Multipartite Entanglement: PPT Mixture

Julia Beliaeva

Plan

1. Definitions
2. Detecting genuine multipartite entanglement with PPT mixtures
3. PPT mixtures and entanglement witnesses
4. Optimization problem
5. Results

Definitions

Multipartite Entanglement of Pure States

[1]

Consider pure states in the composite Hilbert space \(\mathcal{H}_1\otimes \ldots \otimes \mathcal{H}_N\).

Fully separable: \[\ket{\phi^{FS}}_{1|2|\ldots|N} = \ket{\phi_1}\otimes\ket{\phi_2}\otimes\ldots\otimes\ket{\phi_N}.\]

\(m\)-separable across partition \(P_1,\ldots, P_m\), where \(1 < m < N\): \[\ket{\phi^{m-S}}_{P_1|\ldots|P_m} = \ket{\phi_1}_{P_1}\otimes\ket{\phi_2}_{P_2}\otimes\ldots\otimes\ket{\phi_m}_{P_m}.\]

Biseparable across a bipartition \(M|\overline{M}\): \[ \ket{\phi^{BS}}_{M|\overline{M}} = \ket{\alpha}_{M}\otimes\ket{\beta}_{\overline{M}}\qc \varnothing\ne M \subsetneq \{1\ldots N\}. \]

Genuine multipartite entangled (GME): neither fully separable nor \(m\)-separable.

Examples for 3 Qubits: Fully Separable State

\[\ket{0}_A\otimes\ket{0}_B\otimes\ket{0}_C\]

Examples for 3 Qubits: Biseparable States

\[\ket{\Phi^+}_{AB}\otimes\ket{0}_C\]

\[\ket{\Phi^+}_{AC}\otimes\ket{0}_B\]

\[\ket{0}_A\otimes\ket{\Phi^+}_{BC}\]

Examples for 3 Qubits: GME States

\[\begin{aligned} \ket{GHZ} &= \frac{1}{\sqrt{2}}\left(\ket{000} + \ket{111}\right)\\ \ket{W} &= \frac{1}{\sqrt{3}}\left(\ket{001} + \ket{010} + \ket{100}\right) \end{aligned}\]

Multipartite Entanglement of Mixed States

[1]

\[\begin{aligned} \rho^{FS} &= \sum_{i} p_i \ketbra{\phi_i^{FS}}\qc&\text{fully separable}\\ \rho^{m-S} &= \sum_{i} p_i \ketbra{\phi^{m-S}_i}\qc &m-\text{separable}\\ \rho^{BS} &= \sum_{i} p_i \ketbra{\phi^{BS}_i}\qc &\text{biseparable} \end{aligned} \]

States in the mixture can be separable w.r.t. different partitions.

Genuine multipartite entangled: neither fully separable nor \(m\)-separable.

Structure of Tripartite Entanglement

Detecting Genuine Multipartite Entanglement

Given a density matrix \(\rho\), can we determine if the state it describes is genuinely multipartite entangled?

Can We Just Apply PPT Criterion?

[1]

Let’s consider this biseparable state: \[\small \Theta = \frac{1}{3}\left(\ketbra{\Phi^+}_{AB}\otimes \ketbra{0}_C + \ketbra{\Phi^+}_{BC}\otimes \ketbra{0}_A + \ketbra{\Phi^+}_{AC}\otimes \ketbra{0}_B\right)\]

For any bipartition, this state is NPT: \[\Theta^{\top_A}, \Theta^{\top_B}, \Theta^{\top_C} \not\succeq 0\] The naive application of the PPT criterion does not work.

PPT Mixtures

\[\begin{aligned} \rho^{BS} &= \sum_i p_i \sigma_i^{BS}&\quad \sigma_i &= \alpha_{M_i}\otimes\beta_{\overline{M_i}}\\ &\cap &\\ \rho^{PPT Mix} &= \sum_i p_i \sigma_i^{PPT}&\quad \sigma_i^{\top_{M_i}} &\succeq 0 \end{aligned}\]

[2]

PPT Mixtures for Three Parties

Given a density matrix \(\rho\), can we determine if the state it describes is a PPT mixture?

Entanglement Witnesses

Entanglement Witnesses

For a given state \(\rho\) we are looking for hermitian \(W\) (witness)

\[\begin{gathered} \Tr(W\rho) < 0\\ \Tr(W\sigma) \ge 0\quad\text{if $\sigma$ is PPT mixture} \end{gathered}\]

[2]

Decomposable Witnesses

[2]

\(W\) is decomposable w.r.t. bipartition \(M|\overline{M}\) if

\[ \exists P, Q \succeq 0\qq{s.t.}W = P + Q^{\top_M}. \]

Take \(Q = 0\) then \(P\text{ is decomposable }\).

Take \(P = 0\) then \(Q^{\top_M}\text{ is decomposable }\).

The set of decomposable operators is convex.

\(W\) is fully decomposable if it is decomposable w.r.t. all bipartitions \(M|\overline{M}\).

Why Do We Like Decomposable Witnesses?

[4]

Suppose \(W\) is a fully decomposable witness and \(\rho\) is a PPT mixture. Let’s compute \(\Tr(W\rho)\): \[\Tr(W\rho) = \sum_i p_i \Tr(W \sigma_i^{PPT}).\]

Let’s pick some bipartition \(M|\overline{M}\) and corresponding component \(\sigma_i\): \[\begin{gathered} \sigma_i^{\top_M} \succeq 0,\\ W = P + Q^{\top_M}\qc\text{ for some }P,Q\succeq 0. \end{gathered}\]

\[ \Tr(W \sigma_i) = \Tr(P \sigma_i) + \Tr(Q^{\top_M} \sigma_i) = \Tr(P \sigma_i) + \Tr(Q \sigma_i^{\top_M}) \ge 0 \]

Fully decomposable witness can never be negative on a PPT mixture state!

Finding a Fully Decomposable Witnesses

[2]

Instead of \[\begin{gathered} \Tr(W\rho) < 0\\ \Tr(W\sigma) \ge 0\quad\text{if $\sigma$ is PPT mixture} \end{gathered}\]

We can do \[\begin{gathered} \Tr(W\rho) < 0\\ \text{For all bipartitions}\ M|\overline{M}:\\ \quad\exists\ P,Q\text{ s.t. }W = P + Q^{\top_M} \end{gathered}\]

If \(\rho\) is not a PPT mixture, can we always find such witness?

Finding a Fully Decomposable Witness

\[ \begin{gathered} \rho \not\in \left\{\text{PPT Mixtures}\right\}\text{ with witness }W\\ W \not\in S_M\text{, where }S_M=\{\text{decomposable Hermitian operators w.r.t. $M$}\} \end{gathered} \]

[4]

Finding a Fully Decomposable Witness

[4]

There exists Hermitian operator \(\sigma\) that separates \(S_M\) from \(W\): \[\begin{gathered} \Tr(\sigma W) < 0\\ \Tr(\sigma W_M)\ge 0\quad\forall\ W_M \in S_M \end{gathered}\]

\(\sigma\) is positive-semidefinite: \[\forall \ket{v}\quad \bra{v}\sigma\ket{v} = \Tr(\sigma \ketbra{v}) \ge 0\text{ since }\ketbra{v}\in S_M\]

\(\sigma^{\top_M}\) is positive semidefinite: \[\bra{v}\sigma^{\top_M}\ket{v} = \Tr(\sigma^{\top_M} \ketbra{v}) = \Tr(\sigma \ketbra{v}^{\top_M}) \ge 0\text{ since }\ketbra{v}^{\top_M}\in S_M\]

Then \(\sigma\) is a PPT state w.r.t. M and \(\Tr(\sigma W) < 0\), which contradicts to how \(W\) was chosen.

Summary

\[\rho \not\in \left\{\text{PPT Mixtures}\right\}\quad \Leftrightarrow\quad \exists\text{ fully decomposable witness }W\]

[2]

Optimization Problem

[2]

\[ \begin{array}{rl} \min_W & \operatorname{Tr}(W\rho)\\ \text{s.t.} & \operatorname{Tr}(W) = 1\\ & \text{For all subsystem subsets $M$:}\\ &\quad W = P_M + Q_M^{\top_M}\\ &\quad P_M, Q_M \succeq 0 \end{array} \]

We only need to check half the subsystem subsets: \[W = P_M + Q_M^{\top_M} = P_M + \left(Q_M^{\top}\right)^{\top_{\overline{M}}}\]

Implementation with CVXPY

def find_witness(rho: np.ndarray, dims: List[int]) -> Optional[np.ndarray]:
    w = cp.Variable(rho.shape, "w", complex=True, hermitian=True)
    constraints: List[Constraint] = [cp.trace(w) == 1]

    n_subsystems = len(dims)
    subsets = [c for cc in [combinations(range(0, n_subsystems), s)
                            for s in range(1, n_subsystems // 2 + 1)]
               for c in cc]
    for subset in subsets:
        p = cp.Variable(rho.shape, complex=True, hermitian=True)
        q = cp.Variable(rho.shape, complex=True, hermitian=True)
        q_pt = q
        for idx in subset:
            q_pt = partial_transpose(q_pt, tuple(dims), idx)
        constraints.extend([p >> 0, q >> 0, w == p + q_pt])

    problem = cp.Problem(cp.Minimize(cp.real(cp.trace(w @ rho))), 
                         constraints)
    result = problem.solve()
    
    if result < 0 and not np.isclose(result, 0):
        return w.value
    return None

Known Witnesses

[3]

Witness for \(\ketbra{\psi}\):

\[W_{\psi} = \alpha \mathbb{I} - \ketbra{\psi}\]

\[\Tr(W_{\psi} \ketbra{\phi}) = \Tr((\alpha\mathbb{I} - \ketbra{\psi}) \ketbra{\phi}) = \alpha - \abs{\braket{\phi}{\psi}}^2\]

Such witness detects state with squared overlap \(\abs{\braket{\phi}{\psi}}^2 > \alpha\).

We can select \(\alpha\) as maximal squared overlap between biseparable states and \(\ketbra{\psi}\):

\[\begin{aligned} W_{W} &= \frac{2}{3}\mathbb{I} - \ketbra{W}\\ W_{GHZ} &= \frac{1}{2}\mathbb{I} - \ketbra{GHZ} \end{aligned}\]

Comparison with Known Witnesses

• For \(GHZ\) state, the implemented procedure returns the witness very close to \(W_{GHZ}\).
• For \(W\) state the witness is different from \(W_{W}\).

Summary

1. Multipartite states have more separability classes than bipartite states.
2. PPT mixtures give a convex outer approximation to the biseparable set.
3. Non-PPT-mixture states can be detected efficiently using fully decomposable witnesses.

Thank you for listening!

References

1. O. Gühne and G. Toth, “Entanglement detection,” Physics Reports, vol. 474, no. 1–6, pp. 1–75, Apr. 2009, doi: 10.1016/j.physrep.2009.02.004.
2. B. Jungnitsch, T. Moroder, and O. Gühne, “Taming multiparticle entanglement,” Phys. Rev. Lett., vol. 106, no. 19, p. 190502, May 2011, doi: 10.1103/PhysRevLett.106.190502.
3. A. Acin, D. Bruss, M. Lewenstein, and A. Sanpera, “Classification of mixed three-qubit states,” Phys. Rev. Lett., vol. 87, no. 4, p. 040401, July 2001, doi: 10.1103/PhysRevLett.87.040401.
4. M. Lewenstein, “Optimization of entanglement witnesses,” Phys. Rev. A, vol. 62, no. 5, 2000, doi: 10.1103/PhysRevA.62.052310.
5. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Physics Letters A, vol. 223, no. 1–2, pp. 1–8, Nov. 1996, doi: 10.1016/S0375-9601(96)00706-2.

Restricting the Set of Observables

For a given set of observables \(O = \left\{O_1,\ldots,O_k\right\}\), we can look for a witness that can be evaluated with \(O\): \[ \begin{array}{rl} \min_W & \operatorname{Tr}(W\rho)\\ \text{s.t.} & \operatorname{Tr}(W) = 1\\ & W = \sum_i \lambda_i O_i\\ & \text{For all subsystem subsets $M$:}\\ &\quad W = P_M + Q_M^{\top_M}\\ &\quad P_M, Q_M \succeq 0 \end{array} \]

[2]

Negativity

[2]

\[ \begin{array}{rl} N(\rho) &=- \min \operatorname{Tr}(W\rho)\\ \text{s.t.} & \text{For all subsystem subsets $M$:}\\ &\quad W = P_M + Q_M^{\top_M}\\ &\quad \mathbb{I} \ge P_M, Q_M \succeq 0 \end{array} \]

\(N(\rho)\) is an entanglement monotone:

• \(N\left(\rho^{BS}\right) = 0\) for all biseparable \(\rho^{BS}\).
• \(N\left(\Lambda_{LOCC}(\rho)\right) < N\left(\rho\right)\).
• \(N\left(U_{loc}\rho U_{loc}^{\dagger}\right) = N(\rho)\) where \(U_{loc}\) is a local basis change.
• \(N\left(\sum_i p_i \rho_i \right) \le \sum_i p_i N\left(\rho_i \right)\) for convex combinations \(\sum_i p_i \rho_i\).

Negativity

Proving Witness Existence with Hahn-Banach Separation Theorem

Hahn-Banach Separation Theorem
\(W_1\) and \(W_2\) are convex closed sets in real normed vector space. If one of them is compact, then there exists a linear continous functional \(f\) and \(\alpha \in \mathbb{R}\) s.t. for all pairs \(w_1 \in W_1\), \(w_2 \in W_2\) we have \(f(w_1) < \alpha \le f(w_2)\).

• A space of Hermitian operator can be viewed as a real vector space with inner product \(\langle A, B\rangle =\Tr(AB)\).
• A linear functional \(f\) can be represented as \(f(w_1) = \Tr(A w_1)\) for some Hermitian operator \(A\).
• If \(W_1 = \{\rho\}\) and \(W_2 = \{\text{PPT Mixtures}\}\) the conditions of the theorem hold.
• We can select a witness as \(W = A - \alpha \mathbb{I}\).

[5]