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.

\[\begin{split}\begin{equation} \begin{aligned} \textbf{Primal:} \quad & \\ \text{maximize:} \quad & \sum_{j=1}^k \langle P_j, \rho_j \rangle \\ \text{subject to:} \quad & P_1 + \cdots + P_k = \mathbb{I}_{\mathcal{A}} \otimes \mathbb{I}_{\mathcal{B}}, \\ & P_1, \ldots, P_k \in \text{PPT}(\mathcal{A} : \mathcal{B}). \end{aligned} \end{equation}\end{split}\]
\[\begin{split}\begin{equation} \begin{aligned} \textbf{Dual:} \quad & \\ \text{minimize:} \quad & \frac{1}{k} \text{Tr}(Y) \\ \text{subject to:} \quad & Y - \rho_j \geq \text{T}_{\mathcal{A}} (Q_j), \quad j = 1, \ldots, k, \\ & Y \in \text{Herm}(\mathcal{A} \otimes \mathcal{B}), \\ & Q_1, \ldots, Q_k \in \text{Pos}(\mathcal{A} \otimes \mathcal{B}). \end{aligned} \end{equation}\end{split}\]

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.

\[\begin{split}\begin{equation} \begin{aligned} \textbf{Primal:} \quad & \\ \text{maximize:} \quad & \sum_{j=1}^k \langle P_j, \rho_j \rangle \\ \text{subject to:} \quad & P_1 + \cdots + P_k = \mathbb{I}_{\mathcal{A}} \otimes \mathbb{I}_{\mathcal{B}}, \\ & P_1, \ldots, P_k \in \text{PPT}(\mathcal{A} : \mathcal{B}), \\ & \langle P_i, \rho_j \rangle = 0, \quad 1 \leq i, j \leq k, \quad i \not= j. \end{aligned} \end{equation}\end{split}\]
\[\begin{split}\begin{equation} \begin{aligned} \textbf{Dual:} \quad & \\ \text{minimize:} \quad & \frac{1}{k} \text{Tr}(Y) \\ \text{subject to:} \quad & Y - \rho_j \geq \text{T}_{\mathcal{A}} (Q_j), \quad j = 1, \ldots, k, \\ & Y \in \text{Herm}(\mathcal{A} \otimes \mathcal{B}), \\ & Q_1, \ldots, Q_k \in \text{Pos}(\mathcal{A} \otimes \mathcal{B}). \end{aligned} \end{equation}\end{split}\]

Distinguishing four Bell states

Consider the following Bell states:

\[\begin{split}\begin{equation} \begin{aligned} | \psi_0 \rangle = \frac{| 00 \rangle + | 11 \rangle}{\sqrt{2}}, \quad | \psi_1 \rangle = \frac{| 01 \rangle + | 10 \rangle}{\sqrt{2}}, \\ | \psi_2 \rangle = \frac{| 01 \rangle - | 10 \rangle}{\sqrt{2}}, \quad | \psi_3 \rangle = \frac{| 00 \rangle - | 11 \rangle}{\sqrt{2}}. \end{aligned} \end{equation}\end{split}\]

Assuming a uniform probability of selecting from any one of these states, that is, assuming we define an ensemble of Bell states defined as

\[\begin{equation} \mathbb{B} = \left\{ \left(| \psi_0 \rangle, \frac{1}{4} \right), \left(| \psi_1 \rangle, \frac{1}{4} \right), \left(| \psi_2 \rangle, \frac{1}{4} \right), \left(| \psi_3 \rangle, \frac{1}{4} \right) \right\} \end{equation}\]

it holds that

\[\begin{equation} \text{opt}_{\text{PPT}}(\mathbb{B}) = \frac{1}{2}. \end{equation}\]

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

\[\begin{equation} \text{opt}_{\text{LOCC}}(\mathbb{B}) = \frac{1}{2}. \end{equation}\]

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.

\[\begin{split}\rho_0 = |\psi_0\rangle |\psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \quad \rho_1 = |\psi_1 \rangle |\psi_3 \rangle \langle \psi_1 | \langle \psi_3 |, \\ \rho_2 = |\psi_3\rangle |\psi_1 \rangle \langle \psi_3 | \langle \psi_1 |, \quad \rho_3 = |\psi_1 \rangle |\psi_1 \rangle \langle \psi_1 | \langle \psi_1 |, \\\end{split}\]

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.

\[\begin{equation} | \tau_{\epsilon} \rangle = \sqrt{\frac{1+\epsilon}{2}} | 00 \rangle + \sqrt{\frac{1-\epsilon}{2}} | 11 \rangle, \end{equation}\]

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:

\[\begin{equation} \text{opt}_{\text{PPT}}(\eta) = \text{opt}_{\text{SEP}}(\eta) = \frac{1}{2} \left(1 + \sqrt{1 - \epsilon^2} \right) \end{equation}\]

where the ensemble is defined as

\[\begin{equation} \eta = \left\{ | \psi_0 \rangle \otimes | \tau_{\epsilon} \rangle, | \psi_1 \rangle \otimes | \tau_{\epsilon} \rangle, | \psi_2 \rangle \otimes | \tau_{\epsilon} \rangle, | \psi_3 \rangle \otimes | \tau_{\epsilon} \rangle \right\}. \end{equation}\]

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

\[\begin{equation} \sigma_0^{(n)} = \frac{\mathbb{I} \otimes \mathbb{I} + W_n}{n(n+1)} \quad \text{and} \quad \sigma_1^{(n)} = \frac{\mathbb{I} \otimes \mathbb{I} - W_n}{n(n-1)} \end{equation}\]

where

\[W_n = \sum_{i,j=0}^{n-1} | i \rangle \langle j | \otimes | j \rangle \langle i | \in \text{U}\left(\mathbb{C}^n \otimes \mathbb{C}^n\right)\]

is the swap operator defined for some dimension \(n \geq 2\).

It was shown in [Cosentino15] that

\[\begin{equation} \text{opt}_{\text{PPT}}(\eta) = \frac{1}{2} + \frac{1}{n+1}, \end{equation}\]

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.