An elementary introduction to the geometry of quantum states with pictures

TL;DR

This paper provides an elementary, visual introduction to the geometry of quantum states, illustrating high-dimensional convex bodies and their properties, including separability and entanglement, using simple geometric shapes and pictures.

Contribution

It offers a visual and elementary approach to understanding the complex geometry of quantum states, emphasizing cross sections and high-dimensional shapes.

Findings

Quantum state space resembles a high-dimensional ball in most directions.

The space of states is well-approximated by simple geometric shapes like balls and simplices.

Geometric properties of entangled and separable states are discussed.

Abstract

This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of n-qubits, the dimension is exponentially large in n. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra. When the dimension gets large there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.

Figures

Equations

$$\rho (\mathbf{x})=\frac{\text{mathds}1+\mathbf{x}\cdot \mathbf{\sigma }}{2},\mathbf{x}\in {\mathbb{R}}^3,$$

$$D_{N}=\left\{\rho\,\Bigg{|}\,\rho=\sum_{j=1}^{k}p_{j}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|,\ p_{j}\geq 0,\ \sum_{j}p_{j}=1,\ \left|\psi_{j}\right\rangle\in{\cal H},\ \|\psi_{j}\|=1\right\}$$

$$1\ge {\rho }_1\ge \cdots \ge {\rho }_N\ge 0,\sum {\rho }_j=1$$

$$Tr{\left(|\psi ⟩⟨\psi |-\frac{\text{mathds}1}{N}\right)}^2=1-\frac{1}{N}$$

$$B_{gb}=\left\{\rho\,\Bigg{|}\,Tr\left(\rho-\frac{\mathds{1}}{N}\right)^{2}\leq\frac{1}{N-1}-\frac{1}{N}\right\}$$

$$\dim D_N=N^2-1$$

$$\frac{\text{radius of bounding ball of }{D}_N}{\text{radius of inscribed ball of }{D}_N}=N-1$$

$$I(\rho )=\frac{\text{mathds}1}{N}-\left(\rho -\frac{\text{mathds}1}{N}\right)=\frac{2}{N}\text{mathds}1-\rho$$

$$0\leq Tr\big{(}\rho I(\rho)\big{)}=\frac{2}{N}-Tr(\rho^{2})$$

$$2NTr{\left(\rho -\frac{\text{mathds}1}{N}\right)}^2\to 1,\rho \in \partial D_N$$

$$S_{N_{1}\dots N_{n}}=\left\{\rho\ \Bigg{|}\,\rho=\sum_{j=1}^{k}p_{j}\left|\psi_{j}^{(1)}\right\rangle\left\langle\psi_{j}^{(1)}\right|\otimes\dots\otimes\left|\psi_{j}^{(n)}\right\rangle\left\langle\psi_{j}^{(n)}\right|,\,\left|\psi_{j}^{(m)}\right\rangle\in{\cal H}_{N_{m}}\right\}$$

$$S_{N_1,N_2,N_3}\subseteq S_{N_1,N_2\times N_3}\cap S_{N_3,N_1\times N_2}\cap S_{N_2,N_1\times N_3}$$

$$|{\psi }_j^{(m)}⟩⟨{\psi }_j^{(m)}|\mapsto {\left(|{\psi }_j^{(m)}⟩⟨{\psi }_j^{(m)}|\right)}^t$$

$$\dim S_{N_1,\dots ,N_n}=N^2-1$$

$$B_{gb}\subseteq S_{N_1,N_2}$$

$$\frac{\text{radius of bounding ball of }{S}_{N_1,N_2}}{\text{radius of inscribed ball of }{S}_{N_1,N_2}}=N-1$$

$$\rho (\mathbf{x})=\frac{\text{mathds}{1}_4+\sqrt{3}\sum x_{\mu \nu }{\sigma }_{\mu }\otimes {\sigma }_{\nu }}{4},$$

$$4\rho(x,y,z)={\mathds{1}}+\sqrt{3}\big{(}x\sigma_{1}\otimes\sigma_{1}+y\sigma_{2}\otimes\sigma_{2}+z\sigma_{3}\otimes\sigma_{3}\big{)}$$

$$T{r}_2\rho =T{r}_1\rho =\frac{\text{mathds}1}{2}$$

$$Tr(\rho^{2})=\frac{1}{4}+\frac{3}{4}\big{(}x^{2}+y^{2}+z^{2}\big{)}$$

$$({\sigma }_1\otimes {\sigma }_1)({\sigma }_2\otimes {\sigma }_2)=-{\sigma }_3\otimes {\sigma }_3$$

$$x\pm y\mp z\ge -\frac{1}{\sqrt{3}},-x\pm y\pm z\ge -\frac{1}{\sqrt{3}}$$

$$\frac{1}{\sqrt{3}}(1,1,-1),\frac{1}{\sqrt{3}}(1,-1,1)\frac{1}{\sqrt{3}}(-1,1,1),\frac{1}{\sqrt{3}}(-1,-1,-1)$$

$$0\leq Tr\big{(}W\rho(\mathbf{s})\big{)}=\frac{1+3\mathbf{s\cdot w}}{4}\,,$$

$$\rho (\mathbf{x})=\frac{\text{mathds}{1}_N+\sqrt{N-1}\mathbf{x}\cdot \mathbf{\sigma }}{N},\mathbf{x}\in {\mathbb{R}}^{N^2-1}$$

$$Tr({\sigma }_{\alpha }{\sigma }_{\beta })={\delta }_{\alpha \beta }N,\alpha ,\beta \in \{1,\dots ,N^2-1\}$$

$${\sigma }_{\alpha }^t=\pm {\sigma }_{\alpha }$$

$${\sigma }_{\mu }={\sigma }_{{\mu }_1}\otimes \cdots \otimes {\sigma }_{{\mu }_n},{\mu }_j\in \{0,\dots ,3\},\mu \in \{1,\dots ,N^2-1\}$$

$${\sigma }_{\mu }^2=\text{mathds}{1}_N,Tr{\sigma }_{\mu }=0$$

$${\sigma }_{\mu }^t=\pm {\sigma }_{\mu }$$

Full text

*“All convex bodies behave a bit like Euclidean balls.”

Keith Ball*

1. 1 Introduction

2. 1.1 The geometry of quantum states

3. 1.2 The geometry of separable states

4. 2 Two qubits

5. 2.1 Numerical sections for 2 qubits

6. 2.2 A 3-D section through Bell states

7. 3 Basic geometry of Quantum states

8. 3.1 Choosing coordinates

9. 3.2 Most of the unit sphere does not represent states

10. 3.3 Inversion asymmetry

11. 3.4 The inscribed sphere

12. 4 Cross sections

13. 4.1 Cross sections that are N-1 simplexes

14. 4.2 Cross sections that are balls

15. 4.3 Cross sections that are polyhedra and hyper-octahedra

16. 4.4 2D cross sections in the Pauli basis

17. 5 The radius function

18. 5.1 The radius function is N-Lifshitz

19. 6 A tiny ball in most directions

20. 6.1 Application of random matrix theory

21. 6.2 Directions associated with states with substantial purity are rare

22. 7 Separable and entangled states

23. 7.1 Why separability is hard

24. 7.2 Completely separable simplex: Classical bits

25. 7.3 Entangled pure states

26. 7.4 Two types of entangled states

27. 7.5 Entangled low purity states

28. 7.6 The largest ball of bi-partite separable states

29. 7.7 Entanglement witnesses

30. 7.8 Entangled states and witnesses near the Gurvits-Barnum ball

31. 7.9 A Clifford ball of separable states

32. A The average purity of quantum states

33. B The N dimensional unit cube is almost a ball

1 Introduction

1.1 The geometry of quantum states

The set of states of a single qubit is geometrically a ball, the Bloch ball [1]: The density matrix $\rho$ is $2\times 2$:

[TABLE]

with $\boldsymbol{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$, the vector of $2\times 2$ (Hermitian, traceless) Pauli matrices. $\rho\geq 0$, provided $|\mathbf{x}|\leq 1$. The unit sphere, $|\mathbf{x}|^{2}=1$, represents pure states where $\rho$ is a rank one projection. The interior of the ball describes mixed states and the center of the ball the fully mixed state, (Fig. 1).

The geometry of a qubit is not always a good guide to the geometry of general quantum states: $n$-qubits are not represented by $n$ Bloch balls222$n$ Bloch balls describe uncorrelated qubits., and quantum states are not, in general, a ball in high dimensions.

Quantum states are mixtures of pure states. We denote the set of quantum states in an $N\geq 2$ dimensional Hilbert space $\cal H$ by $D_{N}$:

[TABLE]

The representation implies:

• •

The quantum states form a convex set.

• •

The pure states are its extreme points. As we shall see in section 3.2, the set of pure states is a smooth manifold, which is a tiny subset of $\partial D_{N}$ when $N$ is large.

• •

The spectral theorem gives (generically) a distinguished decomposition with $k\leq\dim{\cal H}$.

Fig. 2 shows a three dimensional body whose geometry reflects better the geometry of the space of states $D_{N}$ for general $N$, (and also the spaces of separable states), than the Bloch sphere does.

Choosing a basis in ${\cal H}$, the state $\rho$ is represented by a positive $N\times N$ matrix with unit trace ($N=\dim{\cal H}\geq 2$). In the case of $n$-qubits $N=2^{n}$. Since the sum of two positive matrices is a positive matrix, the positive matrices form a convex cone in $\mathbb{R}^{N^{2}}$, and the positive matrices with unit trace are a slice of this cone. The slice is an $N^{2}-1$ dimensional convex body with the pure states $\left|\psi\right\rangle\left\langle\psi\right|$ as its extreme points and the fully mixed state as its “center of mass”.

The geometric properties of $D_{N}$ can be complicated and, because of the high dimensions involved, counter-intuitive. Even the case of two qubits, where $D_{N}$ is 15 dimensional, is difficult to visualize [2, 3, 4, 5].

In contrast with the complicated geometry of $D_{N}$, the geometry of equivalence classes of quantum states under unitaries, even for large $N$, is simple: It is parametrized by eigenvalues and represented by the $N-1$-simplex, Fig. 3,

[TABLE]

All pure states are represented by the single extreme point, $(1,0,\dots,0)$, and the fully mixed state by the extreme point $(1,\dots,1)/N$. The equivalence classes corresponding to the Bloch ball are represented by an interval (1-simplex) which corresponds to the radius of the Bloch ball.

Clearly, the geometry of $D_{N}$ does not resemble the geometry of the set of equivalence classes: The two live in different dimensions, have different extreme points and $D_{N}$ is, of course, not a polytope.

One of the features of a qubit that holds for any $D_{N}$, is that the pure states are equidistant from the fully mixed state. Indeed

[TABLE]

This implies that $D_{N}$ is contained in a ball, centered at the fully mixed state, whose radius squared is given by the rhs of Eq. (1.3). $D_{N}$ is, however, increasingly unlike a ball when $N$ is large: The largest ball inscribed in $D_{N}$ is the Gurvits-Barnum ball333Gurvits and Barnum define the radius of the ball by its purity, rather than the distance from the maximally mixed states. [6]:

[TABLE]

It is a tiny ball when $N$ is large, whose radius squared is given by the rhs of Eq. (1.4) and centered at the fully mixed state. It follows that:

• •

[TABLE]

• •

Since

[TABLE]

only $D_{2}$ is a ball and $D_{N}$ gets increasingly far from a ball when $N$ is large, Fig. 4.

Another significant difference between a single qubit and the general case is that $D_{N}$ is inversion symmetric only for $N=2$. Indeed, inversion with respect to the fully mixed state is defined by

[TABLE]

Evidently $I$ is trace preserving and $I^{2}=1$. However, it is not positivity preserving unless the purity of $\rho$ is sufficiently small. Indeed, $\rho\geq 0$ and $I(\rho)\geq 0$ imply that444$I(\rho)\geq 0$ iff $\|\rho\|\leq 2/N$.

[TABLE]

The low symmetry together with the large aspect ratio indicate that the geometry of $D_{N}$ may be complicated. It can be visualized, to some extent, by looking at cross sections. As we shall see $D_{N}$ has several cross sections that are simple to describe: Regular simplexes, balls and hyper-octahedra.

$D_{N}$ has a Yin-Yang relation to spheres in high dimensions: As $N$ gets large $D_{N}$ gets increasingly far from a ball as is evidenced by the diverging ratio of the bounding sphere to the inscribed ball. At the same time there is a sense in which the converse is also true: Viewed from the center (the fully mixed state), the distance to $\partial D_{N}$, is the same for almost all directions. In this sense, $D_{N}$ increasingly resembles a ball. The radius of the ball can be easily computed using standard facts from random matrix theory [7], and we find that as $N\to\infty$, for almost all directions,

[TABLE]

The fact that $D_{N}$ is almost a ball is not surprising. In fact, a rather general consequence of the theory of “concentration of measures”, [8, 9, 10], is that sufficiently nice high dimensional convex bodies are essentially balls555Applications of concentration of measure to quantum information are given in e.g. [6, 9, 11]. . $D_{N}$, however, is not sufficiently nice so that one can simply apply standard theorems from concentration of measure. Instead, we use information about the second moment of $D_{N}$ and Hölder inequality to show that the set of directions $\theta$ that allow for states with significant purity has super-exponentially small measure (see section 6.2).

1.2 The geometry of separable states

The Hilbert space of a quantum system partitioned into $n$ groups of (distinguishable) particles has a tensor product structure ${\cal H}={\cal H}_{N_{1}}\otimes\dots\otimes{\cal H}_{N_{n}}$. The set of separable states of such a system, denoted $S_{N_{1}\dots N_{n}}$, is defined by [12],

[TABLE]

where $p_{j}$ are probabilities.

For reasons that we shall explain in section 7, $S_{N_{1}\dots N_{n}}$ are more difficult to analyze than $D_{N}$. They have been studied by many authors from different perspectives [5, 9, 6, 2]. It will be a task with diminishing returns to try and make a comprehensive list of all of the known results. Selected few references are [3, 4, 9, 13, 11, 14, 15, 16, 17]. We shall review, instead, few elementary observations and accompany them by pictures.

The representation in Eq. (1.9) implies that

• •

$S_{N_{1}\dots N_{n}}\subseteq D_{N}$ where $N=\prod_{j}N_{j}$

• •

$S_{N_{1}\dots N_{n}}$ is a convex set with pure-product states as its extreme points.

• •

The finer the partition the smaller the set

[TABLE]

Strict inclusion implies that Alice, Bob and Charlie may have a 3-body entanglement that is not visible in any bi-partite partition.

• •

By Caratheodory theorem, one can always find a representation with $k\leq N^{2}$ in Eq. (1.9). In the case of two qubits a results of Wootters [18] gives $k\leq N$.

• •

$S_{N_{1}\dots N_{n}}$ is invariant under partial transposition, (transposition of any one of its factor), i.e.

[TABLE]

• •

The bounding sphere of $S_{N_{1}\dots N_{n}}$ is the bounding sphere of $D_{N}$.

• •

The separable states are of full measure:

[TABLE]

It is enough to show this for the maximally separable set. For simplicity, consider the case of $n$ qubits. For each qubit ${\mathds{1}+\sigma_{\mu}}$ with $\mu=1,\dots,3$ are linearly independent and positive. The same is true for their tensor products. This gives $4^{n}=N^{2}$ linearly independent separable states spanning a basis in the space of Hermitian matrices.

By a result of [6]:

[TABLE]

for any partition. It implies that:

• •

Since666The balls are centered at the mixed state which is the natural “center of mass” of all states.

[TABLE]

the separable states get increasingly far from a ball when $N$ is large.

We expect that the separable states too are approximated by balls in most directions, but unlike the case of $D_{N}$, we do not know how to estimate the radii of these balls.

We list below few general simple facts about separable states:

• •

$S_{N_{1}N_{2}}\subseteq D_{N}\cap D_{N}^{pt}$ where $D_{N}^{pt}$ is the reflection associated with partial transposition and equality holds when $N_{1}N_{2}\leq 6$.

• •

The classical states, i.e. diagonal density matrices, are represented by and $N-1$ regular simplex, which is a cross section of the separable states. check

• •

States with purity $p\leq 1/(N-1)$, the Gurvits-Barnum ball [6], are separable.

• •

The boundary of the Gurvits-Barnum ball is tangent to $\partial D_{N}$, and also tangent to the entangled states.

In the case of 2-qubits we show pictures of two dimensional random sections and the three dimensional section associated with the Bell states which turn out to coincide with the picture of the SLOCC 777The acronym stands for: Stochastic local operations and classical communication. It is an equivalence class that allows for filtering in addition to local operations. equivalence classes [4, 3].

2 Two qubits

Two qubits give a much better intuition about the geometry of general quantum states than a single qubit. However, as 2 qubits live in 15 dimensions, they are still hard to visualize.

One way to gain insight into the geometry of two qubits is to consider equivalence classes that can be visualized in 3 dimensions [19, 2, 4, 20]). However, as we have noted above, the geometry of equivalence classes is distinct from the geometry of states. An alternate way to visualize 2 qubits is to look at 2 and 3 dimensional cross sections through the space of states.

The states of two qubits can be parametrized by $\mathbf{x}\in\mathbb{R}^{15}$. It is convenient to lable the 15 components of $x$ by $\mathbf{x}_{\mu\nu}$ with $\mu\nu\in({01},\dots,{33})$

[TABLE]

${\sigma_{\mu}}$ are the Pauli matrices. By a 2 dimensional section in the space of two qubits we mean a two dimensional plane in $\mathbf{R}^{15}$ going through the origin.

2.1 Numerical sections for 2 qubits

The 2 dimensional figures 5 and 6 show random sections obtained by numerically testing the positivity and separability of $\rho$, using Mathematica. A generic plane will miss the pure states, which are a set of lower dimension. This situation is shown in Fig. 5.

Fig. 6 shows a two dimensional section obtained by picking two pure states randomly. Since a generic pure state is entangled, the section goes through two pure entangled states lying on the unit circle.

2.2 A 3-D section through Bell states

Consider the 3D cross section given by888A generic two qubits state is SLOCC equivalent to a point of this section, see [4].:

[TABLE]

The section has the property that both subsystems are maximally mixed

[TABLE]

Since the purity is given by

[TABLE]

the pure states lie on the unit sphere and all the states in this section must lie inside the unit ball.

The matrices on the right commute and satisfy one relation

[TABLE]

It follows that $\rho\geq 0$ iff $(x,y,z)$ lie in the intersection of the 4 half spaces:

[TABLE]

This defines a regular tetrahedron with vertices

[TABLE]

The vertices of the tetrahedron lie at the corners of the cube in Fig. 7, at unit distance from the origin. It follows that the vertices of the tetrahedron represent pure states. As the section represents states with maximally mixed subsystems, the four pure states are maximally entangled: They are the 4 Bell states

The pairwise averages of the four corners of the tetrahedron give the $\binom{4}{2}=6$ vertices of the octahedron in Fig. (7). By Eq. (7.3) below, these averages represent separable states. It follows that the octahedron represents separable states.

The cube, represents the trace-normalized entanglement witnesses (see section 7.7). A vector $\mathbf{s}$ inside the octahedron represents the state $\rho(\mathbf{s})$ per Eq. (2.2). Similarly, a vector $\mathbf{w}$ inside the cube represents the witness $W$ (see Eq. (7.24)). Since the cube is the dual of the octahedron we have

[TABLE]

which is the defining relation of witnesses.

3 Basic geometry of Quantum states

3.1 Choosing coordinates

Any Hermitian $N\times N$ matrix with unit trace can be written as:

[TABLE]

$\mathds{1}_{N}=\sigma_{0}$ is the identity matrix and $\boldsymbol{\sigma}=(\sigma_{1},\dots,\sigma_{N^{2}-1})$ a vector of $N^{2}-1$ traceless, Hermitian, mutually orthogonal, $N\times N$ matrices

[TABLE]

This still leaves considerable freedom in choosing the coordinates $\sigma_{\alpha}$ and one may impose additional desiderata. For example:

• •

$\sigma_{\alpha}$ are either real symmetric or imaginary anti-symmetric

[TABLE]

This requirement is motivated by $\rho\geq 0\Longleftrightarrow\rho^{t}\geq 0$

• •

$\sigma_{\alpha}$ for $\alpha\neq 0$ are unitarily equivalent, i.e. are iso-spectral.

A coordinate system that has these properties in $N=2^{n}$ dimensions, is the (generalized) Pauli coordinates:

[TABLE]

$\sigma_{\mu}$ are iso-spectral with eigenvalues $\pm 1$. This follows from:

[TABLE]

The Pauli coordinates behave nicely under transposition:

[TABLE]

In addition, they either commute or anti-commute

[TABLE]

This will prove handy in what follows. One drawback of the Pauli coordinates is that they only apply to Hilbert spaces with special dimensions, namely $N=2^{n}$.

For $N$ arbitrary, one may not be able satisfy all the desiderata simultaneously. In particular, the standard basis

[TABLE]

is iso-spectral with eigenvalues $\{\pm 1,0\}$ and behave nicely under transposition. However, the $Z_{jN}$ coordinates are not mutually orthogonal.

With a slight abuse of notation we redefine

[TABLE]

Note that the Hilbert space and the Euclidean distances are related by scaling

[TABLE]

The basic geometric properties of $D_{N}$ follow from Eq. (1.2):

• •

The fully mixed state, $\mathds{1}/N$, is represented by the origin $\mathbf{x}=0$

• •

The pure states lie on the unit sphere for all $N$. This follows either from Eqs.(1.3,3.10) or, alternatively, from a direct computation of the purity:

[TABLE]

• •

Since the pure states are the extreme points of $D_{N}$:

[TABLE]

• •

Since $\rho\geq 0\Longleftrightarrow\rho^{t}\geq 0$ $D_{N}$ is symmetric under reflection of the “odd” coordinates associated with the anti-symmetric matrices.

• •

Since there is no reflection symmetry for the “even” Pauli coordinates, $\sigma_{\alpha}=\sigma_{\alpha}^{t}$, one does not expect $D_{N}$ to have inversion symmetry in general (as we have seen in Eq. (1.7)).

Let $\theta$ be a point on the unit sphere in $\mathbb{R}^{N^{2}-1}$ and $(r,\theta)$ be the polar representation of $\mathbf{x}$, in particular $r=|\mathbf{x}|$. Denote by $r(\theta)$ the radius function of $D_{N}$, i.e. the distance from the origin of the boundary of $D_{N}$ in the $\theta$ direction. Then

[TABLE]

where $\lambda_{1}(\theta)<0$ is the smallest eigenvalue of

[TABLE]

3.2 Most of the unit sphere does not represent states

For $N=2$, every point of the unit sphere represents a pure state, however, for $N\geq 3$ this is far from being the case. In fact, $\rho(\mathbf{x})$ of Eq. (3.1) is not a positive matrix for most $\mathbf{x}^{2}=1$. This follows from a simple counting argument: Pure states can be written as $\left|\psi\right\rangle\!\left\langle\psi\right|$ with $\left|\psi\right\rangle$ a normalized vector in $\mathbb{C}^{N}$. It follows that

[TABLE]

while

[TABLE]

When $N\geq 3$ pure states make a small subset of the of the unit sphere. When $N$ is large the ratio of dimensions is arbitrarily small.

Since (pure) states make a tiny subset of the unit sphere, spheres with radii close to $1$, should be mostly empty of states. In section 6.2 we shall give a quantitative estimate of this observation.

3.3 Inversion asymmetry

The Hilbert space and the Euclidean space scalar products are related by

[TABLE]

The positivity of $Tr\rho\rho^{\prime}\geq 0$ and Eq. (3.17) say that if both $\mathbf{x}$ and $\mathbf{x^{\prime}}$ correspond to bona-fide states then it must be that

[TABLE]

In particular, no two pure states are ever related by inversion if $N\geq 3$, (see fig. 8).

3.4 The inscribed sphere

The inscribed ball in $D_{N}$, the Gurvits-Barnum ball, is

[TABLE]

It is easy to see that the inscribed ball is at most the Gurvits-Barnum ball since the state

[TABLE]

clearly lies on $\partial D_{N}$. Using Eq. (3.17), one verifies that $|\mathbf{x}_{\psi}|=\frac{1}{N-1}$ saturating Eq. (3.19).

Eq. (3.19) follows from:

[TABLE]

In the last step we used the fact that the positivity of $\rho$ follows from the positivity of $\rho^{\prime}$ by Eq. (3.17).

Remark 3.1**.**

An alternate proof is: By Eq. (3.13) minimizing $r(\theta)$ is like minimizing the smallest eigenvalue $\lambda_{1}(\theta)$ of $S(\theta)=\boldsymbol{\theta}\cdot\boldsymbol{\sigma}$ under the constraints $TrS^{2}(\theta)=N$ and $TrS(\theta)=0$. The minimum occurs when

[TABLE]

This, together with Eq. (3.13), gives $r_{0}^{2}$ for the radius of the inscribed ball of non-negative matrices.

4 Cross sections

$D_{N}$ has few sections that are simple to describe, even when $N$ is large.

4.1 Cross sections that are N-1 simplexes

Let $\mathbf{v}_{j}$, with $j=0,\dots,N-1$ be the (unit) vectors associated with pure states $\rho_{j}=\left|\psi_{j}\right\rangle\!\left\langle\psi_{j}\right|$ corresponding to the orthonormal basis $\{\left|\psi_{j}\right\rangle\}$. Using Eq. (3.17) we find for $Tr\rho_{j}\rho_{k}=\delta_{jk}$

[TABLE]

For a single qubit, $N=2$, orthogonal states are (annoyingly) represented by antipodal points on the Bloch sphere. The situation improves when $N$ gets large: Orthogonal states are represented by almost orthogonal vectors. Moreover, from Eq. (3.1)

[TABLE]

The $N$ vectors $\mathbf{v}_{j}$ define a regular $(N-1)$-simplex, centered at the origin (in $\mathbb{R}^{N^{2}-1}$, see Fig. 9)

[TABLE]

Since the boundary of $C_{N-1}$ represent states that are not full rank, it belongs to the boundary of $D_{N}$ and therefore is an $N-1$ slice of $D_{N}$.

4.2 Cross sections that are balls

Suppose $N=2^{n}$. Consider the largest set of mutually anti-commuting matrices among the $N^{2}-1$ (generalized) Pauli matrices ${\sigma}_{\alpha}$. Since the Pauli matrices include the matrices that span a basis of a Clifford algebra we have at least $\ell$ anti-commuting matrices

[TABLE]

For the anti-commuting $\sigma_{j}$ we have

[TABLE]

The positivity of

[TABLE]

holds iff

[TABLE]

This means that $D_{N}$ has $\ell$ dimensional cross sections that are perfect balls999This and section 6 are reminiscent of Dvoretzki-Milmann theorem [10].. This result extends to $2^{n}\leq N<2^{n+1}$.

4.3 Cross sections that are polyhedra and hyper-octahedra

Consider the set of commuting matrices $\sigma_{\alpha}$. Since these $N-1$ matrices can be simultaneously diagonalized, the positivity condition on the cross-section

[TABLE]

reduces to a set of linear inequalities for $x_{\alpha}$. This defines a polyhedron.

In the case of $n$ qubits, a set of $n$ commuting $\sigma_{\alpha}$ matrices with no relations is:

[TABLE]

The cross section is the intersection of $N$ half-spaces

[TABLE]

The corresponding cross section is a regular $n$ dimensional hyper-octahedron101010See footnote 1.: A regular, convex polytope with $n$ vertices and $N=2^{n}$ hyper-planes. (The dual of the $n$ dimensional cube.)

4.4 2D cross sections in the Pauli basis

Any two dimensional cross section along two Pauli coordinates can be written as:

[TABLE]

By Eq. (3.7), ${\sigma}_{\alpha,\beta}$ either commute or anti-commute. The case that they anti-commute is a special case of the Clifford ball of section 4.2 where positivity implies

[TABLE]

The case that ${\sigma}_{\alpha,\beta}$ commute is a special case of section 4.3 where positivity holds if

[TABLE]

Both are balls, albeit in different metrics, ($\ell^{2}$ and $\ell^{1}$), see Fig. 10.

5 The radius function

By a general principle: “All convex bodies in high dimensions are a bit like Euclidean balls” [21]. More precisely, consider a convex body $C_{N}$ in $N$ dimensions, which contains the origin as an interior point. The radius function of $C_{N}$ is called K-Lifshitz, if

[TABLE]

By a fundamental result in the theory of concentration of measure [21, 8], the radius is concentrated near its median, with a variance that is at most $O(K^{2}/N)$ [21].

$D_{N}$ is a convex body in $N^{2}-1$ dimensions. As we shall see in the next section, $r(\theta)$ turns out to be $N$-Lifshitz. As a consequence, the variance of the distribution of the radius about the mean is only guaranteed to be $O(1)$. This is not strong enough to conclude that $D_{N}$ is almost a ball.

5.1 The radius function is N-Lifshitz

Since $D_{N}$ is convex and $r(\theta)>0$, the radius function is continuous, but not necessarily differentiable. The fact that $D_{N}$ is badly approximated by a ball is reflected in the continuity properties of $r(\theta)$.

Using the notation of section 4.1, let $C_{j}$ be the simplex

[TABLE]

$C_{j}$, for $j<N-1$, is a face of $C_{N-1}$. Denote by $\bar{C}_{j}$ the bari-center of $C_{j}$,

[TABLE]

$\bar{C}_{N-1}=0$ represents the fully mixed state, by Eq. (4.2).

The three points $\bar{C}_{0},\bar{C}_{N-2}$ and $\bar{C}_{N-1}$ define a triangle, shown in Fig. 11. The sides of the triangle can be easily computed, e.g.

[TABLE]

Since $\mathbf{v}_{0}$ represents a pure state. Similarly

[TABLE]

Consider the path from $\bar{C}_{N-2}$ to $\bar{C}_{0}$. The path lies on the boundary of $D_{N}$. Therefore $r(\theta)$ in the figure is the radius function.

By the law of sines

[TABLE]

and

[TABLE]

It follows that when $N$ is large, the radius function has large derivatives near the vertices of the simplex. This reflects the fact that locally $D_{N}$ is not well approximated by a ball.

Remark 5.1**.**

One can show that $Max\,|r^{\prime}(\theta)|\leq O(N)$ is tight. But we shall not pause to give the proof here.

In fact, $r(\theta)$ is, in general, an $(N-1)$-Lifshitz function. This means:

[TABLE]

To show the inequality, let $r_{1,2}$ be two radii to the boundary of $D_{N}$ so that

[TABLE]

as both lie outside the inscribed ball of states, see fig. 12. Let $\tilde{r}_{2}$ denote the intersection of $r_{2}$ with the convex hull of $r_{1}$ and the inscribed ball. Clearly

[TABLE]

with $\alpha$ defined in fig. 12. Since

[TABLE]

we conclude that

[TABLE]

and the claim in Eq. (5.8), follows.

6 A tiny ball in most directions

A basic principle in probability theory asserts that while anything that might happen will happen as the system gets large, certain features become regular, universal, and non-random [22, 8]. We shall use basic results from random matrix theory [7, 23] to show that when $N$ is large $D_{N}$ approaches a ball whose radius is

[TABLE]

Although the radius of the ball is small when $N$ is large, it is much larger than the inscribed ball whose radius is $r_{0}^{2}$.

6.1 Application of random matrix theory

Define a random direction $\boldsymbol{\theta}$ by a vector of iid Gaussian random variables:

[TABLE]

where ${\mathcal{N}}\left[\mu,\sigma^{2}\right]$ denotes the normal distribution with mean $\mu$ and variance $\sigma^{2}$.

$\boldsymbol{\theta}$ has mean unit length

[TABLE]

and small variance

[TABLE]

$S(\boldsymbol{\theta})=\boldsymbol{\theta}\cdot\boldsymbol{\sigma}$ may be viewed as an element of the ensemble of the $N\times N$ traceless Hermitian random matrices.

By Wigner semi-circle law, when $N$ is large, the density $\mu$ of eigenvalues approach a semi-circle

[TABLE]

with edges at

[TABLE]

When $N$ is large, the bottom of the spectrum, $\lambda_{1}(\boldsymbol{\theta})$, a random variable, is close to the bottom edge at $-2$ [23]. The radius function $r(\theta)$ is related to the lowest eigenvalue $\lambda_{1}(\boldsymbol{\theta})$ by Eq. (3.13) and the value for $r_{t}$ in Eq. (6.1) follows111111Since $\boldsymbol{\theta}$ is a unit vector only on average, there is a slight relative ambiguity in $r$. The error is smaller than the $N^{-2/3}$ fluctuation in the Tracy-Widom distribution [23]..

Remark 6.1**.**

To see where the value of $\Lambda$ comes from, observe that by Eq. (6.3)

[TABLE]

6.2 Directions associated with states with substantial purity are rare

Our aim in this section is to show that the probability for finding directions where $r(\theta)\geq r_{0}$ is exponentially small,121212Note that $r_{0}=2r_{t}$ with $r_{t}$ the radius of the ball determined by random matrix theory. This is an artefact of the method we use where the radius of inertia $r_{e}$ plays a role. When $N$ is large $r_{0}\approx r_{e}$. The stronger result should have $r_{0}$ replaced by $r_{t}$ in Eq. (6.6). (see Fig. 13). More precisely:

[TABLE]

In particular, states that lie outside the sphere of radius $r_{0}$ have super-exponentially small measure in the space of directions.

To see this, let $d\mu$ be any (normalized) measure on $D_{N}$. From Eq. (3.11) we get a relations between the average purity and the average radius:

[TABLE]

In the special case that $d\mu$ is proportional to the Euclidean measure in $\mathbb{R}^{N^{2}-1}$, i.e.

[TABLE]

where $d\Omega$ is the angular part, the lhs of Eq. (6.7) is known exactly131313In Appendix A we show how to explicitly compute the average purity for measures obtained by partial tracing. [24]:

[TABLE]

This gives for the radius of inertia of $D_{N}$:

[TABLE]

We use this result to estimate the probability of rare direction that accommodate states with substantial purity. From Eq. (6.10) we have

[TABLE]

Cancelling common terms we find

[TABLE]

By Hölder inequality

[TABLE]

Picking

[TABLE]

gives

[TABLE]

And hence,

[TABLE]

It follows that

[TABLE]

This gives Eq. (6.6).

Remark 6.2**.**

The inequality Eq. (6.6) implies that

[TABLE]

independent of random matrix theory. This implies that $r_{t}\leq r_{0}$, which is weaker by factor 2 compared with Eq. (6.1)).

7 Separable and entangled states

7.1 Why separability is hard

Testing whether $\rho$ is a state involves testing the positivity of its eigenvalues. The cost of this computation is polynomial in $N$. Testing whether $\rho$ is separable is harder. Properly formulated, it is known to be NP-hard, see e.g. the review [25]. Algorithms that attempt to decide whether $\rho$ is separable or not have long running times.

A pedestrian way to see why separability might be a hard decision problem is to consider the toy problem of deciding whether a given point $\mathbf{x}\in\mathbb{R}^{d}$ lies inside a polygon. The polygon is assumed to contain the origin and is given as the intersection of $M$ half-spaces, each of which contains the origin (see fig. 14). This can be formulated as141414$\mathbf{c}_{\alpha}$ is, in general, not normalized to 1.

[TABLE]

To decide if a point $\mathbf{x^{\prime}}$ belongs to the polygon, one needs to test $M$ inequalities. The point is that $M$ can be very large even if $d$ is not. For example, in the poly-octahedron $M=2^{d}$ and the number of inequalities one needs to check is exponentially large in $d$.

Locating a point in a high-dimensional polygon is related to testing for separability [14]: The separable states can be approximated by a polyhedron in $\mathbb{R}^{N^{2}-1}$ whose vertices are chosen from a sufficiently fine mesh of pure product states. Since the number $M$ of half-spaces could, in the general case, be exponentially large in $N$. Testing for separability becomes hard.

Myrheim et. al. [3] gave a probabilistic algorithm that, when successful, represents the input state as a convex combination of product states, and otherwise gives the distance from a nearby convex combination of product states. The algorithm works well for small $N$ and freely available as web applet [26].

7.2 Completely separable simplex: Classical bits

The computational basis vectors are pure products, and are the extreme points of a completely separable $(N-1)$-simplex ($N=2^{n}$). The computational states represent classical bits corresponding to diagonal density matrices:

[TABLE]

The simplex is interpreted as the space of probability distributions for classical $n$ bits strings: $\rho_{\alpha}$ is the probability of the $n$-bits string $\alpha\in\{0,\dots,{N-1}\}$.

7.3 Entangled pure states

Pure bi-partite states can be put into equivalence classes labeled by the Schmidt numbers, [1], leading to a simple geometric description.

Write the bipartite pure state in $\mathbb{C}^{N_{1}}\otimes\mathbb{C}^{N_{2}}$, ($N_{1}\leq N_{2}$), in the Schmidt decomposition [1],

[TABLE]

with $p_{j}\geq 0$ probabilities. The simplex

[TABLE]

has the pure product state as the extreme point

[TABLE]

All other points of the simplex represent entangled states. The extreme point

[TABLE]

is the maximally entangled state. Most pure states are entangled. (In contrast to the density matrix perspective, where by Eq. (1.11), the separable states are of full dimension.)

We denote by $\left|0\right\rangle$ the standard maximally entangled state151515We henceforth set $N_{1}=N_{2}=M$ to simplify the notation.:

[TABLE]

Let $\sigma_{\mu}$ be $M^{2}$ hermitian and mutually orthogonal $M\times M$ matrices i.e.

[TABLE]

The projection $P_{0}=\left|\beta_{0}\right\rangle\!\left\langle\beta_{0}\right|$ can be written in terms of $\sigma_{\mu}$ as:

[TABLE]

In the case of qubits and $N=M^{2}$, a complete set of mutually orthogonal projections on the $N$ maximally entangled states is:

[TABLE]

This is a natural generalization of the Bell basis of two qubits, to $2n$ qubits.

In the two qubits case, $M=2$, an equal mixture of two Bell states, is a separable state161616Here $\sigma_{\mu}$ are the usual Pauli matrices and in particular $\sigma_{0}$ is the identity:

[TABLE]

The two terms on the last line are products of one dimensional projections, and represent together a mixture of pure product states.

7.4 Two types of entangled states

Choosing the basis $\sigma_{\alpha}$ made with either symmetric real or anti-symmetric imaginary matrices, makes partial transposition a reflection in the anti-symmetric coordinates

[TABLE]

Partial transposition [27, 1] distinguishes between two types of entangled states:

• •

$\rho\geq 0$ while $\rho^{pt}$ is not a positive matrix.

• •

Both $\rho,\rho^{pt}\geq 0$ but $\rho$ is not separable.

In the case that $\rho$ is a pure state or171717Simple geometric proofs for two qubits are given in [3, 4]. $N_{1}N_{2}\leq 6$, only the first type exists [16].

The Peres181818Gurvits and Barnum attribute the test to an older 1976 paper of Woronowicz. Apparently, nothing is ever discovered for the first time (M. Berry’s law). entanglement test [27] checks the non-positivity of $\rho^{pt}$ and uncovers entangled state of the first type, (see Fig. 15). Local operations can not convert states of the second kind into states of the first kind since positivity of partial transposition of $\rho$ implies the positivity under partial transposition of a local operation

[TABLE]

In particular, one can not distill Bell pairs from entangled states of the second kind by local operations. This is the reason why states of the second kind are called “bound entangled”: The Bell pairs used to produce them can not be recovered.

7.5 Entangled low purity states

Most pure states are entangled (see section 7.3). One would therefore expect most high purity states to be entangled but we shall not attempt to make this statement quantitative. Entangled states persist to low purity states, this can be seen from the following construction, due to [29], which partitions the Hilbert space into a direct sum of spaces, so the one of the sub-spaces contains only entangled states.

Let

[TABLE]

be a generic set of vectors. Generically, any set of $N_{1}$ vectors $\left|\psi_{j}\right\rangle$ is linearly independent, and so is any set of $N_{2}$ vectors $\left|\phi_{k}\right\rangle$. The linear span of

[TABLE]

is a linear subspace of $\mathbb{C}^{N_{1}}\otimes\mathbb{C}^{N_{2}}=\mathbb{C}^{N_{1}N_{2}}$ which is generically of dimension $N_{1}+N_{2}-1$. Let $P$ denote the projection on this space. The orthogonal complement of $P$ contains no pure product states since

[TABLE]

implies that $\left|\psi\otimes\phi\right\rangle=0$. (A non zero $\left|\psi\right\rangle$ can be orthogonal to at most $N_{1}-1$ vectors $\left|\psi_{j}\right\rangle$, and $\left|\phi\right\rangle$ to at most $N_{2}-1$ vectors $\left|\phi_{j}\right\rangle$. This is not possible for more than $N_{1}+N_{2}-2$ elements $\left|\psi\right\rangle_{j}\otimes\left|\phi\right\rangle_{j}$.)

A state $\rho$ in $\mathds{1}-P$ has the property that $\text{Image}(\rho)$ has no pure product states. However, any separabale state

[TABLE]

(made from linearly independent terms) has in its image any one of the pure product states

[TABLE]

It follows that any state in $\mathds{1}-P$ is entangled. In particular, this is the case for

[TABLE]

whose purity is

[TABLE]

Its distance squared from the fully mixed state is

[TABLE]

It follows that the radius of the largest ball of separable state must be smaller than $O\left(N^{-3/4}\right)$ when $N$ is large.

7.6 The largest ball of bi-partite separable states

The Gurvits-Barnum ball was introduced in section 3.4 as the largest inscribed ball in $D_{N}$:

[TABLE]

Since partial transposition is a reflection in the $\sigma_{\alpha}$ coordinates, and any sphere centered at the origin is invariant under reflection, we have that

[TABLE]

$B_{gb}$ therefore does not contain entangled states that are discoverable by the Peres test.

Gurvits and Barnum replace partial transposition by contracting positive maps of the form $\mathds{1}\otimes\phi(D)$ to show [6], that $B_{gb}$ is a ball of bi-partite separable states

[TABLE]

7.7 Entanglement witnesses

An entanglement witness for a given partition, $(N_{1},\dots,N_{n})$, is a Hermitian matrix $W$ so that

[TABLE]

This definition makes the set of witnesses a convex cone (see Fig. 16).

Remark 7.1**.**

We consider $W\geq 0$ a witness even though it is “dumb” as it does not identify any entangled state. This differs from the definition used in various other places where witnesses are required to be non-trivial, represented by an indefinite $W$. Non-trivial witnesses have the drawback that they do not form a convex cone.

The inequality, Eq (7.22), is sharp for $\rho$ in the interior of $S_{N_{1},\dots N_{n}}$. As the fully mixed state belongs to the interior

[TABLE]

we may normalize witnesses to have a unit trace and represent them, alongside the states, by

[TABLE]

We shall show that:

[TABLE]

This follows from

[TABLE]

and

[TABLE]

Example 7.1**.**

A witness for the partitioning $N=N_{1}N_{2},\ N_{1},N_{2}\geq M>1$, is:

[TABLE]

$S$* is indeed an entanglement witness since*

[TABLE]

Since $Tr\,{S}=M$, the corresponding normalized witness is

[TABLE]

and the equality on the right follows from the first line of Eq. (7.3).

Since partial transposition is an isometry, it is clear that $W_{S}$ lies at the same distance from the maximally mixed state as the pure state $\left|\beta\right\rangle\left\langle\beta\right|$. In particular, the associated vector $\mathbf{w}_{S}$ lies on the unit sphere.

Remark 7.2**.**

In the case that the partitioning is to two isomorphic Hilbert spaces, $N_{1}=N_{2}$, $S$ is the swap. In a coordinate free notation

[TABLE]

$\left|\beta\right\rangle$* is then the Bell state.*

7.8 Entangled states and witnesses near the Gurvits-Barnum ball

Near the boundary of $B_{gb}$ one can find entangled states and (non-trivial) entanglement witnesses, see Fig. (17).

To see this, consider the bi-partitionning $N=M^{2}$. Since $S^{2}=\mathds{1}$ and $S=S^{*}$,

[TABLE]

are orthogonal projections. $P_{S}$ projects on the states that are symmetric under swap, and $P_{A}$ on the anti-symmetric ones. Hence,

[TABLE]

The state

[TABLE]

is entangled with the swap as witness. Indeed,

[TABLE]

When $\varepsilon$ is small, $\rho$ is close to the Gurvits-Barnum ball. One way to see this is to compute its purity

[TABLE]

Using Eq. (3.11) to translate purity to the radius one finds, after some algebra,

[TABLE]

Since partial transposition is an isometry, $\rho^{pt}$ is also near the Gurvits-Barnum ball. It is an entanglement witness for the Bell state:

[TABLE]

and we have used Eq. (7.30) in the last step.

7.9 A Clifford ball of separable states

The separable states, in $\mathbb{C}^{N}=\mathbb{C}^{M}\otimes\mathbb{C}^{M}$ contain a $2\ell-1$ ball with radius $1/\sqrt{N-1}$. $\ell=O(\log M)$, is the (maximal) number of anti-commuting (generalized) Pauli matrices $\sigma_{\mu}$, acting on $\mathbb{C}^{M}$, (see Eq. (4.4)). We call this ball the Clifford ball. It is larger than the Gurvits-Barnum ball, whose radius is $1/(N-1)$, but it lives in a lower dimension.

A standard construction of $2\ell-1$ anti-commuting Pauli matrices, acting on $\mathbb{C}^{N},\ N=M^{2}$, from $\ell$ anti-commuting matrices acting on $\mathbb{C}^{M}$ is

[TABLE]

Consider the $2\ell-1$ dimensional family of quantum states in $\mathbb{C}^{N}$, parametrized by191919Note the change of normalization relative to Eq. (3.1). $\mathbf{a,b}\in\mathbb{R}^{\ell}$

[TABLE]

where $\boldsymbol{\sigma}=(\sigma_{1},\dots,\sigma_{\ell})$ is a vector of $M\times M$ (generalized, anti-commuting) Pauli matrices. We shall show that for $\mathbf{a}^{2}+\mathbf{b}^{2}\leq 1$, with $b_{\ell}=0$, $\rho$ is essentially equivalent to a family of 2-qubit states, which are manifestly separable. Re-scaling the $\mathbf{a,b}$ coordinates to fit with the convention in Eq.(3.1) gives the radius $r_{0}=1/\sqrt{N-1}$.

Note first that since

[TABLE]

we may write

[TABLE]

where $Z$ is the standard $2\times 2$ Pauli matrix. Similarly, since

[TABLE]

we may write

[TABLE]

The numerator in Eq. (7.40) takes the form

[TABLE]

The brackets can be written as:

[TABLE]

$\left(\mathds{1}\pm{|\mathbf{a}|}Z+{|\mathbf{b}|}X\right)\geq 0$ when $\mathbf{a}^{2}+\mathbf{b}^{2}\leq 1$, and the resulting expression for $\rho$ is manifestly separable. Rescaling the coordinates to agree with the normalization in Eq. (3.1), gives the radius $r_{0}=1/\sqrt{N-1}$ for the Clifford ball of separable state.

Acknowledgement

The research has been supported by ISF. We thank Yuval Lemberg for help in the initial stages of this work and Andrzej Kossakowski for his encouragement.

Appendix A The average purity of quantum states

In section 6.2 we quoted a result of [24], Eq. (6.9), which allows to explicitly compute the radius of inertia $r_{e}$ of $D_{N}$ as a rational function of $N$. The aim of this appendix is to give an elementary derivation of this formula.

The measure $d\mu$ on the space of density matrices in section 6.2 is a special case of a more general measures $d\mu_{N,K}$ when $N=K$. The measures $d\mu_{N,K}$ are the induced measure on density matrices acting on $\mathbb{C}^{N}$, obtained from the uniform measure on pure states on $\mathbb{C}^{N}\otimes\mathbb{C}^{K}$ with $K\geq N$, by partial tracing over the second factor202020In the case $K=N-1$ the measure is concentrated on the boundary of $D_{N}$ being proportional to $\delta(\det\rho)$.. They are all simply related [24]

[TABLE]

$Z_{N,K}$ is a normalization factor. The Euclidean measure $d\mu$ manifestly corresponds to the case $N=K$.

The derivation given below of Eq. (6.9) is simpler than the original derivation in [24] in that it avoids the constraint associated with the normalization of the wave functions. This allows to reduce the computation of moments of $\mathbf{x}^{2}$ with respect to the measure $d\mu_{N,K}$ to an exercise in Gaussian integration.

Let $\langle\alpha j|\psi\rangle=\xi_{\alpha j}$ be the amplitudes of the pure state $\left|\psi\right\rangle$ in $\mathbb{C}^{N}\otimes\mathbb{C}^{K}$. The first factor, $\alpha$, is the system and the second, $j$, is the ancila. The density matrix $\rho$ is obtained by partial tracing the ancila:

[TABLE]

where $\xi$ is an $N\times K$ matrix. The requirement $K\geq N$ guarantees that (generically) $\rho$ is full rank.

Choosing $Re\,\xi_{\alpha j}$ and $Im\,\xi_{\alpha j}$ to be normally distributed i.i.d., gives a uniform measure on pure states, $d\mu_{\psi}$, which is unitary invariant under $U(NK)$. The induced measure on the density matrices $d\mu_{N,K}$ is:

[TABLE]

Since the Gaussian measure for $\xi$ allows for states that are not normalized, the measure $d\mu_{N,K}$ allows for any $Tr\rho\geq 0$. This means that to compute the moments of normalized density matrices we need to compute an integral of a ratio:

[TABLE]

Remarkably, this reduces to computation of two Gaussian integrals. The reason for this is that the measure factors into radial and angular part:

[TABLE]

Since $Tr\rho=\|\psi\|^{2}$ is invariant under an arbitrary unitary in $\mathbb{C}^{N}\otimes\mathbb{C}^{K}$, and such unitaries act on the measure $d\mu_{N,K}$ as rotations, $Tr\rho$ depends only on the first, radial, coordinate while $\frac{Tr(\rho^{2})}{(Tr\rho)^{2}}$ depends only on the second, angular, part. Hence

[TABLE]

(Assuming $d\mu_{NK}$ normalized.) The integral in Eq. (A.4) is therefore the ratio of two Gaussian integrals. Wick theorem (for the standard normal distribution) gives

[TABLE]

Similarly

[TABLE]

So finally,

[TABLE]

This reduces to Eq. (6.10) when $N=K$.

The computation of higher moments can be similarly reduced to a (tedious) combinatoric problem.

Appendix B The N dimensional unit cube is almost a ball

The fact that $D_{N}$ looks like a ball in most directions is a general fact about convex bodies in high dimensions. It is instructive to see this happening for the unit cube in $N$ dimensions

[TABLE]

The radius of inertia of $C$ is

[TABLE]

Let us now consider $r(\theta)$, defined as the maximal $r$ that is inside the cube for a given direction $\theta$.

Choose a random direction $\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{N})$ by picking $\theta_{j}$ to be normal iid with

[TABLE]

When $N$ is large there is a “phase transition” in the sense that

[TABLE]

To see this we first observe that the probability that $x\sim{\mathcal{N}}\left(0,\sigma^{2}\right)$ takes values outside the interval $[-x_{0},x_{0}]$ is given by the complementary error function

[TABLE]

Anticipating the result, Eq. (B.4), let us replace $r$ by its re-scaled version $k$:

[TABLE]

For the case at hand

[TABLE]

The probability for $r|\theta_{j}|=|x_{j}|\leq 1/2$ happening for all coordinates simultaneously is

[TABLE]

Since ${\rm erfc}(x)\in[0,1]$ for $x\in[0,\infty)$, the limit $N\to\infty$ tends to

[TABLE]

Since

[TABLE]

Eq. (B.4) follows.