Class
ME, PRE,
Meas. Scheme Values
I
~α {0.06, −0.47 − 0.47i, −0.04 − 0.09i, −0.48, 0.41 − 0.15i, −0.21}
~γ
1
{−0.54 − 1.94i, −2.71 + 2.07i, 2.93 + 1.01i, 0.82 + 0.05i,
0.83 − 2.76i, −0.6 + 1.42i, 0.44 − 1.71i, −1.47 − 0.42i}
~γ
2
{−2.73 − 2.68i, −1.71 − 1.26i, −2.85 + 1.27i, 1.89 + 2.53i,
1.87 + 0.33i, −1.66 + 1.21i, −2.18 + 0.72i, −0.11 + 0.53i}
~γ
3
{−1.18 − 2.69i, 2.01 + 1.03i, 2.73 − 2.71i, −1.11 + 1.35i,
−0.11 − 0.67i, −1.41 − 2.12i, −0.64 + 0.15i, 0.82 − 0.88i}
II
~α {−1.24, −0.79 + 1.17i, 0.06 − 0.09i, 1.39, 0.09 − 1.15 − 0.09i, 1.1}
~γ
1
{2.06 − 0.95i, −2.55 − 0.1i, 0.81 − 0.11i, −1.32 − 2.35i,
−1.33 − 2.9i, 2.1 + 2.11i, 1.02 − 2.99i, −2.72 − 2.74i}
III
~α {0, −0.81i, −2.2i, 0, −0.39i, 0}
~γ
1
{1.22, 0.48, 2.67, −2.1, −1.04, 2.3, −0.01, 0.33}
~γ
2
{2.75, 1.93, −1.93, 2.35, −0.25, −2.87, 2.46, 0.93}
~γ
3
{−2.95, −0.24, 0.08, −1.15, −2.67, −2.3, 2.07, 0.94}
{|φ
k
i}
k
{{0.371, −0.884, 0.283}, {0.046, 0.999, −0.011},
{0.338, −0.795, −0.504}, {−0.651, 0.756, −0.069}}
κ
{{0, 21.777, 16.12, 9.521}, {4.809, 0, 17.162, 1.057},
{5.513, 1.883, 0, 5.155}, {13.97, 1.599, 12.946, 0}}
{
~
β
k
}
k
{{1.143, 1.52, −2.256}, {−2.743, −3.073, −2.626},
{−0.509, −3.317, −2.43}, {0.71, −0.005, 4.22}}
S
1
{{−0.925, 0.093, −0.369}, {0.355, 0.56, −0.748}, {−0.137, 0.823, 0.552}}
S
2
{{0.52, 0.814, 0.257}, {0.41, 0.026, −0.912}, {−0.749, 0.58, −0.32}}
S
3
{{−0.375, 0.784, 0.495}, {−0.348, −0.614, 0.708}, {0.859, 0.093, 0.503}}
S
4
{{0.722, −0.483, 0.495}, {−0.135, −0.8, −0.584}, {−0.678, −0.355, 0.643}}
IV
~α {0, 0, −0.04, 0, 0, 0}
~γ
1
{0, −0.38, 0, 0, 0, −0.38, 0, 0}
~γ
2
{0, 0, 0, 0, 0, 0, 0.43, 0}
{|φ
k
i}
k
{{0.676, −0.372, −0.637}, {−0.423, −0.703, −0.571},
{−0.423, 0.703, −0.571}, {0.676, 0.372, −0.637}}
κ
{{0, 0.002, 0.066, 0.062}, {0.067, 0, 0.005, 0.041},
{0.041, 0.005, 0, 0.041}, {0.062, 0.066, 0.002, 0}}
{
~
β
k
}
k
{{0.004, 0.189, 0.128}, {0.122, 0.199, 0.127},
{−0.199, 0.127, 0.122}, {0.189, 0.004, −0.128}}
S
1
{{0.76, 0.519}, {−0.298, 0.813}, {0.577, −0.264}}
S
2
{{0.313, −0.216}, {0.424, −0.839}, {0.85, 0.499}}
S
3
{{0.424, 0.839}, {−0.85, 0.499}, {−0.313, −0.216}}
S
4
{{0.298, 0.813}, {−0.76, 0.519}, {0.577, 0.264}}
Table 4: An example system for each class of Table 3 is specified. The ME is determined by
~α, ~γ
l
,
while PREs (if they exist) are given in terms of the ensemble states,
{|φ
k
i}
k
=
{|φ
1
i , |φ
2
i , |φ
3
i , |φ
4
i}
and the transition rates
κ ≡ κ
jk
(with the diagonal elements irrelevant and set to zero). We represent
the re3it as
{x, y, z}
with
|φi
=
x |1i
+
y |2i
+
z |3i
. For each PRE, an explicit adaptive measurement
scheme that can realize it is provided via the parameters
~
β
k
,
S
k
. The ME parameters are exact but the
PRE and measurement scheme are necessarily approximations that can be refined to arbitrary accuracy
if desired. Both class III and IV possess the re3it invariant subspace symmetry as the Lindbladians are
real-valued. This leads to a real-valued PRE and measurement scheme. Class IV additionally possesses
the Wigner unitary symmetry
ˆ
U
=
−
2
|2i h2|
, which is reflected in the symmetry of the PRE, as per
Eqs. (8)–(9), and the measurement scheme.
Accepted in Quantum 2019-09-27, click title to verify. Published under CC-BY 4.0. 35