III. COMPOSITE SYSTEMS, PREPARATIONS, AND MEASUREMENTS

A. States of composite systems

1. Schmidt decomposition

[Schmidt 07 a, b], [von Neumann 32] (Sec. VI. 2), [Furry 36 a, b], [Jauch 68] (Sec. 11. 8), [Ballentine 90 a] (Sec. 8. 3), [Albrecht 92] (Secs. II, III and Appendix), [Barnett-Phoenix 92], [Albrecht 93] (Sec. II and Appendix), [Peres 93 a] (Chap. 5), [Elby-Bub 94] (uniqueness of triorthogonal decomposition of pure states), [Albrecht 94] (Appendix), [Mann-Sanders-Munro 95], [Ekert-Knight 95], [Peres 95 c] (Schmidt decomposition of higher order), [Aravind 96], [Linden-Popescu 97] (invariances in Schmidt decomposition under local transformations), [Acín-Andrianov-Costa-(+3) 00] (Schmidt decomposition and classification of three-quantum-bit pure states), [Terhal-Horodecki 00] (Schmidt number for density matrices), [Higuchi-Sudbery 00], [Carteret-Higuchi-Sudbery 00] (multipartite generalisation of the Schmidt decomposition), [Pati 00 c] (existence of the Schmidt decomposition for tripartite system under certain condition).

2. Entanglement measures

[Barnett-Phoenix 91] ("index of correlation"), [Shimony 95], [Bennett-DiVincenzo-Smolin-Wootters 96] (for a mixed state), [Popescu-Rohrlich 97 a], [Schulman-Mozyrsky 97], [Vedral-Plenio-Rippin-Knight 97], [Vedral-Plenio-Jacobs-Knight 97], [Vedral-Plenio 98 a], [DiVincenzo-Fuchs-Mabuchi-(+3) 98], [Belavkin-Ohya 98], [Eisert-Plenio 99] (a comparison of entanglement measures), [Vidal 99 a] (a measure of entanglement is defended which quantifies the probability of success in an optimal local conversion from a single copy of a pure state into another pure state), [Parker-Bose-Plenio 00] (entanglement quantification and purification in continuous-variable systems), [Virmani-Plenio 00] (various entanglement measures do not give the same ordering for all quantum states), [Horodecki-Horodecki-Horodecki 00 a] (limits for entanglement measures), [Henderson-Vedral 00] (relative entropy of entanglement and irreversibility), [Benatti-Narnhofer 00] (on the additivity of entanglement formation), [Rudolph 00 b], [Nielsen 00 c] (one widely used method for defining measures of entanglement violates that dimensionless quantities do not depend on the system of units being used), [Brylinski 00] (algebraic measures of entanglement), [Wong-Christensen 00], [Vollbrecht-Werner 00] (entanglement measures under symmetry), [Hwang-Ahn-Hwang-Lee 00] (two mixed states such that their ordering depends on the choice of entanglement measure cannot be transformed, with unit efficiency, to each other by any local operations), [Audenaert-Verstraete-De Bie-De Moor 00], [Bennett-Popescu-Rohrlich-(+2) 01] (exact and asymptotic measures of multipartite pure state entanglement), [Majewski 01], [Życzkowski-Bengtsson 01] (relativity of pure states entanglement), [Abouraddy-Saleh-Sergienko-Teich 01] (any pure state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are unique), [Vedral-Kashefi 01] (uniqueness of entanglement measure and thermodynamics), [Vidal-Werner 02] (a computable measure of entanglement), [Eisert-Audenaert-Plenio 02].

3. Separability criteria

[Peres 96 d, 97 a, 98 a] (positive partial transposition (PPT) criterion), [Horodecki-Horodecki-Horodecki 96 c], [Horodecki 97], [Busch-Lahti 97], [Sanpera-Tarrach-Vidal 97, 98], [Lewenstein-Sanpera 98] (algorithm to obtain the best separable approximation to the density matrix of a composite system. This method gives rise to a condition of separability and to a measure of entanglement), [Cerf-Adami-Gingrich 97], [Aravind 97], [Majewski 97], [Dür-Cirac-Tarrach 99] (separability and distillability of multiparticle systems), [Caves-Milburn 99] (separability of various states for N qutrits), [Duan-Giedke-Cirac-Zoller 00 a] (inseparability criterion for continuous variable systems), [Simon 00 b] (Peres-Horodecki separability criterion for continuous variable systems), [Dür-Cirac 00 a] (classification of multiqubit mixed states: Separability and distillability properties), [Wu-Chen-Zhang 00] (a necessary and sufficient criterion for multipartite separable states), [Wang 00 b], [Karnas-Lewenstein 00] (optimal separable approximations), [Terhal 01] (review of the criteria for separability), [Chen-Liang-Li-Huang 01 a] (necessary and sufficient condition of separability of any system), [Eggeling-Vollbrecht-Wolf 01] ([Chen-Liang-Li-Huang 01 a] is a reformulation of the problem rather than a practical criterion; reply: [Chen-Liang-Li-Huang 01 b]), [Pittenger-Rubin 01], [Horodecki-Horodecki-Horodecki 01 b] (separability of n-particle mixed states), [Giedke-Kraus-Lewenstein-Cirac 01] (separability criterion for all bipartite Gaussian states), [Kummer 01] (separability for two qubits), [Albeverio-Fei-Goswami 01] (separability of rank two quantum states), [Wu-Anandan 01] (three necessary separability criteria for bipartite mixed states), [Rudolph 02], [Doherty-Parrilo-Spedalieri 02], [Fei-Gao-Wang-(+2) 02], [Chen-Wu 02] (generalized partial transposition criterion for separability of multipartite quantum states).

4. Multiparticle entanglement

[Elby-Bub 94] (uniqueness of triorthogonal decomposition of pure states), [Linden-Popescu 97], [Clifton-Feldman-Redhead-Wilce 97], [Linden-Popescu 98 a], [Thapliyal 99] (tripartite pure-state entanglement), [Carteret-Linden-Popescu-Sudbery 99], [Fivel 99], [Sackett-Kielpinski-King-(+8) 00] (experimental four-particle entanglement), [Carteret-Sudbery 00] (three-qubit pure states are classified by means of their stabilizers in the group of local unitary transformations), [Acín-Andrianov-Costa-(+3) 00] (Schmidt decomposition and classification of three-qubit pure states), [Acín-Andrianov-Jané-Tarrach 00] (three-qubit pure-state canonical forms), [van Loock-Braunstein 00 b] (multipartite entanglement for continuous variables), [Wu-Zhang 01] (multipartite pure-state entanglement and the generalized GHZ states), [Brun-Cohen 01] (parametrization and distillability of three-qubit entanglement).

5. Entanglement swapping

[Yurke-Stoler 92 a] (entanglement from independent particle sources), [Bennett-Brassard-Crépeau-(+3) 93] (teleportation), [Żukowski-Zeilinger-Horne-Ekert 93] (event-ready-detectors), [Bose-Vedral-Knight 98] (multiparticle generalization of ES), [Pan-Bouwmeester-Weinfurter-Zeilinger 98] (experimental ES: Entangling photons that have never interacted), [Bose-Vedral-Knight 99] (purification via ES), [Peres 99 b] (delayed choice for ES), [Kok-Braunstein 99] (with the current state of technology, event-ready detections cannot be performed with the experiment of [Pan-Bouwmeester-Weinfurter-Zeilinger 98]), [Polkinghorne-Ralph 99] (continuous variable ES), [Żukowski-Kaszlikowski 00 a] (ES with parametric down conversion sources), [Hardy-Song 00] (ES chains for general pure states), [Shi-Jiang-Guo 00 c] (optimal entanglement purification via ES), [Bouda-Buzzek 01] (ES between multi-qudit systems), [Fan 01 a, b], [Son-Kim-Lee-Ahn 01] (entanglement transfer from continuous variables to qubits), [Karimipour-Bagherinezhad-Bahraminasab 02 a] (ES of generalized cat states).

6. Entanglement distillation (concentration and purification)

(Entanglement concentration: How to create, using only LOCC, maximally entangled pure states from not maximally entangled ones. Entanglement purification: How to distill pure maximally entangled states out of mixed entangled states. Entanglement distillation means both concentration or purification) [Bennett-Bernstein-Popescu-Schumacher 95] (concentrating partial entanglement by local operations), [Bennett 95 b], [Bennett-Brassard-Popescu-(+3) 96], [Deutsch-Ekert-Jozsa-(+3) 96], [Murao-Plenio-Popescu-(+2) 98] (multiparticle EP protocols), [Rains 97, 98 a, b], [Horodecki-Horodecki 97] (positive maps and limits for a class of protocols of entanglement distillation), [Kent 98 a] (entangled mixed states and local purification), [Horodecki-Horodecki-Horodecki 98 b, c, 99 a], [Vedral-Plenio 98 a] (entanglement measures and EP procedures), [Cirac-Ekert-Macchiavello 99] (optimal purification of single qubits), [Dür-Briegel-Cirac-Zoller 99] (quantum repeaters based on EP), [Giedke-Briegel-Cirac-Zoller 99] (lower bounds for attainable fidelity in EP), [Opatrný-Kurizki 99] (optimization approach to entanglement distillation), [Bose-Vedral-Knight 99] (purification via entanglement swapping), [Dür-Cirac-Tarrach 99] (separability and distillability of multiparticle systems), [Parker-Bose-Plenio 00] (entanglement quantification and EP in continuous-variable systems), [Dür-Cirac 00 a] (classification of multiqubit mixed states: Separability and distillability properties), [Brun-Caves-Schack 00] (EP of unknown quantum states), [Acín-Jané-Dür-Vidal 00] (optimal distillation of a GHZ state), [Cen-Wang 00] (distilling a GHZ state from an arbitrary pure state of three qubits), [Lo-Popescu 01] (concentrating entanglement by local actions-beyond mean values), [Kwiat-Barraza López-Stefanov-Gisin 01] (experimental entanglement distillation), [Shor-Smolin-Terhal 01] (evidence for nonadditivity of bipartite distillable entanglement), [Pan-Gasparoni-Ursin-(+2) 03] (experimental entanglement purification of arbitrary unknown states, Nature).

7. Disentanglement

[Ghirardi-Rimini-Weber 87] (D of wave functions), [Chu 98] (is it possible to disentangle an entangled state?), [Peres 98 b] (D and computation), [Mor 99] (D while preserving all local properties), [Bandyopadhyay-Kar-Roy 99] (D of pure bipartite quantum states by local cloning), [Mor-Terno 99] (sufficient conditions for a D), [Hardy 99 b] (D and teleportation), [Ghosh-Bandyopadhyay-Roy-(+2) 00] (optimal universal D for two-qubit states), [Buzek-Hillery 00] (disentanglers), [Zhou-Guo 00 a] (D and inseparability correlation in a two-qubit system).

8. Bound entanglement

[Horodecki 97], [Horodecki-Horodecki-Horodecki 98 b, 99 a] (a BE state is an entangled mixed state from which no pure entanglement can be distilled), [Bennett-DiVincenzo-Mor-(+3) 99] (unextendible incomplete product bases provide a systematic way of constructing BE states), [Linden-Popescu 99] (BE and teleportation), [Bruß-Peres 00] (construction of quantum states with BE), [Shor-Smolin-Thapliyal 00], [Horodecki-Lewenstein 00] (is BE for continuous variables a rare phenomenon?), [Smolin 01] (four-party unlockable BE state, r_S=¹/₄ е_i=1⁴ |f_iсбf_i| Д|f_iсбf_i|, where f_i are the Bell states), [Murao-Vedral 01] (remote information concentration -the reverse process to quantum telecloning- using Smolin's BE state), [Gruska-Imai 01] (survey, p. 57), [Werner-Wolf 01 a] (BE Gaussian states), [Sanpera-Bruß-Lewenstein 01] (Schmidt number witnesses and BE), [Kaszlikowski-Żukowski-Gnaci\'nski 02] (BE admits a local realistic description).

9. Entanglement as a catalyst

[Jonathan-Plenio 99 b] (using only LOCC one cannot transform |f₁с into |f₂с, but with the assistance of an appropriate entangled state |yс one can transform |f₁с into |f₂с using LOCC in such a way that the state |yс can be returned back after the process: |yс serves as a catalyst for otherwise impossible transformation), [Barnum 99] (quantum secure identification using entanglement and catalysis), [Jensen-Schack 00] (quantum authentication and key distribution using catalysis), [Zhou-Guo 00 c] (basic limitations for entanglement catalysis), [Daftuar-Klimesh 01 a] (mathematical structure of entanglement catalysis), [Anspach 01] (two-qubit catalysis in a four-state pure bipartite system).

B. State determination, state discrimination, and measurement of arbitrary observables

1. State determination, quantum tomography

[von Neumann 31], [Gale-Guth-Trammell 68] (determination of the quantum state), [Park-Margenau 68], [Band-Park 70, 71, 79], [Park-Band 71, 80, 92], [Brody-Meister 96] (strategies for measuring identically prepared particles), [Hradil 97] (quantum state estimation), [Raymer 97] (quantum tomography, review), [Freyberger-Bardroff-Leichtle-(+2) 97] (quantum tomography, review), [Chefles-Barnett 97 c] (entanglement and unambiguous discrimination between non-orthogonal states), [Hradil-Summhammer-Rauch 98] (quantum tomography as normalization of incompatible observations).

2. Generalized measurements, positive operator-valued measurements (POVMs), discrimination between non-orthogonal states

[Neumark 43, 54] (representation of a POVM by a projection-valued measure -a von Neumman measure- in an extended higher dimensional Hilbert space; see also [Nagy 90]), [Berberian 66] (mathematical theory of POVMs), [Jauch-Piron 67] (POVMs are used in a generalized analysis of the localizability of quantum systems), [Holevo 72, 73 c, 82], [Benioff 72 a, b, c], [Ludwig 76] (POVMs), [Davies-Lewis 70] (analysis of quantum observables in terms of POVMs), [Davies 76, 78], [Helstrom 76], [Ivanovic 81, 83, 93], [Ivanovic 87] (how discriminate unambiguously between a pair of non-orthogonal pure states -the procedure has less than unit probability of giving an answer at all-), [Dieks 88], [Peres 88 b] (IDP: Ivanovic-Dieks-Peres measurements), [Peres 90 a] (Neumark's theorem), [Peres-Wootters 91] (optimal detection of quantum information), [Busch-Lahti-Mittelstaedt 91], [Bennett 92 a] (B92 quantum key distribution scheme: Using two nonorthogonal states), [Peres 93 a] (Secs. 9. 5 and 9. 6), [Busch-Grabowski-Lahti 95], [Ekert-Huttner-Palma-Peres 94] (application of IDP to eavesdropping), [Massar-Popescu 95] (optimal measurement procedure for an infinite number of identically prepared two-level systems: Construction of an infinite POVM), [Jaeger-Shimony 95] (extension of the IDP analysis to two states with a priori unequal probabilities), [Huttner-Muller-Gautier-(+2) 96] (experimental unambiguous discrimination of nonorthogonal states), [Fuchs-Peres 96], [Lütkenhaus 96] (POVMs and eavesdropping), [Brandt-Myers 96, 99] (optical POVM receiver for quantum cryptography), [Grossman 96] (optical POVM; see appendix A of [Brandt 99 b]), [Myers-Brandt 97] (optical implementations of POVMs), [Brandt-Myers-Lomonaco 97] (POVMs and eavesdropping), [Fuchs 97] (nonorthogonal quantum states maximize classical information capacity), [Biham-Boyer-Brassard-(+2) 98] (POVMs and eavesdropping), [Derka-Buzek-Ekert 98] (explicit construction of an optimal finite POVM for two-level systems), [Latorre-Pascual-Tarrach 98] (optimal, finite, minimal POVMs for the cases of two to seven copies of a two-level system), [Barnett-Chefles 98] (application of the IDP to construct a Hardy type argument for maximally entangled states), [Chefles 98] (unambiguous discrimination between multiple quantum states), [Brandt 99 b] (review), [Nielsen-Chuang 00], [Chefles 00 b] (overview of the main approaches to quantum state discrimination), [Sun-Hillery-Bergou 01] (optimum unambiguous discrimination between linearly independent nonorthogonal quantum states), [Sun-Bergou-Hillery 01] (optimum unambiguous discrimination between subsets of non-orthogonal states), [Peres-Terno 02].

3. State preparation and measurement of arbitrary observables

[Fano 57], [Fano-Racah 59], [Wichmann 63] (density matrices arising from incomplete measurements), [Newton-Young 68] (measurability of the spin density matrix), [Swift-Wright 80] (generalized Stern-Gerlach experiments for the measurement of arbitrary spin operators), [Vaidman 88] (measurability of nonlocal states), [Ballentine 90 a] (Secs. 8. 1-2, state preparation and determination), [Phoenix-Barnett 93], [Popescu-Vaidman 94] (causality constraints on nonlocal measurements), [Reck-Zeilinger-Bernstein-Bertani 94 a, b] (optical realization of any discrete unitary operator), [Cirac-Zoller 94] (theoretical preparation of two particle maximally entangled states and GHZ states with atoms), [Żukowski-Zeilinger-Horne 97] (realization of any photon observable, also for composite systems), [Weinacht-Ahn-Bucksbaum 99] (real experiment to control the shape of an atomic electron's wavefunction), [Hladký-Drobný-Buzek 00] (synthesis of arbitrary unitary operators), [Klose-Smith-Jessen 01] (measuring the state of a large angular momentum).

4. Stern-Gerlach experiment and its successors

[Gerlach-Stern 21, 22 a, b], (SGI: Stern-Gerlach interferometer; a SG followed by an inverted SG:) [Bohm 51] (Sec. 22. 11), [Wigner 63] (p. 10), [Feynman-Leighton-Sands 65] (Chap. 5); [Swift-Wright 80] (generalized SG experiments for the measurement of arbitrary spin operators), (coherence loss in a SGI:) [Englert-Schwinger-Scully 88], [Schwinger-Scully-Englert 88], [Scully-Englert-Schwinger 89]; [Summhammer-Badurek-Rauch-Kischko 82] (experimental "SGI" with polarized neutrons), [Townsend 92] (SG, Chap. 1, SGI, Chap. 2), [Platt 92] (modern analysis of a SG), [Martens-de Muynck 93, 94] (how to measure the spin of the electron), [Batelaan-Gay-Schwendiman 97] (SG for electrons), [Venugopalan 97] (decoherence and Schrödinger's-cat states in a SG experiment), [Patil 98] (SG according to QM), [Hannout-Hoyt-Kryowonos-Widom 98] (SG and quantum measurement theory), [Shirokov 98] (spin state determination using a SG), [Garraway-Stenholm 99] (observing the spin of a free electron), [Amiet-Weigert 99 a, b] (reconstructing the density matrix of a spin s through SG measurements), [Reinisch 99] (the two output beams of a SG for spin 1/2 particles should not show interference when appropriately superposed because an entanglement between energy level and path selection occurs), [Schonhammer 00] (SG measurements with arbitrary spin), [Gallup-Batelaan-Gay 01] (analysis of the propagation of electrons through an inhomogeneous magnetic field with axial symmetry: A complete spin polarization of the beam is demonstrated, in contrast with the semiclassical situation, where the spin splitting is blurred), [Berman-Doolen-Hammel-Tsifrinovich 02] (static SG effect in magnetic force microscopy), [Batelaan 02].

5. Bell operator measurements

[Michler-Mattle-Weinfurter-Zeilinger 96] (different interference effects produce three different results, identifying two out of the four Bell states with the other two states giving the same third measurement signal), [Lütkenhaus-Calsamiglia-Suominen 99] (a never-failing measurement of the Bell operator of a two two-level bosonic system is impossible with beam splitters, phase shifters, delay lines, electronically switched linear elements, photo-detectors, and auxiliary bosons), [Vaidman-Yoran 99], [Kwiat-Weinfurter 98] ("embedded" Bell state analysis: The four polarization-entangled Bell states can be discriminated if, simultaneously, there is an additional entanglement in another degree of freedom -time-energy or momentum-), [Scully-Englert-Bednar 99] (two-photon scheme for detecting the four polarization-entangled Bell states using atomic coherence), [Paris-Plenio-Bose-(+2) 00] (nonlinear interferometric setup to unambiguously discriminate the four polarization-entangled EPR-Bell photon pairs), [DelRe-Crosignani-Di Porto 00], [Vitali-Fortunato-Tombesi 00] (with a Kerr nonlinearity), [Andersson-Barnett 00] (Bell-state analyzer with channeled atomic particles), [Tomita 00, 01] (solid state proposal), [Calsamiglia-Lütkenhaus 01] (maximum efficiency of a linear-optical Bell-state analyzer), [Kim-Kulik-Shih 01 a] (teleportation experiment of an unknown arbitrary polarization state in which nonlinear interactions are used for the Bell state measurements and in which all four Bell states can be distinguished), [Kim-Kulik-Shih 01 b] (teleportation experiment with a complete Bell state measurement using nonlinear interactions).