Distinguishing quantum states via PPT measurements
In this section we will be investigation how to make use of the
qustop
package to optimally distinguish quantum states via PPT
measurements.
Minimum-error
In [Cosentino13], an semidefinite program formulation whose optimal value corresponds to the optimal probability of distinguishing a quantum state from an ensemble using PPT measurements with minimum error was provided. The primal and dual problems of this SDP are defined as follows.
Unambiguous
In [Cosentino13], an semidefinite program formulation whose optimal value corresponds to the optimal probability of distinguishing a quantum state from an ensemble using PPT measurements unambiguously was provided. The primal and dual problems of this SDP are defined as follows.
Distinguishing four Bell states
Consider the following Bell states:
Assuming a uniform probability of selecting from any one of these states, that is, assuming we define an ensemble of Bell states defined as
it holds that
We can observe this using qustop
as follows.
1from toqito.states import bell
2
3from qustop import Ensemble, OptDist, State
4
5# Construct the corresponding density matrices of the Bell states.
6dims = [2, 2]
7ensemble = Ensemble(
8 [
9 State(bell(0), dims),
10 State(bell(1), dims),
11 State(bell(2), dims),
12 State(bell(3), dims),
13 ],
14 [1 / 4, 1 / 4, 1 / 4, 1 / 4],
15)
16res = OptDist(ensemble, "ppt", "min-error")
17res.solve()
18
19# 0.5000000000530641
20print(res.value)
Indeed, a stronger statement is known to hold for \(\mathbb{B}\), that is
Recall that for any ensemble \(\eta\), it holds that \(\text{opt}_{\text{LOCC}}(\eta) < \text{opt}_{\text{PPT}}(\eta)\).
Four indistinguishable orthogonal maximally entangled states
In [YDY12] the following ensemble of states was shown not to be perfectly distinguishable by PPT measurements, and therefore also indistinguishable via LOCC measurements.
While it was known that perfect distinguishability could not be achieved, the actual value and bound of optimal distinguishability was not known. It was shown in [Cosentino13] and later extended in [CR13] that the optimal probability of distinguishing the above ensemble via a PPT measurement should yield an optimal probability of 7/8.
1import numpy as np
2from toqito.states import bell
3
4from qustop import Ensemble, OptDist, State
5
6# Define the maximally entangled states from arXiv1107.3224
7dims = [2, 2, 2, 2]
8ensemble = Ensemble(
9 [
10 State(np.kron(bell(0), bell(0)) * np.kron(bell(0), bell(0)).conj().T),
11 State(np.kron(bell(2), bell(1)) * np.kron(bell(2), bell(1)).conj().T),
12 State(np.kron(bell(3), bell(1)) * np.kron(bell(3), bell(1)).conj().T),
13 State(np.kron(bell(1), bell(1)) * np.kron(bell(1), bell(1)).conj().T),
14 ]
15)
16
17# The min-error probability of distinguishing via PPT
18# is equal to 7/8.
19res = OptDist(ensemble, "ppt", "min-error")
20res.solve()
21
22# 0.87500000060847
23print(res.value)
In was also shown in [Cosentino13] that the optimal probability of distinguishing this ensemble unambiguously when making use of PPT measurements was equal to 3/4.
1import numpy as np
2from toqito.states import bell
3
4from qustop import Ensemble, OptDist, State
5
6# Define the maximally entangled states from arXiv1107.3224
7dims = [2, 2, 2, 2]
8ensemble = Ensemble(
9 [
10 State(
11 np.kron(bell(0), bell(0)) * np.kron(bell(0), bell(0)).conj().T,
12 dims,
13 ),
14 State(
15 np.kron(bell(2), bell(1)) * np.kron(bell(2), bell(1)).conj().T,
16 dims,
17 ),
18 State(
19 np.kron(bell(3), bell(1)) * np.kron(bell(3), bell(1)).conj().T,
20 dims,
21 ),
22 State(
23 np.kron(bell(1), bell(1)) * np.kron(bell(1), bell(1)).conj().T,
24 dims,
25 ),
26 ]
27)
28
29# The unambiguous probability of distinguishing via PPT
30# is equal to 3/4.
31res = OptDist(ensemble, "ppt", "unambiguous")
32res.solve()
33
34# 0.7499999975754753
35print(res.value)
Entanglement cost of distinguishing Bell states
One may ask whether the ability to distinguish a state can be improved by making use of an auxiliary resource state.
for some \(\epsilon \in [0,1]\).
Distinguishing four Bell states
It was shown in [BCJRWY15] that the probability of distinguishing four Bell states with a resource state via PPT measurements is given by the closed-form expression:
where the ensemble is defined as
Using qustop
, we may encode this scenario as follows.
1import numpy as np
2from toqito.states import basis, bell
3
4from qustop import Ensemble, OptDist, State
5
6e_0, e_1 = basis(2, 0), basis(2, 1)
7
8eps = 0.5
9tau = np.sqrt((1 + eps) / 2) * np.kron(e_0, e_0) + np.sqrt(
10 (1 - eps) / 2
11) * np.kron(e_1, e_1)
12
13dims = [2, 2, 2, 2]
14ensemble = Ensemble(
15 [
16 State(np.kron(bell(0), tau), dims),
17 State(np.kron(bell(1), tau), dims),
18 State(np.kron(bell(2), tau), dims),
19 State(np.kron(bell(3), tau), dims),
20 ],
21 [1 / 4, 1 / 4, 1 / 4, 1 / 4],
22)
23
24ppt_res = OptDist(ensemble, "ppt", "min-error")
25ppt_res.solve()
26
27sep_res = OptDist(ensemble, "sep", "min-error")
28sep_res.solve()
29
30eq = 1 / 2 * (1 + np.sqrt(1 - eps ** 2))
31
32# 0.9330127018922193
33print(eq)
34# 0.9330127016540999
35print(ppt_res.value)
36# 0.9330127016540999
37print(sep_res.value)
Note that [BCJRWY15] also proved the same closed-form expression for when Alice and Bob make use of separable measurements. More on that in the tutorial on distinguishing via separable measurements.
Werner hiding pairs
In [TDL01] and [DLT02], a quantum data hiding protocol that encodes a classical bit in a Werner hiding pair was provided.
A Werner hiding pair is defined by
where
is the swap operator defined for some dimension \(n \geq 2\).
It was shown in [Cosentino15] that
where \(\eta = \left\{\sigma_0, \sigma_1\right\}\). Using qustop
,
we may encode this scenario as follows.
1import numpy as np
2from toqito.perms import swap_operator
3
4from qustop import Ensemble, OptDist, State
5
6dim = 2
7sigma_0 = (
8 np.kron(np.identity(dim), np.identity(dim)) + swap_operator(dim)
9) / (dim * (dim + 1))
10sigma_1 = (
11 np.kron(np.identity(dim), np.identity(dim)) - swap_operator(dim)
12) / (dim * (dim - 1))
13
14ensemble = Ensemble([State(sigma_0, [dim, dim]), State(sigma_1, [dim, dim])])
15
16expected_val = 1 / 2 + 1 / (dim + 1)
17
18res = OptDist(ensemble, "ppt", "min-error")
19res.solve()
20
21# opt_ppt \approx 0.8333333333668715
22print(res.value)
23# Closed-form expression is : 1/2 + 1/(dim+1) = 0.8333333333333333
24print(expected_val)
References
- TDL01
Terhal, Barbara M., David P. DiVincenzo, and Debbie W. Leung. “Hiding bits in Bell states.” Physical review letters 86.25 (2001): 5807.
- DLT02
DiVincenzo, David P., Debbie W. Leung, and Barbara M. Terhal. “Quantum data hiding.” IEEE Transactions on Information Theory 48.3 (2002): 580-598.
- Cosentino15
Cosentino, Alessandro “Quantum state local distinguishability via convex optimization”. University of Waterloo, Thesis https://uwspace.uwaterloo.ca/handle/10012/9572
- Cosentino13(1,2,3,4)
Cosentino, Alessandro, “Positive-partial-transpose-indistinguishable states via semidefinite programming”, Physical Review A 87.1 (2013): 012321. https://arxiv.org/abs/1205.1031
- CR13
Cosentino, Alessandro and Russo, Vincent “Small sets of locally indistinguishable orthogonal maximally entangled states”, Quantum Information & Computation, Volume 14, https://arxiv.org/abs/1307.3232
- YDY12
Yu, Nengkun, Runyao Duan, and Mingsheng Ying. “Four locally indistinguishable ququad-ququad orthogonal maximally entangled states.” Physical review letters 109.2 (2012): 020506. https://arxiv.org/abs/1107.3224
- BCJRWY15(1,2)
Bandyopadhyay, Somshubhro, Cosentino, Alessandro, Johnston, Nathaniel, Russo, Vincent, Watrous, John, & Yu, Nengkun. “Limitations on separable measurements by convex optimization”. IEEE Transactions on Information Theory 61.6 (2015): 3593-3604.