# Quantum Incompatibility in Collective Measurements

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## Abstract

**:**

## 1. Introduction

## 2. Quantum Observables

**Definition**

**1.**

- (i)
- $\mathsf{A}(x)\ge 0$ for all $x\in \Omega $;
- (ii)
- ${\sum}_{x\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\Omega}\mathsf{A}(x)=\mathrm{\U0001d7d9}$.

- (i′)
- $\mathsf{A}(X)\ge 0$ for all $X\subseteq \Omega $;
- (ii′)
- $\mathsf{A}(\Omega )=\mathrm{\U0001d7d9}$;
- (iii′)
- $\mathsf{A}(X\cup Y)=\mathsf{A}(X)+\mathsf{A}(Y)$ for all $X,Y\subseteq \Omega $, such that $X\cap Y=\mathsf{\varnothing}$.

## 3. k-Compatibility of Observables

#### 3.1. Definition

**Definition**

**2.**

#### 3.2. Basic Properties

- Any collection of n observables is n-compatible.

- Any subset of a k-compatible set of observables is k-compatible.

- Any collection of k-compatible observables is ${k}^{\prime}$-compatible for all ${k}^{\prime}\ge k$.

**Proposition**

**1.**

**Proof.**

#### 3.3. Index of Incompatibility

**Definition**

**3.**

- (i)
- $1\le \mathfrak{i}(\mathcal{A})\le \#\mathcal{A}$;
- (ii)
- if $\mathcal{A}\subseteq \mathcal{B}$, then $\mathfrak{i}(\mathcal{A})\le \mathfrak{i}(\mathcal{B})$;
- (iii)
- $\mathfrak{i}({\mathcal{A}}_{1}\cup {\mathcal{A}}_{2})\le \mathfrak{i}({\mathcal{A}}_{1})+\mathfrak{i}({\mathcal{A}}_{2})$;
- (iv)
- $\mathfrak{i}(\mathcal{A})=1$ if and only if $\mathcal{A}$ is compatible.

## 4. Compatibility Stack

#### 4.1. Definition

**Definition**

**4.**

- (S1)
- each ${H}_{k}=(V,{E}_{k})$ is a joint measurability hypergraph,
- (S2)
- ${E}_{1}$ contains all singleton sets and ${E}_{n}={2}^{V}$ and
- (S3)
- if $\mathcal{A}\in {E}_{k}$ and $\mathcal{B}\in {E}_{l}$, then $\mathcal{A}\cup \mathcal{B}\in {E}_{k\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}l}$.

**Proposition**

**2.**

- (1)
- ${E}_{1}\subseteq {E}_{2}\subseteq \cdots \subseteq {E}_{n}$;
- (2)
- For each $k=1,\dots ,n$, the set ${E}_{k}$ contains all subsets of V of order k.

**Proof.**

#### 4.2. Compatibility Stacks with Three Vertices

#### 4.3. Compatibility Stacks with Four Vertices

- If two sets composed of reciprocal pairs have index 1, then the set of all four vertices has index ≤ 2.

## 5. Structure of k-Copy Joint Observables

#### 5.1. Symmetric Product

#### 5.2. Structure Theorem

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Example**

**1.**

## 6. Three Qubit Observables

#### 6.1. 2-Copy Joint Observables from Mixing

**Theorem**

**2.**

- (1)
- ${\mathsf{X}}_{a}$ and ${\mathsf{Y}}_{b}$ are compatible if and only if ${a}^{2}+{b}^{2}\le 1$.
- (2)
- ${\mathsf{X}}_{a}$, ${\mathsf{Y}}_{b}$ and ${\mathsf{Z}}_{c}$ are compatible if and only if ${a}^{2}+{b}^{2}+{c}^{2}\le 1$.

- (a)
- $a=b=c=1/\sqrt{3}$, with joint observable$$\mathsf{G}(x,y,z)=\frac{1}{8}\left(\right)open="["\; close="]">\mathrm{\U0001d7d9}+\frac{1}{\sqrt{3}}(x{\sigma}_{x}+y{\sigma}_{y}+z{\sigma}_{z})$$
- (b)
- $a=b=c=1/\sqrt{2}$, with joint observables ${\mathsf{G}}_{\pi /4}^{1,2}$, ${\mathsf{G}}_{\pi /4}^{2,3}$ and ${\mathsf{G}}_{\pi /4}^{1,3}$ for the corresponding pairs of observables;
- (c)
- $a=b=4/5$ and $c=3/5$, with joint observable ${\mathsf{G}}_{\gamma}^{1,3}$ (having $sin\gamma =3/5$ and $cos\gamma =4/5$) for observables ${\mathsf{X}}_{a}$ and ${\mathsf{Z}}_{c}$ and joint observable ${\mathsf{G}}_{\beta}^{2,3}$ (having $sin\beta =4/5$ and $cos\beta =3/5$) for observables ${\mathsf{Y}}_{b}$ and ${\mathsf{Z}}_{c}$;
- (d)
- $a=4/5$, $b=1$ and $c=3/5$, with joint observable ${\mathsf{G}}_{\gamma}^{1,3}$ (having $sin\gamma =3/5$ and $cos\gamma =4/5$) for observables ${\mathsf{X}}_{a}$ and ${\mathsf{Z}}_{c}$.

**Proposition**

**3.**

**Proof.**

- (e)
- $a=b=c=(1+\sqrt{2})/3$.

#### 6.2. Optimal 2-Copy Joint Observable

**Theorem**

**3.**

**Proposition**

**4.**

- (i)
- ${\mathsf{G}}^{\wedge}(g.x)=U(g){\mathsf{G}}^{\wedge}(x)U{(g)}^{*}$ for all $x\in \Omega $ and $g\in G$;
- (ii)
- ${\mathsf{G}}^{\wedge}(X)=\mathsf{G}(X)$ for all $X\in \mathcal{F}$.

**Proof.**

**Proposition**

**5.**

**Proof.**

**Remark**

**1.**

- (f)
- $a=b=c=1$,

## 7. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Kunjwal, R.; Heunen, C.; Fritzi, T. Quantum realization of arbitrary joint measurability structures. Phys. Rev. A
**2014**, 89, 052126. [Google Scholar] [CrossRef] - Busch, P.; Grabowski, M.; Lahti, P. Operational Quantum Physics, 2nd ed.; Springer-Verlag: Berlin, Germany, 1997. [Google Scholar]
- Lahti, P. Coexistence and joint measurability in quantum mechanics. Int. J. Theor. Phys.
**2003**, 42, 893–906. [Google Scholar] [CrossRef] - Busch, P. Unsharp reality and joint measurements for spin observables. Phys. Rev. D
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**Figure 1.**Visualization of an incompatibility stack for four observables $\mathsf{A}$, $\mathsf{B}$, $\mathsf{C}$ and $\mathsf{D}$. On the left is a compact representation in terms of indexes, i.e., the smallest possible k-compatibility for a given subset of observables. On the right is the same stack in terms of hypergraphs ${H}_{k}$, where each hypergraph is represented by different color ($k=1$ is orange; $k=2$ is blue; $k=3$ is green; and $k=4$ is red).

**Figure 2.**All possible compatibility stacks with three vertices. The orange color marks index 1; blue marks index 2; and green marks index 3. Whereas (

**a**) depicts the most compatible case, when all of the measurements can be performed on a single copy of a state; (

**f**) depicts the worst case where for each measurement, we need an extra copy of the state. The cases (

**b**)–(

**e**) represent all the intermediate possibilities.

**Figure 3.**Figures (

**a**) and (

**b**) depict two examples of impossible k-compatibility relations for three observables. The orange color marks index 1; blue marks index 2; and green marks index 3.

**Figure 4.**Compatibility stacks with four vertices can be represented by colored tetrahedrons. As before, orange color marks index 1; blue marks index 2; and green marks index 3. In addition, index 4 is marked by red color. The case (

**a**) represents the most compatible case, where we need only a single copy of a state to measure all four observables; whereas the case (

**b**) is in this respect the worst one, as we need a new copy of a state for each measurement. The cases (

**c**) and (

**d**) show some intermediate possibilities of compatibility stacks.

**Figure 5.**The index of incompatibility $\mathfrak{i}(\{{\mathsf{X}}_{a},{\mathsf{Y}}_{a},{\mathsf{Z}}_{a}\})$ as a function of the noise parameter a for three noisy orthogonal qubit observables.

**Table 1.**All possible compatibility stacks enumerated by their bulk index (number of copies of a system required to measure all four observables; colors in parentheses represent indexes from previous figures) and the number of edges with index 2 (how many pairs of observables are not 1-compatible). Altogether, 34 different stacks (up to trivial permutations) are possible.

# of Index 2 Edges ► | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

Bulk Index ▼ | |||||||

1 (orange) | 1 | – | – | – | – | – | – |

2 (blue) | 5 | 3 | 3 | 4 | 2 | 1 | 1 |

3 (green) | – | – | – | 3 | 2 | 3 | 5 |

4 (red) | – | – | – | – | – | – | 1 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Carmeli, C.; Heinosaari, T.; Reitzner, D.; Schultz, J.; Toigo, A.
Quantum Incompatibility in Collective Measurements. *Mathematics* **2016**, *4*, 54.
https://doi.org/10.3390/math4030054

**AMA Style**

Carmeli C, Heinosaari T, Reitzner D, Schultz J, Toigo A.
Quantum Incompatibility in Collective Measurements. *Mathematics*. 2016; 4(3):54.
https://doi.org/10.3390/math4030054

**Chicago/Turabian Style**

Carmeli, Claudio, Teiko Heinosaari, Daniel Reitzner, Jussi Schultz, and Alessandro Toigo.
2016. "Quantum Incompatibility in Collective Measurements" *Mathematics* 4, no. 3: 54.
https://doi.org/10.3390/math4030054