ICTS 2025 - Tensor norms for quantum entanglement

My name is Ion Nechita. I am a researcher at the CNRS, LPT Toulouse, part of the Tiq-Toqs team. Here's my webpage and my X account.

Most of the material in these lectures is from standard textbooks and review papers. Some of my favorites are:

Alice and Bob meet Banach by Guillaume Aubrun and Stanislav Szarek
The Theory of Quantum Information by John Watrous
Graphical tensor notation for interpretability by Jordan Taylor

I will be happy to discuss with you in person, or by email.

Overview of the three lectures

The goal of these lectures is to introduce you to the framework of tensor norms and to their applications in quantum information theory.

Here are some results I will present in detail:

given two vector spaces, each one endowed with a norm, there are several natural ways of extending these norms the tensor product of these spaces. Among those, there is a minimal choice (called the injective norm), and a maximal choice (called the projective norm)
in the case of matrices, the operator norm and the nuclear norm are, respectively, the smallest and the largest tensor norm one can put on the tensor product between the input and output spaces
in the case of a tensor product of Hilbert spaces ( normed spaces), the injective norm of a tensor is closely related to the geometric measure of entanglement of the multipartite quantum state the tensor defines
understanding the entanglement of multipartite pure states is active area of research; the most entangled states are known only in several special cases. There are new surprising results about the entanglement of random multipartite pure quantum states
in the case of multipartite mixed quantum states, the projective tensor product of the matrix spaces endowed with the nuclear norm is relevant to entanglement
the two most important entanglement criteria for mixed quantum states (the PPT criterion and the realignment criterion) can be readily found from the tensor norm formulation of entanglement, by trying to reduce the number of tensor products in the formulas
one can study other entanglement criteria in this framework, using the notion of entanglement testers

You will learn about the following topics:

normed spaces
a bit of convex geometry to describe their unit balls
classical examples: spaces, Schatten classes
duality of normed spaces; polarity for convex sets
tensors
tensor rank
graphical notation for tensors
tensor norms; the minimal (injective) and maximal (projective) tensor norms
geometric measure of entanglement for pure quantum states
entanglement for mixed quantum states

Normed spaces

A normed space is a vector space over or endowed with a norm
The norm map has the following properties:

positive homogeneity:
triangle inequality:
faithfulness: .

To a normed space we associate its unit ball
Importantly, one can recover the norm from unit ball as follows:
where is the scaling of the unit ball by a (positive) factor .

The unit ball is a convex set that is centrally symmetric: . From the two formulas above we can see that there is a one-to-one correspondance between norms and centrally symmetric convex sets.

Remark

Studying normed spaces is, loosely speaking, equivalent to studying the convex geometry of their unit balls.

Example: spaces

These are the most important normed spaces that you will encounter. In the real case, we have
with
There are three important cases:

the norm:
the or the euclidean norm:
the norm:

The complex case is very similar, adding absolute values in the formulas above when needed. Of special importance in the complex situation is the case : pure quantum states are unit-norm vectors in the Hilbert space ; here, is the number of degrees of freedom of the system we model:
The unit balls of the three spaces above are called, respectively:

In the case of , they look as follows:
Pasted image 20250201122850.png

In :
Pasted image 20250201123611.png

Example: spaces

The Schatten classes are the equivalent of the spaces introduced previously for matrices. Here, the vector space is a matrix space, usually

real matrices
complex matrices
self-adjoint complex matrices , i.e. complex matrices with ; this is a -vector space
symmetric real matrices :

The Schatten norms can either be defined as
or as the norm of the singular value vector of as
We write . The Schatten norms can be seen as non-commutative analogues of the norms: if , then .

Let us recall here the singular value decomposition of matrices, arguably the most important result in linear algebra. Given a matrix , there exist:

unitary matrices
non-negative numbers called the singular values of
such that
where we set .
Geometrically, the singular values measure how the operator distorts vectors. For example, in the case of , a linear operator send the unit ball of (a circle), to an ellipse. The two singular values of are the large, respectively the small, semi-axes of the ellipse.

Going back to the Schatten norm, let us emphasize the three most important cases:

, the nuclear norm, or the trace norm
, the Frobenius norm
, the operator norm or simply, THE norm

Question

What are

your favorite
other important

norms I could have mentioned?

Answer in class

The diamond norm.

Duality

The notion of duality fundamental in the study of normed spaces (or, more generally, Banach spaces). The corresponding notion for convex sets, polarity, plays an equally important role in convex geometry.

At the level of vector spaces, recall that the dual space contains the linear forms on :
At the level of normed spaces, to a norm on , we associate a dual norm on by
The normed space is called the dual space of .

A scalar product naturally induces a map by
The standard scalar products on , , induce the following fundamental isomorphisms of normed spaces:
where are conjugate coefficients:

At the level of convex sets, the corresponding notion is defined as follows. Given a convex set , define its dual (or polar set):
We have the following correspondance:
In particular:

the euclidean balls (disk, sphere, etc) are self-dual
the dual of the (hyper)cube is the cross-polytope.

The duality relation has many interesting properties. For example, it naturally defines a bijection between extremal points of the primal set and facets of the dual set. In the case of balls:

ball ofUnit	Number of extremal points	Number of facets
, cross-polytope
, euclidean ball
, hypercube

Mixed quantum states

In standard quantum mechanics textbooks, quantum states are represented by unit vectors inside a complex Hilbert space . In many applications, the Hilbert space is composed of two parts: a subsystem of interest and an environment that is generally much larger than :
We are interested on observables defined on the system of interest and wish to discard the environment of which we have no knowledge. If the total Hilbert space is in a (pure) state , the average value of some observable is given by
where is the reduced density matrix of on the system :
Hence, all the observable averages pertaining to the system of interest can be computed from the reduced density matrix . Since the global state is a unit vector, is a positive semidefinite, unit trace matrix.

A mixed quantum state (or a density matrix) is a positive semidefinite operator of trace . We denote the set of density matrices on a Hilbert space of dimension by
We recall that a self-adjoint matrix is called positive semidefinite if all its eigenvalues are non-negative, or, equivalently,

Apart the normalization condition (which, for most problems, is irrelevant and/or easy to deal with), the main feature of (mixed) quantum states is the positive semidefinite condition above. Importantly for us, this condition can be stated with the help of the (or nuclear) norm of the density matrix, as follows.

Remark

A self-adjoint matrix is positive semidefinite iff

The case of qubit density matrices () is quite remarkable. The space of self-adjoint matrices of size has dimension (as a real vector space). In the case , a standard basis of the space is given by the Pauli matrices

Any such matrix of unit trace can be thus decomposed as
for some real parameters . The semidefinite condition reads
Geometrically, this means that the set of qubit density matrices is precisely the unit ball of the euclidean space ; this is known as the Bloch ball:
Pasted image 20250201125459.png

More generally, we have the following isomorphism of normed spaces:

Application: distinguishing quantum states

Consider the following state discrimination scenario:

Fix two arbitrary quantum states and a parameter
Alice prepares a random quantum state as follows:
- with probability , Alice prepares
- with probability , Alice prepares
Alice sends the quantum state to Bob
Bob's goal is to determine whether Alice prepared the state or

Bob receives an unkown quantum state and has to decide whether or . The most general strategy Bob can use is for him to perform a quantum measurement and to declare that

Alice had sent if the measurement outcome corresponds to the operator
Alice had sent if the measurement outcome corresponds to the operator

Using a measurement defined by an operator as above, the probability that Bob guesses correctly is
Bob's best strategy is obtained by optimizing over the measurement he performs. The following result is due to Holevo and Helstrom.

Theorem

Bob's best guess proability is

Proof

Let us first prove that the RHS of the equation above provides an upper bound for the success probability of any strategy Bob might use. The form of the optimal will follow from the computations below.

Let a quantum effect and compute:
Let us perform the following optimization, from some general self-adjoint operator :
where is the (unique) decomposition of as a difference of positive semidefinite operators with orthogonal supports. The optimal above is the projection on the support of .

Back to our state distinguishability problem, we have obtained
Use the fact that
and, in general :
to compute
from which the conclusion follows easily.

Note that in the unbiased case we have two extreme cases:

: Bob's best strategy is a random coin flip
: Bob can perfectly distinguish the two quantum states by performing a projective measurement on their supports.

Remark

The Holevo-Helstrom theorem shows that the nuclear norm (or the 1-norm, or Schatten's 1-norm) is the "correct" norm to use when trying to measure how different two quantum states are.

The nuclear norm is contracting under quantum channels (trace preserving, completely positive maps):
In particular, this means that applying a quantum channel to a pair of states renders them less distinguishable.

Question

How about more than two states? Alice produces, with probability , one of given quantum states . What is Bob's optimal strategy and his maximum success probability? Is the latter given by a norm involving the states ?

Hint

Discriminating between multiple states and the pretty good measurement.

Question

Consider the scenario with two possible states but now Alice prepares copies of the state and sends to Bob. How well can Bob do in this situation, in the regime where ?

Hint

The relative entropy appears as a large deviations exponent:
Pasted image 20250203114831.png
from The tangled state of quantum hypothesis testing.

Tensors

A tensor can mean many things, depending on the person you ask.

Here are DeepSeek-R1-Distill-Qwen-7B's answers when prompted as follows:

what is a tensor? give me a one-sentence definition
``A tensor is a mathematical object representing a multi-dimensional array of numerical values, which generalizes scalars, vectors, and matrices into higher dimensions, often used in physics and machine learning.
you are a nobel prize winning physicist. what is a tensor? give me a one-sentence definition
``A tensor is a mathematical generalization of scalars, vectors, and matrices, representing multi-dimensional quantities with specific transformation rules under coordinate changes, essential for describing complex physical systems involving multiple dimensions or directions.
you are an abel prize winning mathematician. what is a tensor? give me a one-sentence definition
``A tensor is a versatile mathematical object that generalizes concepts from vector calculus, serving as a bridge between geometry and algebra by efficiently representing complex, multi-dimensional data and transformations. It can be thought of as an extension of vectors and matrices into higher dimensions, used across various fields including physics, engineering, and machine learning to model intricate systems and phenomena with precision.
you are a turing prize winning computer scientist. what is a tensor? give me a one-sentence definition
``A tensor is a mathematical object that extends the concept of scalars, vectors, and matrices to an arbitrary number of dimensions, enabling efficient representation and manipulation of multidimensional numerical data.

In this lecture, a tensor will simply be an element of a tensor product space
where are (real or complex) vector spaces. The integer is called the order of the tensor .

Here is one way (computer science-y and not very nice from a mathematical point of view) to define the tensor product of vector spaces. For every , consider a basis
of the vector space . Then, the formal elements
form a basis of the tensor product vector space ; above, the index runs over the cartesian product
More generally, the space is spanned by the set of simple tensors, i.e. elements of the form
where, for all , .

You are familiar with the tensor product space which is the Hilbert space of two qubits. If we denote by the bases of the two copies of , then a basis of is given by
This is the point of view that is taken in computer science and programming: tensors are arrays of numbers that are indexed by -tuples of integers.

Here is a more mathematical definition of the tensor product; we state it for two vector spaces for the sake of readability. Let and be two vector spaces. The tensor product is a vector space equipped with a bilinear map such that for any other vector space and any bilinear map , there exists a unique linear map making the following diagram commute:
The rank of the matrix multiplication tensor represents the minimal number of scalar multiplications needed to compute the product of two matrices. In more detail, if the tensor can be expressed as a sum of simple (rank-1) tensors, then there exists an algorithm that multiplies matrices using only multiplications. This directly ties the tensor’s rank to the efficiency of matrix multiplication algorithms: a lower tensor rank means fewer multiplications, which in turn leads to a faster algorithm compared to the standard method. This insight is the foundation behind fast matrix multiplication techniques like Strassen’s algorithm, which reduce the overall computational complexity by exploiting low-rank decompositions.
Pasted image 20250203233300.png

This universality property ensures that the tensor product captures in a minimal and unique way all bilinear information from into a linear structure on .

Tensors can be seen as higher order matrices. Indeed, we have the following hierarchy of mathematical objects as a function of the order :

: scalars, in or
: vectors, ,
: matrices, ,
: tensors of higher order, , .

Duality and linear maps

We said earlier that matrices are order 2 tensors. This is true if one considers matrices as an array of numbers, i.e. a linear map expressed in some basis.

From a theoretical perspective, expressing linear maps using coordinates is not recommended. Instead, consider a linear transformation between two vector spaces
We claim that we can see, canonically, the map as an element of the tensor product ; in other words, there is an isomorphism:
There are several ways to see this.

Firstly, let us use coordinates. If and are, respectively, bases for the vector spaces and , then one can write the action of as
Equivalently, this corresponds to the following decomposition of in the product basis of :

Secondly, there is a more mathematically elegant way to understand the isomorphism above. Consider the evaluation tensor
defined canonically, on simple tensors, as
Given a vector , the output can be seen as
Thirdly, the isomorphism can be understood at the level of rank one matrices, which correspond to unit tensors. Let us consider here the real case, to avoid the complications of complex conjugates. A rank one matrix corresponds canonically to the tensor , where now is seen as the linear map

More generally, linear applications
are canonically associated to tensors

Example: the unit tensor

The unit tensor is defined, for some order and local space dimension , as
having coordinates

For , we get the all ones vector
For , we get the identity matrix

For , using bra-ket notation, we have the tensor
Identifying one of the copies of with its dual, we recognize the copy tensor
that creates two copies of the input classical information. The map
does not copy quantum information perfectly, see the no-cloning theorem.

Example: GHZ and W states

The GHZ (Greenberger-Horne-Zeilinger) state is one of the paradigmatic examples of a maximally entangled state in the multipartite setting. It is defined by
In tensor notation we often write this compactly as
Notice that the GHZ state is a superposition of two simple tensors, and is equal to the unit tensor with and :
The W state, is another important 3-qubit state. It is given by
In terms of tensor components, the W state has nonzero coefficients only for those indices with exactly one occurrence of the state . That is
Note that both of these tensors are symmetric: they are invariant under leg permutation. More precisely, they are invariant under the action of the symmetric group . In the general setting, acts on as follows:
If for all we say that is symmetric, .

Graphical notation for tensors

The following is (slightly) adapted from an answer on math.stackexchange by Jordan Taylor, see also the arXiv paper, Wikipedia, or Hand-waving and interpretive dance: an introductory course on tensor networks.

In graphical notation, tensors are represented as shapes (small disks below) with legs sticking out of them. A vector can be represented as a shape with one leg, a matrix can be represented as a shape with two legs, and so on:

Pasted image 20250203152625.png

Each leg corresponds to an index of the tensor - specifying an integer value for each leg of the tensor addresses a number inside of it:

Pasted image 20250203152652.png

where 0.157 happens to be the number in the position of the tensor (note that computer scientists index arrays starting at when coding in python). The amount of memory required to store a tensor grows exponentially with the number of legs, so tensors with lots of legs are usually represented only implicitly: decomposed as operations between many smaller tensors.

Connecting the legs of two tensors indicates a tensor contraction (also known as an Einstein summation). Here are the most common kinds of contractions between vectors and matrices:

Pasted image 20250203153001.png

In every case you can tell how many legs (i.e. what is the order) the resulting tensor will have by the number of uncontracted "free" legs on the left column.

But graphical notation is most useful for representing unfamiliar operations between many tensors. One example in this direction is
which can be represented in graphical notation as

Pasted image 20250203153222.png

The middle part of the graphical notation here shows that the number in each position of the final matrix can be calculated with a sum over every possible indexing of the internal legs and , where each term in the sum consists of three numbers being multiplied (though in practice the contraction should be calculated in a much more efficient way).

Graphical notation really comes into its own when dealing with larger networks of tensors. For example, consider the contraction
which is tedious to parse: indices must be matched up across tensors, and it is not immediately clear what kind of tensor (eg. number, vector, matrix ...) the result will be. But in graphical notation this is

Pasted image 20250203153630.png

and we can immediately see which tensors are to be contracted, and that the result will be a single number (no un-contracted, or dangling legs). Contractions like this can be performed in any order. Some contraction orders are much more efficient than others, but they all get the same answer eventually.

Question

What is the most efficient way to multiply matrices , where ? The answer should depend on the dimensions .

Remark

In the general case, contracting a tensor network is known to be a computationally intractable problem: it is -hard. The parameter which controls the difficulty of the problem is the treewidth of the network.

Using the graphical notation, we can denote the unit tensor by a dot :

Pasted image 20250203155921.png

Example: the maximally entangled state

The maximally entangled state (or the Bell state) is the pure bipartite state given by
For qubits () this reads
Graphically, the pure maximally entangled state can be represented as

Pasted image 20250203212539.png

The corresponding maximally entangled density matrix
can be represented graphically as

Pasted image 20250203213343.png

Question

Show graphically that the partial trace of the maximally entangled density matrix is a maximally mixed state.

Example: the matrix multiplication tensor

The matrix multiplication tensor is a compact way of encoding the bilinear operation of matrix multiplication. It serves as a bridge between the algebraic description of multiplying matrices and the study of tensor rank, which is deeply connected to the complexity of matrix multiplication algorithms.

Consider matrices:

with entries (for and ),
with entries (for and ).
The matrix product is given component-wise by: This bilinear map can be represented by a tensor
and is defined by where:
is the standard basis for matrices (with a 1 in the entry and 0 elsewhere),
is the corresponding dual basis.
In the case where , we simply write for the matrix multiplication tensor of matrices.

Let us now use the isomorphism we have seen earlier (assume also that the matrices are real valued):
implemented by and similarly for the other tensor spaces to write
with

where the sign denotes equality up to tensor factor permutation.

Warning

Note that the tensor products in the first and second operation above are not the same!

Graphically, we find the form of the tensor starting from the definition of matrix multiplication

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This implies that

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Entanglement for pure states

In quantum information theory, the study of entanglement in multipartite systems is essential for understanding non-classical correlations between subsystems. A multipartite pure state belonging to the composite Hilbert space

is said to be entangled if it cannot be expressed as a simple tensor product of states from each individual subsystem. In other words, if there are no states such that

then exhibits entanglement; otherwise it is called separable. One recognizes separable states as simple tensors inside the tensor product space of the local Hilbert spaces .

We shall discuss next two ways of quantifying how entangled a pure multipartite quantum state is:

a discrete measure, the tensor rank
a continuous measure, the injective norm.

Tensor rank

Not all elements can be written as a tensor product of elements . One requires in general sums of such elements:
The minimum positive integer for which such a decomposition exists is called the tensor rank of and is denoted by . We can reformulate the separability problem for pure multipartite quantum states as:
In conclusion, the tensor rank is a discrete measure of the entanglement present in a pure multipartite quantum state.

It the bipartite case, unit rank tensors corresponds to unit rank matrices , hence
where is the matrix version of the 2-tensor and is the usual matrix rank.

For example, one can easily show that, for three qubit states,
hence both states are entangled.

Fact

In his seminal work, Håstad (1990) proved that the decision problem of determining whether a given tensor (of order 3) has rank at most is -complete when the field over which the field is defined is finite or when . This result shows that, unlike the matrix rank problem (which is solvable in polynomial time), the tensor rank problem is computationally intractable in general.

Here is another aspect in which tensor rank is very different than the usual matrix rank. In matrix () case, for any , the set
is closed, since it is the intersection of closed sets:
However, this is not true for tensors, as it can be seen from the following example:
Hence, we have written , having rank , as a limit of rank 2 tensors, proving that the set of tensors having rank is not closed.

The rank of the MaMu tensor

Let us focus in this section on square matrices: . The rank of the matrix multiplication tensor is the minimum such that there exist a decomposition
where the tensor products separate the different matrix algebras. Equivalently, such a decomposition can be see as
where collect the vectors , see the graphical representation below:

Pasted image 20250204082611.png

The rank of the matrix multiplication tensor represents the minimal number of scalar multiplications needed to compute the product of two matrices. In more detail, if the tensor can be expressed as a sum of simple (rank-1) tensors, then there exists an algorithm that multiplies matrices using only multiplications. This directly ties the tensor’s rank to the efficiency of matrix multiplication algorithms: a lower tensor rank means fewer multiplications, which in turn leads to a faster algorithm compared to the standard method.

Let us now focus on the case: how many multiplications are needed to compute the product of two matrices? The standard algorith needs 4 multiplications, as it provides a decomposition with 8 terms:

Pasted image 20250125165052.png

In his breakthrough paper from 1969, Volker Strassen showed that the rank of the tensor is at mostx 7:

Pasted image 20250125165238.png

Later in 1971, Winograd showed that decomposition with only 6 term cannot exist, hence
Strassen's algorithm can be used not only for matrices, but also for larger matrices, by decomposing them into blocks. This reduces the complexity of matrix multiplication (of two matrices) from
The asymptotic rank of a tensor is a measure of its complexity when one considers Kronecker tensor powers, and it is defined as

where denotes the tensor rank of . The Kroneker tesor product of two tensors and of order is a tensor of the same order , obtained by grouping the tensor factors as follows: if
then the Kronecker product is the tensor

In the case of the matrix multiplication tensor, the asymptotic rank is deeply connected to the matrix multiplication exponent defined byThis means that the best-known algorithms for multiplying matrices achieve a complexity of scalar multiplications. Current research has established that . Computing the actual value of (thought to be ) is one of the most important open problems in computer science.

Pasted image 20250204083538.png

Recently, progress on similar questions have been achieved by machine learning systems. AlphaTensor improved the best know upper bound for the multiplication of matrices over , lowering the upper bound for the rank from to .

The injective tensor norm as an entanglement measure

One of the key tools for quantifying the entanglement in multipartite pure states is the injective tensor norm. This norm measures the maximum overlap of the state with any product state. Formally, the injective tensor norm is defined as

Here, each is a normalized state in . The norm has the following important properties:

If is a separable (i.e., non-entangled) state, then there exists a product state such that $$
|\langle \psi | \phi_1 \otimes \phi_2 \otimes \cdots \otimes \phi_N \rangle| = 1, |\psi|_{\epsilon} = 1.
$$
For an entangled state, no product state can perfectly overlap with , so $$
|\psi|_{\epsilon} < 1. \left|\psi-\varphi{1} \otimes \ldots \otimes \varphi{n}\right|^{2}=2-2\left|\left\langle \psi \mid \varphi{1} \otimes \ldots \otimes \varphi{n} \right\rangle\right|^{2}
$$

This property makes the injective tensor norm a natural candidate for quantifying how "far" a state is from being separable.

In the bipartite case (), the injective tensor norm of a 2-tensor can be written as
where the supremum is taken over all unit vectors , and is the image of the 2-tensor under the isomorphism
One recognizes above the optimization problem for the operator norm of :
When discussing the general framework of tensor norms, we shall see that the injective norm discussed here is actually the injective tensor product of the norms on each tensor factor Hilbert space.

Question

What is the injective norm of the GHZ and W states for three qubits? Which one is further away from the set of separable states?

The geometric measure of entanglement (GME) is another important concept used to quantify the entanglement of multipartite pure states. It is defined based on the maximal overlap between the state and the set of all product states. One common definition of the geometric measure of entanglement is

The underlying idea is as follows: the closer the state is to being separable (i.e., the larger the injective tensor norm ), the smaller its geometric measure of entanglement, and vice versa. The paper Additivity and non-additivity of multipartite entanglement measures is a great ressource for all question relative to the GME.

Tensor norms

The goal of this section is to show how to construct larger normed spaces out of smaller ones using the tensor product construction.

Motivation: Let be normed spaces.

What norm on

For example: $\|(x, y)\|=\max \left(\|x\|_{X},\|y\|_{Y}\right)$ or $\|(x, y)\|=\|x\|_{X}+\|y\|_{Y}$

What norm on ?

Remember that any norm space comes with a dual , where
and, for ,

Definition

A tensor norm on is a norm such that it agrees with the product on the norms, at the primal and dual levels:

Let us start with a simple example. Take the two spaces and . On , consider the norm . This is indeed a tensor norm:

follows from the first point since the norm is self-dual.

Definition

Let , be given normed spaces. The injective tensor norm is defined by
and the projective tensor norm is defined by

We write

Let us consider some simple examples. Given a linear map , we define the induced norm by

This norm turns into a normed space. Recall that on the vector space level, . On the level of the norms, we have:

Hence, we have shown the following equality of normed spaces.

Important

Induced norms are injective tensor products:

In particular for , we get

The operator norm is the injective tensor product norm of the euclidean (i..e. ) norms on the input and output spaces.

Let us consider now the nuclear norm (or the trace norm) . We claim that it is the projective tensor product norm of the corresponding spaces: . Note that

where is the polar decomposition.

Show :
$$
\begin{aligned}
A & =\sum s{i} a{i} \otimes b{i}^{*} \
& =\sum s{i}\left|a{i}\right\rangle\left\langle b{i}\right|
\end{aligned}
- Show : for all and consider the optimal decomposition giving the norm: . We have

proving the second inequality.

Therefore, the nuclear norm is the projective tensor product norm of the euclidean (i..e. ) norms on the input and output spaces.

Conclusion

x=\sum{i=1}^{n} \ket{i} \otimes x{i}=\left(\begin{array}{c}
x{1} \
x{2} \
\vdots \
x_{n}
\end{array}\right)

\begin{aligned}
|x|{\ell{\infty}^n \otimes{\varepsilon} X} & = \sup {|\alpha|{\ell{1}^{n} \leq 1}, , |\beta|_{X^{*}} \leq 1}\langle\alpha \otimes \beta, x\rangle \

& =\sup{|\beta|{X^{*}} \leq 1,, i \in[n],, \varepsilon= \pm 1}\left\langle\varepsilon e{i} \otimes \beta, x\right\rangle \
& =\sup {|\beta|{X^{\circ}} \leq 1, i \in[n]}\left|\left\langle e{i} \otimes \beta, x\right\rangle\right| \

& =\sup {i \in[n]} \underbrace{\sup {|\beta|{X^{\circ}} \leq 1}\left|\left\langle\beta, x{i}\right\rangle\right|}{\left.| \beta, x{i}\right\rangle} \

& =\max {i \in[n]}\left|x{i}\right|_{X} .
\end{aligned}

Exercise

Prove this theorem.

The following result shows that the injective (resp. the projective) tensor norm is the smallest (resp. the largest) tensor norm on .

Theorem

A norm on is a tensor norm if and only if for all ,

We need to prove two inequalities:

for some . But

. Take . Then

Thus

Let us revisit the example of the injective norm, used to measure the entanglement of pure quantum states. Recall its definition:

By the previous observations, the injective tensor product norm being the smallest tensor norm, we have

with equality if and only if the state is separable.

Questions

Find or for . This corresponds to the most entangled -partite pure quantum state.

Entanglement of mixed states

Recall that mixed quantum states are represented by density matrices :
The set of density matrices is convex and compact and can be thought of as the quantum analogue of the probability simplex

A density matrix is said to be separable if it can be written as a convex combination of product density matrices:

with and . Otherwise is said to be entangled.

Our goal in this section is to characterize the separability of a density matrix using an appropriate tensor norm. Note that the definition of separability is based on the notion of positivity. Recall the following useful characterization of positivity in terms of the trace norm:

In the multipartite case this reads:

We come now to the main result of this section.

Theorem

Consider a density matrix . The following are equivalent:

(1) is separable
(2)
(3) ,
where we write and . In words: the projective tensor product of trace norms characterizes separability.

In order to keep the notation concise, let us write

Let us prove the different implications. We start with (1) : separable implies that we have a decomposition with positive , hence

On the other hand, is a tensor norm and thus showing that and finishing the proof of the first implication.

The implication is easy: , since the infimum in the non-selfadjoint projective norm is taken over a larger set of decompositions.

For the final implication start from with . The matrices do not need to be positive; we can show that actually, up to a small modification, they are. Since we get

where we use and . Hence, from the equality conditions, we obtain that for all and

with and . Combining all the phases, we get

where . From the equality conditions, we also get , proving the final implication and the theorem

Let us make on final important observation. Since

we can further decompose the norms and write

expressing separability in terms of the projective tensor product of spaces.

Entanglement criteria

Unlike the situation for pure states where deciding separability is easy, Gurvits has shown that the membership problem for the set of separable states is -hard, even in the bipartite () case. This means that there are probably no efficient algorithms that decide whether a given density matrix is separable or entangled.

In order to prove that a given state is entangled, one needs to use entanglement criteria: easily computable conditions that imply entanglement, without the converse being guaranteed. We discuss next the most important entanglement criteria.

The positive partial transposition (or PPT) criterion states that
or, equivalently,
Above, the symbol denotes the transposition map. The partially transposed state can be seen graphically as:

In coordinates, we have
The proof of the PPT condition is straightforward: if is separable, then it admits a decomposition
with positive semidefinite matrices , . Then,
Importantly, the PPT criterion is equivalent to separability for Hilbert space dimensions . For larger dimensions, one can construct PPT entangled states.

The second most important entanglement criterion is the realignment criterion:
where the realignment operation is defined in coordinates by
and as a digram:

Note that is not a selfadjoint matrix in general.

Question

Prove the realignment criterion.

These two entanglement criteria were invented long before the use of tensor norms in quantum information theory. We shall show next that they appear very naturally in this framework, as a relaxation of the projective norm approach to entanglement.

Recall that a bipartite quantum state is a tensor with 4 legs

and that the separability of can be expressed in terms of the projective tensor product
of 4 spaces corresponding to the four tensor legs. We know how to compute the projective tensor product of two spaces: this gives the trace norm; we do not know (i.e. we do not have formulas, or efficient algorithms) for four spaces.

Idea

Grop the four spaces into two groups of two, and use .

There are such choices, let us explore them one by one.

Choice 1:
We have

so nothing interesting, we recover the trace norm of .

Choice 2: .
We have

So if then is entangled. But , so , which is the partial transposition criterion.

Choice 3: . Then
We have

where is the realignment. If then and thus is entangled. We have recovered thus the realignment criterion.

Entanglement testers

The notion of entanglement testers comes from the previous observations by extending and generalizing them to the multipartite setting and more general operations on the matrix.

Let and consider a partite quantum state .
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Recall that

A linear map such that is called an entanglement tester. Testers will be applied to each of the spaces as follows:

The following is the key observation:

Hence, we obtain the following entanglement criterion induced by the testers :

Let us consider some examples. The map given by

is a tester, since

and thus

The entanglement criterion we obtain in the bipartite case is the realignment criterion from before:

One can recover some entanglement criteria studied in the literature using more general testers of the form , where are arbitrary matrices.

Entanglement testers are powerful enough to detect pure state entanglement:

Theorem

If is pure entangled, i. e. , then there exist testers such that