Degree of twist

Entanglement measures in quantum mechanics quantify how much entanglement is contained in a quantum state . Formally, an entanglement measure is any non-negative function of a state that cannot increase under local operations and classic communication (LOCC) (so-called monotony) and is zero for separable (non-entangled) states. In the general case of mixed states, an entanglement measure is a function of the density matrix of the state.

Classification

Entanglement is a rich and complex property. There are therefore different degrees of twisting, some of which characterize different types of twisting. There are also different ways of defining twisting dimensions. On the one hand, there are operational entanglement measures, such as the distillable entanglement or the entanglement costs. Furthermore, there are abstractly defined dimensions such as those based on convex roof constructions (e.g. concurrence and formation entanglement) or based on the distance to separable states, e.g. B. the relative entropy entropy or entanglement robustness.

Entanglement between two systems

The entanglement between two systems (bipartite entanglement) is the fundamental, best-studied case. Maximum entanglement is only possible between two systems (monogamy of entanglement). So far, a large number of different degrees of twisting between two systems is known. These are generally not ordered linearly, that is, there are entanglement measures VM1 and VM2 and states ρ and σ such that

${\displaystyle \operatorname {VM1} (\rho )<\operatorname {VM2} (\rho )}$ and ${\displaystyle \operatorname {VM1} (\sigma )>\operatorname {VM2} (\sigma )}$

Entropy of entanglement

For a pure state of a system of two subsystems which is Verschränkungsentropie (ger .: entropy of entanglement ) the standard measure of entanglement. It is given by the Von Neumann entropy , applied to the reduced mixture of states of one subsystem after the other has been eliminated: ${\displaystyle |\psi \rangle }$ ${\displaystyle V_{E}}$ ${\displaystyle S(\rho )=-\operatorname {Tr} \left(\rho \ln \,\rho \right)}$

${\displaystyle V_{E}(|\psi \rangle \langle \psi |)=S(\rho _{A})=S(\rho _{B})}$,

where the reduced state is in the first (or in the second) system and denotes the partial track over the subsystem . In the case of a pure overall condition, this measure corresponds in particular to the measures "entanglement costs" and "distillable entanglement" defined below. ${\displaystyle \rho _{A}=\mathrm {tr} _{B}(|\psi \rangle \langle \psi |)}$ ${\displaystyle \rho _{B}=\mathrm {tr} _{A}(|\psi \rangle \langle \psi |)}$ ${\displaystyle \mathrm {tr} _{x}}$ ${\displaystyle x=A,B}$

The measure can be generalized from pure to mixed conditions via a convex roof construction . Then the “formation entanglement” defined below is obtained.

Concurrence

The concurrence is zero for all separable states and one for a maximally entangled 2-qubit state. For pure 2-qubit states , the concentration is analytically defined as ${\displaystyle |\psi \rangle =\alpha |{\uparrow }{\uparrow }\rangle +\beta |{\uparrow }{\downarrow }\rangle +\gamma |{\downarrow }{\uparrow }\rangle +\delta |{\downarrow }{\downarrow }\rangle }$

${\displaystyle {\mathcal {C}}=2|\alpha \delta -\beta \gamma |}$

The definition exists for more general mixed 2-qubit states ${\displaystyle \rho }$

${\displaystyle {\mathcal {C}}=\max({\sqrt {\lambda _{1}}}-{\sqrt {\lambda _{2}}}-{\sqrt {\lambda _{3}}}-{\sqrt {\lambda _{4}}},0)}$,

where in descending order are the eigenvalues ​​of the matrix with the Pauli-y matrix . ${\displaystyle \lambda _{i}}$ ${\displaystyle \varrho =\rho \cdot (\sigma ^{y}\otimes \sigma ^{y})\cdot \rho ^{*}\cdot (\sigma ^{y}\otimes \sigma ^{y})}$ ${\displaystyle \sigma ^{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}}$

Formation entanglement

The formation entanglement measures how much entropy entropy is necessary on average to create the state by mixing pure states. This dimension is defined as the convex roof of entropy of entanglement

${\displaystyle E_{F}(\rho )=\mathrm {inf} \left\{\sum _{i}p_{i}V_{E}(|\psi _{i}\rangle \langle \psi _{i}|):\rho =\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|,p_{i}\geq 0\right\}}$

and can be expressed for bipartite systems by the Concurrence : ${\displaystyle {\mathcal {C}}}$

${\displaystyle E_{F}=h\left({\frac {1+{\sqrt {\mathcal {C}}}}{2}}\right)}$

with the binary entropy function . ${\displaystyle h(x):=-x\log _{2}(x)-(1-x)\log _{2}(1-x)}$

Entanglement costs

The entanglement costs for a state describe the ratio in the limit of large numbers, how many maximally entangled qubit pairs would be required to produce copies of the state . ${\displaystyle \rho }$ ${\displaystyle m/n}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle \rho }$

${\displaystyle E_{c}(\rho )=\inf\{E\mid \forall \epsilon >0,\delta >0,\exists m,n,\Lambda ,|E-{\frac {m}{n}}|\leq \delta {\text{ und }}D(\Lambda (P_{+}^{\otimes m}),\rho ^{\otimes n})\leq \epsilon \}}$

negativity

The negativity is an easy predictable Verschränkungsmaß. means that a state is entangled, whereby there are also entangled states for systems larger than two qubits . The following applies to a general state ${\displaystyle {\mathcal {N}}}$ ${\displaystyle {\mathcal {N}}>0}$ ${\displaystyle {\mathcal {N}}=0}$ ${\displaystyle \rho }$

${\displaystyle {\mathcal {N}}(\rho )=||\rho ^{pt}||_{1}-1}$.

Here is the 1- norm (sum of the amounts of all eigenvalues) and denotes the partially transposed matrix (i.e. in the subspace of a subsystem under consideration) . The negativity thus follows directly on from the Peres-Horodecki (PPT) entanglement criterion. ${\displaystyle ||\cdot ||_{1}}$ ${\displaystyle \rho ^{pt}}$

Logarithmic negativity

Logarithmic negativity , analogous to negativity, is defined as ${\displaystyle E_{N}}$

${\displaystyle E_{N}(\rho )=\log _{2}(||\rho ^{pt}||_{1})=\log _{2}({\mathcal {N}}(\rho )+1)}$.

This has the advantage over the negativity that it is additive for tensor products: . The logarithmic negativity is an upper bound for the distillable entanglement. ${\displaystyle E_{N}(\rho \otimes \sigma )=E_{N}(\rho )+E_{N}(\sigma )}$

Distillable entanglement

Distillable entanglement is defined as the (asymptotic) number of maximally entangled qubit pairs that can be produced (distilled) from the state using LOCC operations.

Relative entropy of entanglement

The relative entropy of entanglement

${\displaystyle E_{R}(\rho )=\inf _{\sigma }S(\rho ||\sigma )=-\operatorname {Tr} (\rho _{A}\log _{2}\rho _{A})}$

where the conditional Von Neumann entropy S is defined as

${\displaystyle S(\rho )=-\operatorname {Tr} (\rho \log _{2}\rho )}$.

using the matrix logarithm for base 2. The reduced density matrix of the overall system can be calculated using the partial track formation . ${\displaystyle \rho _{AB}}$ ${\displaystyle \rho _{A}=\operatorname {Tr} _{B}\rho _{AB}}$

Entanglement robustness

Entanglement robustness measures how much noise would have to be added to make a state separable. The robustness of entanglement

${\displaystyle R(\rho )=s}$

is defined as the smallest for which the state ${\displaystyle s}$

${\displaystyle \xi ={\frac {1}{1+s}}(\rho +s\sigma )}$

is separable with any separable state . ${\displaystyle \sigma }$

Squashed entanglement

The Squashed Entanglement (as "crushed entanglement"), even CMI-entanglement (for Conditional Mutual Information , engl. Conditional Mutual information ) is derived from the classical information theory. The squashed entanglement between two subsystems A and B is defined as

${\displaystyle E_{\mathrm {CMI} }(\rho _{A,B})={\frac {1}{2}}\min _{\rho _{A,B,\Lambda }\in K}S(A:B|\Lambda )}$

With K as the set of all density matrices so that the partial trace over the third subsystem again corresponds to the bipartite system . Here is the quantum conditioned transinformation, defined as ${\displaystyle \rho _{A,B,\Lambda }}$ ${\displaystyle \Lambda }$ ${\displaystyle \rho _{A,B}=\operatorname {Tr} _{\Lambda }(\rho _{A,B,\Lambda })}$ ${\displaystyle S(A:B|\Lambda )}$

${\displaystyle S(A:B|\Lambda )=S(\rho _{A,\Lambda })+S(\rho _{B,\Lambda })-S(\rho _{\Lambda })-S(\rho _{A,B,\Lambda })}$

and is the Von Neumann entropy of a density matrix , depending on the indices of the entire system or of a subsystem after the formation of the partial trace over the other subsystems. ${\displaystyle S(\rho )}$ ${\displaystyle S(\rho )=-\operatorname {Tr} (\rho \log _{2}\rho )}$

For pure states, the squashed entanglement corresponds to the formation entanglement.

Entanglement of several systems

The quantification of the entanglement between three or more subsystems is fundamentally a complex mathematical topic and the subject of current research. It is known that there are different types of entanglement, for example paired entanglement between two sub-systems or entanglement between all sub-systems, which, however, is less strong between pairs.

Tangle

The tangle describes the entanglement of three systems A, B, C

${\displaystyle \tau (A:B:C)=\tau (A:BC)-\tau (AB)-\tau (AC)}$

using the 2-Tangles on the right-hand side, which are each the square of the Concurrence.

Formation entanglement

For any state , the formation entanglement is defined as: ${\displaystyle \rho }$ ${\displaystyle E_{F}}$

${\displaystyle E_{F}(\rho )=\inf _{p_{k},|\phi _{k}\rangle }\sum _{k}p_{k}S((\rho _{A})_{k})}$

For bipartite systems, this definition is simplified to the above-mentioned analytical formula.

Individual evidence

1. ^ ^{A b} C. H. Bennett, HJ Bernstein, S. Popescu, B. Schumacher: Concentrating partial entanglement by local operations . In: Phys. Rev. A . tape 53 , 1996, pp. 2046 , doi : 10.1103 / PhysRevA.53.2046 , arxiv : quant-ph / 9511030 .
2. ↑ There are numerous other measures, but entropy of entanglement is the only thing that has the additivity, monotonicity and continuity properties desired by such measures, cf. G. Vidal: On the continuity of asymptotic measures of entanglement . 2002, arxiv : quant-ph / 0203107 . and R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki: Quantum entanglement . In: Rev. Mod. Phys. tape 81 , 2009, p. 865 , p. 912f, sections XV.D and XV.E , arxiv : quant-ph / 0702225 .
3. ^ A. Uhlmann: Fidelity and Concurrence of conjugated states . In: Phys. Rev. A . tape 62 , 2000, pp. 032307 , arxiv : quant-ph / 9909060 . and R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki: Quantum entanglement . In: Rev. Mod. Phys. tape 81 , 2009, p. 865 , section XV.C.2, p. 911 , arxiv : quant-ph / 0702225 .
4. ↑ Hill, S., Wooters, WK: Entanglement of a Pair of Quantum Bits. In: Physical Review Lett. No. 78, 1997, pp. 5022-5025. arxiv : quant-ph / 9703041 .
5. ^ CH Bennett, DP DiVincenzo, JA Smolin, WK Wootters: Mixed-state entanglement and quantum error correction . In: Phys. Rev. A . tape 54 , 1996, pp. 3824 , arxiv : quant-ph / 9604024 .
6. ^ Bennett, CH, Bernstein, HJ, Popescu, S., & Schumacher, B .: Concentrating partial entanglement by local operations. In: Physical Review A. No. 53, 1996, pp. 2046-2052. arxiv : quant-ph / 9511030
7. ^ Vidal, G., & Tarrach, R .: Robustness of entanglement. In: Physical Review A. No. 59, 1999, pp. 141-155. arxiv : quant-ph / 9806094
8. ↑ Cerf, NJ, Adami, C .: Quantum Mechanics of Measurement. arxiv : quant-ph / 9605002 .
9. ^ Coffman, V., Kundu, J., Wootters, WK: Distributed entanglement. In: Physical Review A. No. 61, 2000, 052306. arxiv : quant-ph / 9907047 .