bibkey: simnacher2021separable authors: Timo Simnacher; Jakub Czartowski; Konrad Szymański; Karol Życzkowski year: 2021 title: “Confident entanglement detection via the separable numerical range” doi: 10.1103/PhysRevA.104.042420 url: https://arxiv.org/abs/2107.04365v1 claim: “For Hermitian A_1, …, A_k the restricted numerical range L_X(A_1, …, A_k) is the set of vectors (Tr rho A_1, …, Tr rho A_k) over states rho in X; L is the numerical range over all states and L_Sep the separable numerical range. The minimal volume ratio mu_{n,d,k} is the minimum over A_1, …, A_k of vol L_Sep / vol L (Definition 2). Proposition 7 proves mu_{2,1} >= sqrt(2) - 1, and Conjecture 8 states that for a single measurement on a two-qubit system the minimal volume ratio is mu_{2,1} = 1/2, which the projector onto (|00> + |11>)/sqrt(2) attains.” strata_touched:

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Confident entanglement detection via the separable numerical range

Timo Simnacher, Jakub Czartowski, Konrad Szymański, Karol Życzkowski, Phys. Rev. A 104, 042420 (2021), arXiv:2107.04365v1 (quant-ph). Quotations are from the arXiv v1 source, with the source’s local macros \trace, \dR, \Sep, \vol, \bm, \ket and \bra written in standard TeX.

Definition 1:

Let $A_1,\dots ,A_k$ be Hermitian operators. Then, the set

followed by the display $L_X(A_1,\dots ,A_k)=\{(\mathrm{Tr}\rho A_1,\dots ,\mathrm{Tr}\rho A_k)\in {\mathbb{R}}^k|\rho \in X\}$, and

is called the \textit{joint (restricted) numerical range} of $A_1,\dots ,A_k$ where the set $X$ restricts the accessible states. If $X$ is the set of all quantum states, $L$ is simply the (joint) numerical range of $A_1,\dots ,A_k$; if it is the set of all separable quantum states, $L_{\mathrm{Sep}}$ is called the separable (joint) numerical range of $A_1,\dots ,A_k$.

The next sentence:

Furthermore, the Euclidean volume of $L_X(A_1,\dots ,A_k)$ is denoted by $\mathrm{vol}{L}_X$.

Definition 2:

For $k$ independent measurements on a quantum system consisting of $n$ particles and local dimensions $\mathbf{d}=(d_1,\dots ,d_n)$, we denote the minimal volume ratio of the separable numerical range compared to the standard numerical range as

followed by the display ${\mu }_{n,\mathbf{d},k}={\min }_{A_1,\dots ,A_k}\frac{\mathrm{vol}{L}_{\mathrm{Sep}}(A_1,\dots ,A_k)}{\mathrm{vol}L(A_1,\dots ,A_k)}$, and

If $d_1=\cdots =d_n=d$, we just write $d$ instead of $\mathbf{d}$. Further, if $n=2$, we omit the corresponding subscript.

Proposition 7:

It holds that ${\mu }_{2,1}\ge \sqrt{2}-1\approx 0.41$. Moreover, this is the best bound achievable when only absolutely separable states are considered.

Conjecture 8:

For a single measurement on a two-qubit system, the minimal volume ratio is ${\mu }_{2,1}=\frac{1}{2}$.

The sentences after it:

This value is for example reached by $A=|{\varphi }^{+}⟩⟨{\varphi }^{+}|$ with eigenvalues 0 and 1, being the projector onto the maximally entangled state $|{\varphi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$.

and

Thus, it follows that ${\mu }_{2,1}\le \frac{1}{2}$.

The encoding reads a two-qubit state as a positive semidefinite matrix of trace $1$ indexed by Fin 2 × Fin 2, a separable state as a finite sum of Kronecker products of positive semidefinite $2\times 2$ factors with trace $1$, the Euclidean volume of a subset of $\mathbb{R}$ as its Lebesgue measure, and the minimum as the least element of the set of ratios over Hermitian $A$ with $\mathrm{vol}L(A)\ne 0$ (for scalar $A$ both volumes vanish).

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.104.042420 (Phys. Rev. A 104, 042420, published 2021-10-25).
• URL: https://arxiv.org/abs/2107.04365v1 (the only version, 2021-07-09; source SNRJNRratio.tex, md5 9de8e3a2899bc229ba5e7fa6140ca52f): Definition 1 (l. 153–162), the volume sentence (l. 163), Definition 2 (l. 199–207), Proposition 7 (l. 308–310), Conjecture 8 (l. 312–315), the Bell-projector bound (l. 316–318) and the conclusion (l. 521). The theorem environments share one counter, which numbers the conjecture 8.