tutorials.ppt.rst

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.

$$\begin{aligned} \begin{equation} \begin{aligned} \textbf{Primal:} \quad & \\\ \text{maximize:} \quad & \sum_{j=1}^k \langle P_j, \rho_j \rangle \\\ \text{subject to:} \quad & P_1 + \cdots + P_k = \mathbb{I}_{\mathcal{A}} \otimes \mathbb{I}_{\mathcal{B}}, \\\ & P_1, \ldots, P_k \in \text{PPT}(\mathcal{A} : \mathcal{B}). \end{aligned} \end{equation} \end{aligned}$$

$$\begin{aligned} \begin{equation} \begin{aligned} \textbf{Dual:} \quad & \\\ \text{minimize:} \quad & \frac{1}{k} \text{Tr}(Y) \\\ \text{subject to:} \quad & Y - \rho_j \geq \text{T}_{\mathcal{A}} (Q_j), \quad j = 1, \ldots, k, \\\ & Y \in \text{Herm}(\mathcal{A} \otimes \mathcal{B}), \\\ & Q_1, \ldots, Q_k \in \text{Pos}(\mathcal{A} \otimes \mathcal{B}). \end{aligned} \end{equation} \end{aligned}$$

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.

$$\begin{aligned} \begin{equation} \begin{aligned} \textbf{Primal:} \quad & \\\ \text{maximize:} \quad & \sum_{j=1}^k \langle P_j, \rho_j \rangle \\\ \text{subject to:} \quad & P_1 + \cdots + P_k = \mathbb{I}_{\mathcal{A}} \otimes \mathbb{I}_{\mathcal{B}}, \\\ & P_1, \ldots, P_k \in \text{PPT}(\mathcal{A} : \mathcal{B}), \\\ & \langle P_i, \rho_j \rangle = 0, \quad 1 \leq i, j \leq k, \quad i \not= j. \end{aligned} \end{equation} \end{aligned}$$

$$\begin{aligned} \begin{equation} \begin{aligned} \textbf{Dual:} \quad & \\\ \text{minimize:} \quad & \frac{1}{k} \text{Tr}(Y) \\\ \text{subject to:} \quad & Y - \rho_j \geq \text{T}_{\mathcal{A}} (Q_j), \quad j = 1, \ldots, k, \\\ & Y \in \text{Herm}(\mathcal{A} \otimes \mathcal{B}), \\\ & Q_1, \ldots, Q_k \in \text{Pos}(\mathcal{A} \otimes \mathcal{B}). \end{aligned} \end{equation} \end{aligned}$$

Distinguishing four Bell states

Consider the following Bell states:

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

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.

../examples/opt_dist/ppt/min_error/four_bell_states.py

Indeed, a stronger statement is known to hold for 𝔹, that is

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

Recall that for any ensemble η, it holds that opt_LOCC(η) < opt_PPT(η).

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.

$$\begin{aligned} \rho_0 = |\psi_0\rangle |\psi_0 \rangle \langle \psi_0 | \langle \psi_0 |, \quad \rho_1 = |\psi_1 \rangle |\psi_3 \rangle \langle \psi_1 | \langle \psi_3 |, \\\ \rho_2 = |\psi_3\rangle |\psi_1 \rangle \langle \psi_3 | \langle \psi_1 |, \quad \rho_3 = |\psi_1 \rangle |\psi_1 \rangle \langle \psi_1 | \langle \psi_1 |, \\\ \end{aligned}$$

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.

../examples/opt_dist/ppt/min_error/indstinguishable_mes.py

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.

../examples/opt_dist/ppt/unambiguous/indstinguishable_mes.py

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 ϵ ∈ [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

η = {|ψ₀⟩⊗|τ_ϵ⟩,|ψ₁⟩⊗|τ_ϵ⟩,|ψ₂⟩⊗|τ_ϵ⟩,|ψ₃⟩⊗|τ_ϵ⟩}.

Using qustop, we may encode this scenario as follows.

../examples/opt_dist/ppt/min_error/entanglement_cost_four_bell_states.py

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 ≥ 2.

It was shown in [Cosentino15] that

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

where η = {σ₀,σ₁}. Using qustop, we may encode this scenario as follows.

../examples/opt_dist/ppt/min_error/werner_hiding_pair.py

References

BCJRWY15

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.

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

Cosentino13

Cosentino, Alessandro, "Positive-partial-transpose-indistinguishable states via semidefinite programming", Physical Review A 87.1 (2013): 012321. https://arxiv.org/abs/1205.1031

Cosentino15

Cosentino, Alessandro "Quantum state local distinguishability via convex optimization". University of Waterloo, Thesis https://uwspace.uwaterloo.ca/handle/10012/9572

DLT02

DiVincenzo, David P., Debbie W. Leung, and Barbara M. Terhal. "Quantum data hiding." IEEE Transactions on Information Theory 48.3 (2002): 580-598.

TDL01

Terhal, Barbara M., David P. DiVincenzo, and Debbie W. Leung. "Hiding bits in Bell states." Physical review letters 86.25 (2001): 5807.

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