Abstract

The additivity property of quantum channels is still an open question and an exciting subject of current research. There are some nonunital channels for which strict additivity is known, although the general rule for nonunital quantum channels is still not proven. We analyze the additivity of the amplitude damping channel, which is an important channel in physical implementations and optical communications. The effect of amplitude damping has great importance in optical communications, since this channel model describes energy dissipation. We show an efficient information computational geometric method to analyze the additivity property of the amplitude damping quantum channel, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as a distance measure.

© 2010 Optical Society of America

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  1. M. B. Ruskai, “Some open problems in quantum information theory,” arXiv:0708.1902, 2007.
  2. M. B. Hastings, “A counterexample to additivity of minimum output entropy,” arXiv:0809.3972, 2009.
  3. F. Brandao, M. Horodecki, “On Hastings’ counterexamples to the minimum output entropy additivity conjecture,” arXiv:0907.3210, 2009.
  4. B. W. Schumacher, M. Westmoreland, “Relative entropy in quantum information theory,” arXiv:quant-ph/0004045, 2000.
  5. B. W. Schumacher, M. Westmoreland, “Optimal signal ensembles,” arXiv:quant-ph/9912122, 1999.
  6. P. Shor, “Additivity of the classical capacity of entanglementbreaking quantum channels,” J. Math. Phys., vol. 246, no. 3, pp. 453–472, 2004.
    [CrossRef]
  7. C. King, “Additivity for unital qubit channels,” J. Math. Phys., vol. 43, pp. 4641–4653, 2002.
    [CrossRef]
  8. M. Fukuda, C. King, D. K. Moser, “Comments on Hastings’ additivity counterexamples,” Commun. Math. Phys.,vol. 296, no. 1, pp. 111–143, 2010.
    [CrossRef]
  9. N. Datta, A. S. Holevo, Y. Suhov, “A quantum channel with additive minimum output entropy,” arXiv:quant-ph/0403072, 2004.
  10. K. Matsumoto, F. Yura, “Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels,” J. Phys. A, vol. 37, pp. L167–L171, 2004.
    [CrossRef]
  11. M. M. Wolf, J. Eisert, “Classical information capacity of a class of quantum channels,” New J. Phys., vol. 7, no. 93, 2005.
    [CrossRef]
  12. M. Hayashi, H. Nagaoka, “General formulas for capacity of classical-quantum channels,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1753–1768, 2003.
    [CrossRef]
  13. M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.
  14. K. Kato, M. Oto, H. Imai, K. Imai, “Voronoi diagrams for pure 1-qubit quantum states,” arXiv:quant-ph/0604101, 2006.
  15. J. A. Cortese, “The Holevo-Schumacher-Westmoreland channel capacity for a class of qudit unital channels,” arXiv:quant-ph/0211093, 2002.
  16. F. Nielsen, J.-D. Boissonnat, R. Nock, “On Bregman Voronoi diagrams,” in Proc. of the 18th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2007, pp. 746–755.
  17. F. Nielsen, R. Nock, “Approximating smallest enclosing balls with applications to machine learning,” Int. J. Comput. Geom. Appl., vol. 19, no. 5, pp. 389–414, Oct. 2009.
    [CrossRef]
  18. L. Gyongyosi, S. Imre, “Novel geometrical solution to additivity problem of classical quantum channel capacity,” in 33rd IEEE Sarnoff Symp., Princeton, NJ, 2010.
  19. L. Gyongyosi, S. Imre, “Computational geometric analysis of physically allowed quantum cloning transformations for quantum cryptography,” in Proc. of the 4th WSEAS Int. Conf. on Computer Engineering and Applications (CEA ’10). Cambridge, MA, 2010, pp. 121–126.
  20. J.-D. Boissonnat, C. Wormser, M. Yvinec, “Curved Voronoi diagrams,” in Effective Computational Geometry for Curves and Surfaces, J.-D. Boissonnat and M. Teillaud, Eds. Springer-Verlag, 2007, pp. 67–116.
  21. F. Aurenhammer, R. Klein, “Voronoi diagrams,” in Handbook of Computational Geometry, J. Sack and G. Urrutia, Eds. Elsevier Science Publishing, 2000, chap. V, pp. 201–290.
    [CrossRef]
  22. R. Nock, F. Nielsen, “Fitting the smallest enclosing Bregman ball,” in Proc. ECML, 2005, pp. 649–656.
  23. S. Imre, F. Balázs, Quantum Computing and Communications—An Engineering Approach. Wiley, 2005.
  24. M. B. Ruskai, S. Szarek, E. Werner, “An analysis of completely-positive trace-preserving maps on 2 by 2 matrices,” arXiv:quant-ph/0101003, 2001.
  25. M. Badoiu, S. Har-Peled, P. Indyk, “Approximate clustering via core-sets,” in Proc. 34th ACM Symp. on Theory of Computing, 2002, pp. 250–257.

2010 (1)

M. Fukuda, C. King, D. K. Moser, “Comments on Hastings’ additivity counterexamples,” Commun. Math. Phys.,vol. 296, no. 1, pp. 111–143, 2010.
[CrossRef]

2009 (1)

F. Nielsen, R. Nock, “Approximating smallest enclosing balls with applications to machine learning,” Int. J. Comput. Geom. Appl., vol. 19, no. 5, pp. 389–414, Oct. 2009.
[CrossRef]

2005 (2)

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

M. M. Wolf, J. Eisert, “Classical information capacity of a class of quantum channels,” New J. Phys., vol. 7, no. 93, 2005.
[CrossRef]

2004 (2)

K. Matsumoto, F. Yura, “Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels,” J. Phys. A, vol. 37, pp. L167–L171, 2004.
[CrossRef]

P. Shor, “Additivity of the classical capacity of entanglementbreaking quantum channels,” J. Math. Phys., vol. 246, no. 3, pp. 453–472, 2004.
[CrossRef]

2003 (1)

M. Hayashi, H. Nagaoka, “General formulas for capacity of classical-quantum channels,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1753–1768, 2003.
[CrossRef]

2002 (1)

C. King, “Additivity for unital qubit channels,” J. Math. Phys., vol. 43, pp. 4641–4653, 2002.
[CrossRef]

Aurenhammer, F.

F. Aurenhammer, R. Klein, “Voronoi diagrams,” in Handbook of Computational Geometry, J. Sack and G. Urrutia, Eds. Elsevier Science Publishing, 2000, chap. V, pp. 201–290.
[CrossRef]

Badoiu, M.

M. Badoiu, S. Har-Peled, P. Indyk, “Approximate clustering via core-sets,” in Proc. 34th ACM Symp. on Theory of Computing, 2002, pp. 250–257.

Balázs, F.

S. Imre, F. Balázs, Quantum Computing and Communications—An Engineering Approach. Wiley, 2005.

Boissonnat, J.-D.

J.-D. Boissonnat, C. Wormser, M. Yvinec, “Curved Voronoi diagrams,” in Effective Computational Geometry for Curves and Surfaces, J.-D. Boissonnat and M. Teillaud, Eds. Springer-Verlag, 2007, pp. 67–116.

F. Nielsen, J.-D. Boissonnat, R. Nock, “On Bregman Voronoi diagrams,” in Proc. of the 18th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2007, pp. 746–755.

Eisert, J.

M. M. Wolf, J. Eisert, “Classical information capacity of a class of quantum channels,” New J. Phys., vol. 7, no. 93, 2005.
[CrossRef]

Fukuda, M.

M. Fukuda, C. King, D. K. Moser, “Comments on Hastings’ additivity counterexamples,” Commun. Math. Phys.,vol. 296, no. 1, pp. 111–143, 2010.
[CrossRef]

Gyongyosi, L.

L. Gyongyosi, S. Imre, “Computational geometric analysis of physically allowed quantum cloning transformations for quantum cryptography,” in Proc. of the 4th WSEAS Int. Conf. on Computer Engineering and Applications (CEA ’10). Cambridge, MA, 2010, pp. 121–126.

L. Gyongyosi, S. Imre, “Novel geometrical solution to additivity problem of classical quantum channel capacity,” in 33rd IEEE Sarnoff Symp., Princeton, NJ, 2010.

Har-Peled, S.

M. Badoiu, S. Har-Peled, P. Indyk, “Approximate clustering via core-sets,” in Proc. 34th ACM Symp. on Theory of Computing, 2002, pp. 250–257.

Hayashi, M.

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

M. Hayashi, H. Nagaoka, “General formulas for capacity of classical-quantum channels,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1753–1768, 2003.
[CrossRef]

Imai, H.

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

Imre, S.

S. Imre, F. Balázs, Quantum Computing and Communications—An Engineering Approach. Wiley, 2005.

L. Gyongyosi, S. Imre, “Novel geometrical solution to additivity problem of classical quantum channel capacity,” in 33rd IEEE Sarnoff Symp., Princeton, NJ, 2010.

L. Gyongyosi, S. Imre, “Computational geometric analysis of physically allowed quantum cloning transformations for quantum cryptography,” in Proc. of the 4th WSEAS Int. Conf. on Computer Engineering and Applications (CEA ’10). Cambridge, MA, 2010, pp. 121–126.

Indyk, P.

M. Badoiu, S. Har-Peled, P. Indyk, “Approximate clustering via core-sets,” in Proc. 34th ACM Symp. on Theory of Computing, 2002, pp. 250–257.

King, C.

M. Fukuda, C. King, D. K. Moser, “Comments on Hastings’ additivity counterexamples,” Commun. Math. Phys.,vol. 296, no. 1, pp. 111–143, 2010.
[CrossRef]

C. King, “Additivity for unital qubit channels,” J. Math. Phys., vol. 43, pp. 4641–4653, 2002.
[CrossRef]

Klein, R.

F. Aurenhammer, R. Klein, “Voronoi diagrams,” in Handbook of Computational Geometry, J. Sack and G. Urrutia, Eds. Elsevier Science Publishing, 2000, chap. V, pp. 201–290.
[CrossRef]

Matsumoto, K.

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

K. Matsumoto, F. Yura, “Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels,” J. Phys. A, vol. 37, pp. L167–L171, 2004.
[CrossRef]

Moser, D. K.

M. Fukuda, C. King, D. K. Moser, “Comments on Hastings’ additivity counterexamples,” Commun. Math. Phys.,vol. 296, no. 1, pp. 111–143, 2010.
[CrossRef]

Nagaoka, H.

M. Hayashi, H. Nagaoka, “General formulas for capacity of classical-quantum channels,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1753–1768, 2003.
[CrossRef]

Nielsen, F.

F. Nielsen, R. Nock, “Approximating smallest enclosing balls with applications to machine learning,” Int. J. Comput. Geom. Appl., vol. 19, no. 5, pp. 389–414, Oct. 2009.
[CrossRef]

R. Nock, F. Nielsen, “Fitting the smallest enclosing Bregman ball,” in Proc. ECML, 2005, pp. 649–656.

F. Nielsen, J.-D. Boissonnat, R. Nock, “On Bregman Voronoi diagrams,” in Proc. of the 18th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2007, pp. 746–755.

Nock, R.

F. Nielsen, R. Nock, “Approximating smallest enclosing balls with applications to machine learning,” Int. J. Comput. Geom. Appl., vol. 19, no. 5, pp. 389–414, Oct. 2009.
[CrossRef]

R. Nock, F. Nielsen, “Fitting the smallest enclosing Bregman ball,” in Proc. ECML, 2005, pp. 649–656.

F. Nielsen, J.-D. Boissonnat, R. Nock, “On Bregman Voronoi diagrams,” in Proc. of the 18th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2007, pp. 746–755.

Ruskai, M. B.

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

Shimono, T.

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

Shor, P.

P. Shor, “Additivity of the classical capacity of entanglementbreaking quantum channels,” J. Math. Phys., vol. 246, no. 3, pp. 453–472, 2004.
[CrossRef]

Wolf, M. M.

M. M. Wolf, J. Eisert, “Classical information capacity of a class of quantum channels,” New J. Phys., vol. 7, no. 93, 2005.
[CrossRef]

Wormser, C.

J.-D. Boissonnat, C. Wormser, M. Yvinec, “Curved Voronoi diagrams,” in Effective Computational Geometry for Curves and Surfaces, J.-D. Boissonnat and M. Teillaud, Eds. Springer-Verlag, 2007, pp. 67–116.

Yura, F.

K. Matsumoto, F. Yura, “Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels,” J. Phys. A, vol. 37, pp. L167–L171, 2004.
[CrossRef]

Yvinec, M.

J.-D. Boissonnat, C. Wormser, M. Yvinec, “Curved Voronoi diagrams,” in Effective Computational Geometry for Curves and Surfaces, J.-D. Boissonnat and M. Teillaud, Eds. Springer-Verlag, 2007, pp. 67–116.

Commun. Math. Phys. (1)

M. Fukuda, C. King, D. K. Moser, “Comments on Hastings’ additivity counterexamples,” Commun. Math. Phys.,vol. 296, no. 1, pp. 111–143, 2010.
[CrossRef]

IEEE Trans. Inf. Theory (1)

M. Hayashi, H. Nagaoka, “General formulas for capacity of classical-quantum channels,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1753–1768, 2003.
[CrossRef]

Int. J. Comput. Geom. Appl. (1)

F. Nielsen, R. Nock, “Approximating smallest enclosing balls with applications to machine learning,” Int. J. Comput. Geom. Appl., vol. 19, no. 5, pp. 389–414, Oct. 2009.
[CrossRef]

J. Math. Phys. (2)

P. Shor, “Additivity of the classical capacity of entanglementbreaking quantum channels,” J. Math. Phys., vol. 246, no. 3, pp. 453–472, 2004.
[CrossRef]

C. King, “Additivity for unital qubit channels,” J. Math. Phys., vol. 43, pp. 4641–4653, 2002.
[CrossRef]

J. Phys. A (1)

K. Matsumoto, F. Yura, “Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels,” J. Phys. A, vol. 37, pp. L167–L171, 2004.
[CrossRef]

New J. Phys. (1)

M. M. Wolf, J. Eisert, “Classical information capacity of a class of quantum channels,” New J. Phys., vol. 7, no. 93, 2005.
[CrossRef]

Quantum Inf. Comput. (1)

M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, “Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures,” Quantum Inf. Comput., vol. 5, pp. 032–040, 2005.

Other (17)

K. Kato, M. Oto, H. Imai, K. Imai, “Voronoi diagrams for pure 1-qubit quantum states,” arXiv:quant-ph/0604101, 2006.

J. A. Cortese, “The Holevo-Schumacher-Westmoreland channel capacity for a class of qudit unital channels,” arXiv:quant-ph/0211093, 2002.

F. Nielsen, J.-D. Boissonnat, R. Nock, “On Bregman Voronoi diagrams,” in Proc. of the 18th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2007, pp. 746–755.

N. Datta, A. S. Holevo, Y. Suhov, “A quantum channel with additive minimum output entropy,” arXiv:quant-ph/0403072, 2004.

M. B. Ruskai, “Some open problems in quantum information theory,” arXiv:0708.1902, 2007.

M. B. Hastings, “A counterexample to additivity of minimum output entropy,” arXiv:0809.3972, 2009.

F. Brandao, M. Horodecki, “On Hastings’ counterexamples to the minimum output entropy additivity conjecture,” arXiv:0907.3210, 2009.

B. W. Schumacher, M. Westmoreland, “Relative entropy in quantum information theory,” arXiv:quant-ph/0004045, 2000.

B. W. Schumacher, M. Westmoreland, “Optimal signal ensembles,” arXiv:quant-ph/9912122, 1999.

L. Gyongyosi, S. Imre, “Novel geometrical solution to additivity problem of classical quantum channel capacity,” in 33rd IEEE Sarnoff Symp., Princeton, NJ, 2010.

L. Gyongyosi, S. Imre, “Computational geometric analysis of physically allowed quantum cloning transformations for quantum cryptography,” in Proc. of the 4th WSEAS Int. Conf. on Computer Engineering and Applications (CEA ’10). Cambridge, MA, 2010, pp. 121–126.

J.-D. Boissonnat, C. Wormser, M. Yvinec, “Curved Voronoi diagrams,” in Effective Computational Geometry for Curves and Surfaces, J.-D. Boissonnat and M. Teillaud, Eds. Springer-Verlag, 2007, pp. 67–116.

F. Aurenhammer, R. Klein, “Voronoi diagrams,” in Handbook of Computational Geometry, J. Sack and G. Urrutia, Eds. Elsevier Science Publishing, 2000, chap. V, pp. 201–290.
[CrossRef]

R. Nock, F. Nielsen, “Fitting the smallest enclosing Bregman ball,” in Proc. ECML, 2005, pp. 649–656.

S. Imre, F. Balázs, Quantum Computing and Communications—An Engineering Approach. Wiley, 2005.

M. B. Ruskai, S. Szarek, E. Werner, “An analysis of completely-positive trace-preserving maps on 2 by 2 matrices,” arXiv:quant-ph/0101003, 2001.

M. Badoiu, S. Har-Peled, P. Indyk, “Approximate clustering via core-sets,” in Proc. 34th ACM Symp. on Theory of Computing, 2002, pp. 250–257.

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Figures (10)

Fig. 1
Fig. 1

The analyzed quantum channel model and entropies.

Fig. 2
Fig. 2

Measurement setting for additivity analysis of quantum channel capacity.

Fig. 3
Fig. 3

The triangle of points corresponding to the vertex c, which is (a) the center of its circumcenter and (b) a quantum Delaunay tessellation.

Fig. 4
Fig. 4

Intersection of the quantum informational ball and channel ellipsoid of the amplitude damping channel.

Fig. 5
Fig. 5

The smallest enclosing ball of a set of balls in the quantum space.

Fig. 6
Fig. 6

Smallest quantum informational balls for (a) quantum channel N 1 and (b) channel N 2 .

Fig. 7
Fig. 7

The radii of the smallest enclosing quantum informational balls for channels for product state inputs for (a) channel N 1 and (b) channel N 2 .

Fig. 8
Fig. 8

The radii of the smallest enclosing quantum informational balls for entangled inputs for (a) channel N 1 and (b) channel N 2 .

Fig. 9
Fig. 9

Sum of radii in quantum informational “superball” representation, for (a) unentangled and (b) entangled input states.

Fig. 10
Fig. 10

The sum of radii for product state inputs and for entangled inputs.

Tables (1)

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Table 1 Algorithm

Equations (28)

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r * = max { all possible x i } H ( X ) H ( X | Y ) .
C ( N ) = max { all possible p i and ρ i } X output = max p 1 , , p n , ρ 1 , , ρ n S ( N ( i = 1 n p i ( ρ i ) ) ) + i = 1 n p i S ( N ( ρ i ) ) ,
C ( N ) = r * = max p 1 , , p n , ρ 1 , , ρ n S ( N ( i = 1 n p i ( ρ i ) ) ) + i = 1 n p i S ( N ( ρ i ) ) .
C ( N ) = D ( ρ k σ ) ,
D ( ρ k σ ) = Tr [ ρ k log ( ρ k ) ρ k log ( σ ) ] .
C ( N ) = r * = min { σ } max { ρ } D ( N ( ρ ) N ( σ ) ) .
C ( N ) = r * = min { σ } max { ρ } D ( ρ σ ) .
k p k D ( ρ k σ ) = k ( p k Tr [ ρ k log ( ρ k ) ] p k Tr [ ρ k log ( σ ) ] ) = k ( p k Tr [ ρ k log ( ρ k ) ] ) Tr [ k ( p k ρ k log ( σ ) ) ] = k ( p k Tr [ ρ k log ( ρ k ) ] ) Tr [ σ log ( σ ) ] = S ( σ ) k p k S ( ρ k ) = X .
C ( N ) = max { all possible p k , ψ k } k p k D ( N ( ψ k ) N ( ψ ) ) ,
C ( N 1 N 2 ) = C ( N 1 ) + C ( N 2 ) ,
C ( N 1 N 2 ) > C ( N 1 ) + C ( N 2 )
C ENT. ( N 1 N 2 ) ? C PROD. ( N 1 ) + C PROD. ( N 2 ) ,
S ( ρ ) = Tr ( ρ log ρ ) .
S ( ρ ) = S ( λ ) = i = 1 d λ i log λ i ,
D ( ρ σ ) = Tr [ ρ ( log ρ log σ ) ] .
F ( ρ ) = S ( ρ ) = Tr ( ρ log ρ ) .
D ( ρ σ ) = F ( ρ ) F ( σ ) ρ σ , F ( σ ) ,
vo ( ρ ) = { x | d ( x , ρ i ) d ( x , ρ j ) S { ρ } } ,
d L ( ρ , x i ) = ρ x i 2 r i 2 .
2 x , σ ρ + ρ , ρ σ , σ + r σ 2 r ρ 2 = 0 .
N ( ρ ) = i A i ρ A i ,
A 1 = [ p 0 0 1 ] , and A 2 = [ 0 0 1 p 0 ] ,
r out = ( r out ( x ) r out ( y ) r out ( z ) ) = ( 1 p 0 0 0 1 p 0 0 0 1 p 2 ) ( r in ( x ) r in ( y ) r in ( z ) ) + ( 0 0 p 2 ) .
t x = 0 , t y = 0 , t z = 1 p ,
λ x = p , λ y = p , λ z = p ,
d ( c , S ) ( 1 + E ) r ,
max i { 1 , , n } D F ( c , b i ) = max i { 1 , , n } ( D F ( c , S i ) + r i ) .
c * = arg min c F B ( c ) .