Abstract

Entangled measurement is a crucial tool in quantum technology. We propose a new entanglement measure of multi-mode detection, which estimates the amount of entanglement that can be created in a measurement. To illustrate the proposed measure, we perform quantum tomography of a two-mode detector that is comprised of two superconducting nanowire single photon detectors. Our method utilizes coherent states as probe states, which can be easily prepared with accuracy. Our work shows that a separable state such as a coherent state is enough to characterize a potentially entangled detector. We investigate the entangling capability of the detector in various settings. Our proposed measure verifies that the detector makes an entangled measurement under certain conditions, and reveals the nature of the entangling properties of the detector. Since the precise characterization of a detector is essential for applications in quantum information technology, the experimental reconstruction of detector properties along with the proposed measure will be key features in future quantum information processing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M. G. Genoni, and M. Barbieri, “Entangling measurements for multiparameter estimation with two qubits,” Quantum Sci. Technol. 3(1), 01LT01 (2018).
[Crossref]

2017 (1)

V. Ansari, G. Harder, M. Allgaier, B. Brecht, and C. Silberhorn, “Temporal-mode measurement tomography of a quantum pulse gate,” Phys. Rev. A 96(6), 063817 (2017).
[Crossref]

2015 (1)

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
[Crossref]

2014 (3)

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86(1), 153–185 (2014).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5(1), 3532 (2014).
[Crossref]

2013 (1)

2012 (5)

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14(8), 085001 (2012).
[Crossref]

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6(6), 374–379 (2012).
[Crossref]

L. Zhang, A. Datta, H. Coldenstrodt-Ronge, X.-M. Jin, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Recursive quantum detector tomography,” New J. Phys. 14(11), 115005 (2012).
[Crossref]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

L. Zhang, H. B. Coldenstrodt-Ronge, A. Datta, G. Puentes, J. S. Lundeen, X.-M. Jin, B. J. Smith, M. B. Plenio, and I. A. Walmsley, “Mapping coherence in measurement via full quantum tomography of a hybrid optical detector,” Nat. Photonics 6(6), 364–368 (2012).
[Crossref]

2011 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

2009 (3)

H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,” Nat. Phys. 5(1), 19–26 (2009).
[Crossref]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: theory and practice,” New J. Phys. 11(9), 093038 (2009).
[Crossref]

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, Ch. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

2007 (2)

2006 (2)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrödinger Kittens for Quantum Information Processing,” Science 312(5770), 83–86 (2006).
[Crossref]

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Moelmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97(8), 083604 (2006).
[Crossref]

2002 (1)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

2001 (4)

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref]

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86(5), 910–913 (2001).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188–5191 (2001).
[Crossref]

1999 (2)

A. Luis and L. L. Sánchez-Soto, “Complete Characterization of Arbitrary Quantum Measurement Processes,” Phys. Rev. Lett. 83(18), 3573–3576 (1999).
[Crossref]

D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature 402(6760), 390–393 (1999).
[Crossref]

1998 (2)

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental Entanglement Swapping: Entangling Photons That Never Interacted,” Phys. Rev. Lett. 80(18), 3891–3894 (1998).
[Crossref]

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

1996 (2)

A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77(8), 1413–1415 (1996).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

1993 (2)

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, ““Event-Ready-Detectors” Bell Experiment via Entanglement Swapping,” Phys. Rev. Lett. 71(26), 4287–4290 (1993).
[Crossref]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Allgaier, M.

V. Ansari, G. Harder, M. Allgaier, B. Brecht, and C. Silberhorn, “Temporal-mode measurement tomography of a quantum pulse gate,” Phys. Rev. A 96(6), 063817 (2017).
[Crossref]

Ansari, V.

V. Ansari, G. Harder, M. Allgaier, B. Brecht, and C. Silberhorn, “Temporal-mode measurement tomography of a quantum pulse gate,” Phys. Rev. A 96(6), 063817 (2017).
[Crossref]

Ashhab, S.

I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86(1), 153–185 (2014).
[Crossref]

Barbieri, M.

E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M. G. Genoni, and M. Barbieri, “Entangling measurements for multiparameter estimation with two qubits,” Quantum Sci. Technol. 3(1), 01LT01 (2018).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5(1), 3532 (2014).
[Crossref]

Bennett, C. H.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Bouwmeester, D.

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental Entanglement Swapping: Entangling Photons That Never Interacted,” Phys. Rev. Lett. 80(18), 3891–3894 (1998).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Brecht, B.

V. Ansari, G. Harder, M. Allgaier, B. Brecht, and C. Silberhorn, “Temporal-mode measurement tomography of a quantum pulse gate,” Phys. Rev. A 96(6), 063817 (2017).
[Crossref]

Brida, G.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14(8), 085001 (2012).
[Crossref]

Briegel, H. J.

H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,” Nat. Phys. 5(1), 19–26 (2009).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188–5191 (2001).
[Crossref]

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86(5), 910–913 (2001).
[Crossref]

Briegel, H.-J.

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Browne, D. E.

H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,” Nat. Phys. 5(1), 19–26 (2009).
[Crossref]

Chen, J.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nat. Photonics 6(6), 374–379 (2012).
[Crossref]

Chuang, I. L.

D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature 402(6760), 390–393 (1999).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Ciavarella, L.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14(8), 085001 (2012).
[Crossref]

Cirac, J. I.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref]

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Coldenstrodt-Ronge, H.

C. M. Natarajan, L. Zhang, H. Coldenstrodt-Ronge, G. Donati, S. N. Dorenbos, V. Zwiller, I. A. Walmsley, and R. H. Hadfield, “Quantum detector tomography of a time-multiplexed superconducting nanowire single-photon detector at telecom wavelengths,” Opt. Express 21(1), 893–902 (2013).
[Crossref]

L. Zhang, A. Datta, H. Coldenstrodt-Ronge, X.-M. Jin, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Recursive quantum detector tomography,” New J. Phys. 14(11), 115005 (2012).
[Crossref]

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, Ch. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: theory and practice,” New J. Phys. 11(9), 093038 (2009).
[Crossref]

Coldenstrodt-Ronge, H. B.

L. Zhang, H. B. Coldenstrodt-Ronge, A. Datta, G. Puentes, J. S. Lundeen, X.-M. Jin, B. J. Smith, M. B. Plenio, and I. A. Walmsley, “Mapping coherence in measurement via full quantum tomography of a hybrid optical detector,” Nat. Photonics 6(6), 364–368 (2012).
[Crossref]

Crepeau, C.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Curty, M.

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

Datta, A.

P. C. Humphreys, B. J. Metcalf, T. Gerrits, T. Hiemstra, A. E. Lita, J. Nunn, S. W. Nam, A. Datta, W. S. Kolthammer, and I. A. Walmsley, “Tomography of photon-number resolving continuous-output detectors,” New J. Phys. 17(10), 103044 (2015).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5(1), 3532 (2014).
[Crossref]

L. Zhang, A. Datta, H. Coldenstrodt-Ronge, X.-M. Jin, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Recursive quantum detector tomography,” New J. Phys. 14(11), 115005 (2012).
[Crossref]

L. Zhang, H. B. Coldenstrodt-Ronge, A. Datta, G. Puentes, J. S. Lundeen, X.-M. Jin, B. J. Smith, M. B. Plenio, and I. A. Walmsley, “Mapping coherence in measurement via full quantum tomography of a hybrid optical detector,” Nat. Photonics 6(6), 364–368 (2012).
[Crossref]

Degiovanni, I. P.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14(8), 085001 (2012).
[Crossref]

Donati, G.

M. D. Vidrighin, G. Donati, M. G. Genoni, X. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5(1), 3532 (2014).
[Crossref]

C. M. Natarajan, L. Zhang, H. Coldenstrodt-Ronge, G. Donati, S. N. Dorenbos, V. Zwiller, I. A. Walmsley, and R. H. Hadfield, “Quantum detector tomography of a time-multiplexed superconducting nanowire single-photon detector at telecom wavelengths,” Opt. Express 21(1), 893–902 (2013).
[Crossref]

Dorenbos, S. N.

Duan, L.-M.

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Figures (6)

Fig. 1.
Fig. 1. Two-mode measurement on two bipartite entangled states (i.e., entanglement swapping). Each circle and link between circle represent a quantum state and quantum entanglement, respectively. The inseparability of a two-mode detector is directly related to the maximum entanglement between modes $A$ and $D$, created by the measurement on modes $B$ and $C$.
Fig. 2.
Fig. 2. Characterization of multi-mode detector. (a) Abstract illustration of two-mode quantum detector tomography. Input states are two-mode coherent states $\left |{\alpha }\right \rangle \left |{\beta }\right \rangle$. $\alpha$ and $\beta$ are complex amplitudes of each coherent state. (b) Experimental setup. (c) Details of input states preparation. Input coherent states are horizontally and vertically polarization two-modes $\left |{\alpha }\right \rangle _H\left |{\beta }\right \rangle _V$ within an optical beam. PBS, Polarization Beam Splitter; H, Half Wave Plate; Q, Quarter Wave Plate; ND, Neutral Density Filter; SNSPD, Superconducting Nanowire Single Photon Detector. Optical beams are transmitted through single-mode optical fibers except inside a fiber bench.
Fig. 3.
Fig. 3. Example of an entanglement swapping setup by single-rail encoding. Mode $A$ ($C$) and mode $B$ ($D$) are encoded as horizontally and vertically polarized beams, respectively. Our two-mode quantum detector described in Fig. 2(b) can be used as a projective measurement on modes $B$ and $C$.
Fig. 4.
Fig. 4. Reconstructed two-mode detector POVMs corresponding to “on”. (a) Separable measurement with $L=0$ (no ND filter). (b) Entangled measurement with $L=1$ (blocking the reflected beam from PBS$_2$). (c)-(e) Other measurements with ND filters $L=16.6\%, 49.1\%, 75.4\%$. (f) Measurement without PBS$_2$. We omit the subscriptions denoting polarization modes for simplicity, $\left |{k,l}\right \rangle \equiv \left |{k}\right \rangle _H\left |{l}\right \rangle _V$ and $\langle {m,n}| \equiv \mbox {}_H\! \langle {m}|\mbox {}_V\!\langle {n}|$.
Fig. 5.
Fig. 5. Photon detection probabilities of the reconstructed POVMs for a maximally mixed state. (i) Experimental results for various channel losses. Theoretical prediction curves (ii) with actual quantum efficiencies and (iii) ideal quantum efficiencies.
Fig. 6.
Fig. 6. Logarithmic negativities of the POVMs. (i) Bound for non-entangled measurement. (ii) Experimental results for various channel losses. Theoretical prediction curves (iii) with actual quantum efficiencies and (iv) ideal quantum efficiencies. (v) Experimental result without PBS$_2$.

Tables (1)

Tables Icon

Table 1. Input states preparation in the experiment. $\theta _Q$ and $\theta _H$ are the rotating angles of QWP and HWP, respectively. The units of powers $P$, $|\alpha |^{2}$, and $|\beta |^{2}$ are the photon numbers per wave packet.

Equations (37)

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M ( Π ^ A , B ( i ) ) E ( Π ^ A , B ( i ) T r [ Π ^ A , B ( i ) ] ) ,
M ( Π ^ B , C ( i ) ) = E ( ρ ^ A , D ( i ) ) ,
| | α | , | β | e i δ = exp [ 1 2 ( | α | 2 + | β | 2 ) ] m , n | α | m | β | n e i n δ m ! n ! | m , n .
{ ( | α | , | β | , δ ) } = { ( 0 , 0 , ) , ( P 1 2 , P 1 2 , m 1 π 4 ) , ( P 1 , 0 , ) , ( 0 , P 1 , ) , ( P 2 4 , 3 P 2 4 , m 2 π 2 ) , ( 3 P 2 4 , P 2 4 , m 2 π 2 ) | m 1 = { 0 , 1 , , 7 } , m 2 = { 0 , 1 , 2 , 3 } } ,
| ψ M = ( U ^ A V ^ B ) | Φ + A , B
= 1 d μ , ν , n u μ , n v ν , n | μ , ν A , B ,
ρ ^ = | ψ 1 ψ 1 | A , B | ψ 2 ψ 2 | C , D ,
| ψ 1 A , B = 1 d ( U ^ A ( 1 ) V ^ B ( 1 ) ) n 1 | n 1 , n 1 A , B ,
| ψ 2 C , D = 1 d ( U ^ C ( 2 ) V ^ D ( 2 ) ) n 2 | n 2 , n 2 C , D ,
Π ^ B , C = i , j , k , l π i , j , k , l | i , j k , l | B , C .
Tr B , C [ ρ ^ Π ^ B , C ] = s , t s , t | B , C ρ ^ Π ^ B , C | s , t B , C = s , t , i , j , k , l s , t | B , C ρ ^ π i , j , k , l | i , j B , C k , l | s , t = i , j , k , l k , l | B , C ρ ^ π i , j , k , l | i , j B , C = 1 d 2 i , j , k , l , n 1 , n 2 , n 1 , n 2 π i , j , k , l ( U ^ A ( 1 ) V ^ D ( 2 ) ) [ k , l | B , C ( V ^ B ( 1 ) U ^ C ( 2 ) ) | n 1 , n 2 B , C ] × [ n 1 , n 2 | B , C ( V ^ B ( 1 ) U ^ C ( 2 ) ) | i , j B , C ] [ | n 1 , n 2 n 1 , n 2 | A , D ( U ^ A ( 1 ) V ^ D ( 2 ) ) ] ,
n 1 , n 2 k , l | B , C ( V ^ B ( 1 ) U ^ C ( 2 ) ) | n 1 , n 2 B , C | n 1 , n 2 A , D = n 1 , n 2 v k , n 1 ( 1 ) u l , n 2 ( 2 ) | n 1 , n 2 A , D = ( V ^ A U ^ D ) | k , l A , D ,
n 1 , n 2 n 1 , n 2 | A , D n 1 , n 2 | B , C ( V ^ B ( 1 ) U ^ C ( 2 ) ) | i , j B , C = i , j | A , D ( V ^ A U ^ D ) .
Tr B , C [ ρ ^ Π ^ B , C ] = 1 d 2 i , j , k , l π i , j , k , l ( U ^ A ( 1 ) V ^ A V ^ D ( 2 ) U ^ D ) × | k , l i , j | A , D ( U ^ A ( 1 ) V ^ A V ^ D ( 2 ) U ^ D ) , = U ( Π ^ A , D ) T U ,
Π ^ A , D = i , j , k , l π i , j , k , l | i , j k , l | A , D ,
U U ^ A ( 1 ) V ^ A V ^ D ( 2 ) U ^ D .
ρ ^ A , D = Tr B , C [ ρ ^ Π ^ B , C ] Tr [ ρ ^ Π ^ B , C ] = U ( Π ^ A , D ) T U Tr [ Π ^ A , D ] .
E ( ρ ^ A , D ) = E ( U ( Π ^ A , D ) T U Tr [ Π ^ A , D ] ) = E ( Π ^ B , C Tr [ Π ^ B , C ] ) = M ( Π ^ B , C ) .
| α , β e i δ N = exp [ 1 2 ( | α | 2 + | β | 2 ) ] m = 0 N n = 0 m α n β ( m n ) e i ( m n ) δ n ! ( m n ) ! | n , m n .
Π ^ = n = 0 N k = 0 n l = 0 n π k , n k , l , n l | k , n k l , n l | .
S N = { ( α v ( s ) , β v ( s ) , 2 π 2 s + 1 m ) | s = { 0 , 1 , , N } , 0 v N s , 0 m 2 s , ( v , m ) Z } .
p ( α , β e i δ ) = α , β e i δ | Π ^ | α , β e i δ
= n = 0 N k = 0 n l = 0 n π k , n k , l , n l e α 2 β 2 α k + l β 2 n k l k ! ( n k ) ! l ! ( n l ) ! e i δ ( k l )
= Δ = N N n = | Δ | N k = max ( 0 , Δ ) min ( n , n + Δ ) π k , n k , k Δ , n + Δ k C k , Δ ( α , β ) e i δ Δ ,
1 2 π 0 2 π p ( α , β e i δ ) e i t δ d δ = n = | t | N k = max ( 0 , t ) min ( n , n + t ) π k , n k , k t , n + t k C k , t ( α , β ) .
1 ( 2 s + 1 ) m = 0 2 s p ( α , β e i 2 π 2 s + 1 m ) e i t 2 π 2 s + 1 m = A t ( s ) ( α , β ) + B t ( s ) ( α , β ) ,
A t ( s ) ( α , β ) = n = | t | N k = max ( 0 , t ) min ( n , n + t ) π k , n k , k t , n + t k C k , t ( α , β ) ,
B t ( s ) ( α , β ) = ( u 0 , | Δ | N ) Δ = t + ( 2 s + 1 ) u n = | Δ | N k = max ( 0 , Δ ) min ( n , n + Δ ) π k , n k , k Δ , n + Δ k C k , Δ ( α , β ) .
A N ( N ) ( α 0 ( N ) , β 0 ( N ) ) = π N , 0 , 0 , N C N , N ( α 0 ( N ) , β 0 ( N ) ) ,
B N ( N ) ( α 0 ( N ) , β 0 ( N ) ) = 0.
| α , β U ^ H W P ( θ H ) U ^ Q W P ( θ Q ) | P , 0 ,
( α β ) = U H W P ( θ H ) U Q W P ( θ Q ) ( P 0 ) ,
U H W P ( θ H ) = ( cos 2 θ H sin 2 θ H sin 2 θ H cos 2 θ H ) ,
U Q W P ( θ Q ) = 1 2 ( 1 i cos 2 θ Q i sin 2 θ Q i sin 2 θ Q 1 + i cos 2 θ Q ) .
| α | = P 2 ( 1 + cos 2 θ Q cos 2 ( 2 θ H θ Q ) ) ,
| β | = P 2 ( 1 cos 2 θ Q cos 2 ( 2 θ H θ Q ) ) ,
tan δ = tan 2 θ Q sin 2 ( 2 θ H θ Q ) ,