Numerical range of a Hermitian operator is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers, for instance, the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of -positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
REFERENCES
1.
A.
Horn
and C. R.
Johnson
, Topics in Matrix Analysis
(Cambridge University Press
, Cambridge
, 1994
).2.
K. E.
Gustafson
and D. K. M.
Rao
, Numerical Range: The Field of Values of Linear Operators and Matrices
(Springer-Verlag
, New York
, 1997
).3.
T.
Ando
and C. K.
Li
, “Special Issue: The numerical range and numerical radius
,” edited by T.
Ando
and C. K.
Li
, special issue of Linear Multilinear Algebra
37
, 1
(1994
).4.
C. K.
Li
, Linear Multilinear Algebra
37
, 51
(1994
).5.
6.
M.
Marcus
and B.
Wang
, Linear Multilinear Algebra
9
, 111
(1980
).7.
N.
Bebiano
, C. K.
Li
, and J.
da Providencia
, Linear Multilinear Algebra
29
, 195
(1991
).8.
C. K.
Li
and A.
Zaharia
, Trans. Am. Math. Soc.
354
, 807
(2002
).9.
D. W.
Kribs
, A.
Pasieka
, M.
Laforest
, C.
Ryan
, and M. P.
Silva
, Linear Multilinear Algebra
57
, 491
(2009
).10.
11.
J.
Holbrook
, “Real numerical range
,” Mathematics in Experimental Quantum Information Processing Workshop
, Institute for Quantum Computing, Waterloo, 10–14 August 2009
.12.
N. R.
Silva
, “Numerical ranges and minimal delity guarantees
,” Mathematics in Experimental Quantum Information Processing Workshop
, Institute for Quantum Computing, Waterloo, 10–14 August 2009
.13.
I.
Bengtsson
and K.
Życzkowski
, Geometry of Quantum States. An Introduction to Quantum Entanglement
(Cambridge University Press
, Cambridge
, 2006
).14.
N.
Johnston
and D.
Kribs
, J. Math. Phys.
51
, 082202
(2010
).15.
G.
Dirr
, U.
Helmke
, M.
Kleinsteuber
, and T.
Schulte-Herbrüggen
, Linear Multilinear Algebra
56
, 27
(2008
).16.
R.
Duan
, Y.
Feng
, and M.
Ying
, Phys. Rev. Lett.
100
, 020503
(2008
).17.
T.
Schulte-Herbrüggen
, G.
Dirr
, U.
Helmke
, and S. J.
Glaser
, Linear Multilinear Algebra
56
, 3
(2008
).18.
T.
Schulte-Herbrüggen
, S. J.
Glaser
, G.
Dirr
, and U.
Helmke
, Rev. Math. Phys.
22
, 597
(2010
).19.
B.
Terhal
and P.
Horodecki
, Phys. Rev. A
61
, 040301
(2000
).20.
D. W.
Kribs
, Linear Algebr. Appl.
400
, 147
(2005
).21.
H.
Langer
, A.
Markus
, A.
Matsaev
, and C.
Tretter
, Linear Algebr. Appl.
330
, 89
(2001
).22.
Z.
Puchała
, P.
Gawron
, J. A.
Miszczak
, Ł.
Skowronek
, M. -D.
Choi
, and K.
Życzkowski
(in press).23.
R. T.
Farouki
, H. P.
Moon
, and B.
Ravani
, Geom. Dedic.
85
, 283
(2001
).24.
R. T.
Farouki
and H.
Pottmann
, Reliab. Comput.
8
, 43
(2002
).25.
T.
Cubitt
, A.
Montanaro
, and A.
Winter
, J. Math. Phys.
49
, 022107
(2008
).26.
Ł.
Skowronek
and K.
Życzkowski
, J. Phys. A: Math. Theor.
42
, 325302
(2009
).27.
B.
Kraus
and J. I.
Cirac
, Phys. Rev. A
63
, 062309
(2001
).28.
H. -J.
Sommers
and K.
Życzkowski
, J. Phys. A
37
, 8457
(2004
).29.
Ł.
Skowronek
, E.
Størmer
, and K.
Życzkowski
, J. Math. Phys.
50
, 062106
(2009
).30.
A.
Jamiolkowski
, Rep. Math. Phys.
3
, 275
(1972
).31.
M. -D.
Choi
, Linear Algebr. Appl.
10
, 285
(1975
).32.
M.
Horodecki
, P.
Horodecki
, and R.
Horodecki
, Phys. Lett. A
223
, 1
(1996
).33.
J.
Sperling
and W.
Vogel
, Phys. Rev. A
79
, 022318
(2009
).34.
G.
Sarbicki
, J. Phys. A
41
, 375303
(2008
).35.
A.
Sanpera
, R.
Tarrach
, and G.
Vidal
, Phys. Rev. A
58
, 826
(1998
).36.
K. S.
Ranade
and M.
Ali
, Open Syst. Inf. Dyn.
14
, 371
(2007
).37.
D. P.
DiVincenzo
, P. W.
Shor
, J. A.
Smolin
, B. M.
Terhal
, and A. V.
Thapliyal
, Phys. Rev. A
61
, 062312
(2000
).38.
L.
Clarisse
, Phys. Rev. A
71
, 032332
(2005
).39.
Ł.
Pankowski
, M.
Piani
, M.
Horodecki
, and P.
Horodecki
, IEEE Trans. Inf. Theory
56
, 4085
(2010
).40.
D.
Petz
, Quantum Information Theory and Quantum Statistics
Theoretical and Mathematical Physics
(Springer-Verlag
, Berlin
, 2008
).41.
C.
King
and M. B.
Ruskai
, IEEE Trans. Inf. Theory
47
, 192
(2001
).42.
C.
King
, J. Math. Phys.
43
, 4641
(2002
).43.
M. B.
Hastings
, Nat. Phys.
5
, 255
(2009
).44.
J.
Walgate
, A. J.
Short
, L.
Hardy
, and V.
Vedral
, Phys. Rev. Lett.
85
, 4972
(2000
).45.
A.
Uhlmann
, Rep. Math. Phys.
9
, 273
(1976
).46.
R.
Jozsa
, J. Mod. Opt.
41
, 2315
(1994
).47.
D.
Markham
, J. A.
Miszczak
, Z.
Puchala
, and K.
Życzkowski
, Phys. Rev. A
77
, 042111
(2008
).48.
T. -C.
Wei
and P. M.
Goldbart
, Phys. Rev. A
68
, 042307
(2003
).49.
M.
Hayashi
, D.
Markham
, M.
Murao
, M.
Owari
, and S.
Virmani
, J. Math. Phys.
50
, 122104
(2009
).50.
T. -C.
Wei
and P. M.
Goldbart
, “Matrix permanent and quantum entanglement of permutation invariant states
,” e-print arXiv:0905.0012.51.
R.
Hübener
, M.
Kleinmann
, T. -C.
Wei
, and O.
Gühne
, Phys. Rev. A
80
, 032324
(2009
).52.
P.
Badziąg
, P.
Deuar
, M.
Horodecki
, P.
Horodecki
, and R.
Horodecki
, J. Mod. Opt.
49
, 1289
(2002
).53.
H.
Maassen
and B.
Kümmerer
. Institute of Mathematical Statistics
, Lecture Notes—Monograph Series
Vol. 48
(Institute of Mathematical Statistics
, Beachwood, OH, USA
, 2006
), p. 252
.54.
K.
Majgier
, H.
Maassen
, and K.
Życzkowski
, Quantum Inf. Process.
9
, 343
(2010
).55.
C. H.
Bennett
, D. P.
DiVincenzo
, J. A.
Smolin
, and W. K.
Wootters
, Phys. Rev. A
54
, 3824
(1996
).56.
E.
Knill
and R.
Laflamme
, Phys. Rev. A
55
, 900
(1997
).57.
M. -D.
Choi
, D. W.
Kribs
, and K.
Życzkowski
, Linear Algebr. Appl.
418
, 828
(2006
).58.
M.
Demianowicz
and P.
Horodecki
(unpublished).59.
F.
Mintert
and K.
Życzkowski
, Phys. Rev. A
69
, 022317
(2004
).60.
A.
Horn
and C. R.
Johnson
, Matrix Analysis
(Cambridge University Press
, Cambridge
, 1985
).© 2010 American Institute of Physics.
2010
American Institute of Physics
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