray diffraction , 2010 .

[24]  James D. Louck,et al.  Angular Momentum in Quantum Physics: Theory and Application , 1984 .

[25]  Augusto Smerzi,et al.  Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation. , 2011, Physical review letters.

[26]  D. Leibfried,et al.  Toward Heisenberg-Limited Spectroscopy with Multiparticle Entangled States , 2004, Science.

[27]  P. Windpassinger,et al.  Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit , 2008, Proceedings of the National Academy of Sciences.

[28]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[29]  P. Zoller,et al.  Many-particle entanglement with Bose–Einstein condensates , 2000, Nature.

[30]  Ruth J. Muschel,et al.  Corrigendum: Cancer cells that survive radiation therapy acquire HIF-1 activity and translocate toward tumour blood vessels , 2013, Nature Communications.

[31]  L. Pezzè,et al.  Entanglement, nonlinear dynamics, and the heisenberg limit. , 2007, Physical review letters.

[33]  Holger F. Hofmann,et al.  All path-symmetric pure states achieve their maximal phase sensitivity in conventional two-path interferometry , 2008, 0812.0044.

[34]  S. Lloyd,et al.  Quantum metrology. , 2005, Physical review letters.

[35]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[36]  H. Cramér.  Mathematical methods of statistics , 1947 .

[37]  H M Wiseman,et al.  Entanglement-enhanced measurement of a completely unknown phase , 2010, CLEO/QELS: 2010 Laser Science to Photonic Applications.

[38]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[39]  M. Oberthaler,et al.  Nonlinear atom interferometer surpasses classical precision limit , 2010, Nature.

[40]  Keiji Sasaki,et al.  Beating the Standard Quantum Limit with Four-Entangled Photons , 2007, Science.

[41]  M. Paris.  Quantum estimation for quantum technology , 2008, 0804.2981.

[42]  Matthias D. Lang,et al.  Optimal quantum-enhanced interferometry using a laser power source. , 2013, Physical review letters.

[43]  Klauder,et al.  SU(2) and SU(1,1) interferometers. , 1986, Physical review. A, General physics.

[44]  James K Thompson,et al.  Conditional spin squeezing of a large ensemble via the vacuum Rabi splitting. , 2011, Physical review letters.

[45]  I. Walmsley,et al.  Experimental quantum-enhanced estimation of a lossy phase shift , 2009, 0906.3511.

[46]  T. Monz,et al.  14-Qubit entanglement: creation and coherence. , 2010, Physical review letters.

[47]  M. Brando,et al.  Ferromagnetic quantum criticality in the quasi-one-dimensional heavy fermion metal YbNi4P2 , 2011, 1108.4274.

[48]  W. Ertmer,et al.  Twin Matter Waves for Interferometry Beyond the Classical Limit , 2011, Science.

[49]  A Smerzi,et al.  Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry. , 2007, Physical review letters.

[50]  Kaushik P. Seshadreesan,et al.  Phase estimation at the quantum Cramér-Rao bound via parity detection , 2013 .

[51]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.