Coupling is a powerful technique in probability theory, with which random variables can be linked to or compared with each other. It has been widely used in the studies of random walks and Markov chains, interacting particle systems and diffusions, just name a few, in order to establish limit theorems about them, to develop approximations for them, or to derive correlation inequalities between them.
Recently, a very successful application of coupling in computer science was discovered by Barthe et al.  that it can serve as a solid mathematical foundation for defining the semantics of probabilistic relational Hoare logic. This discovery enables them to develop a series of powerful proof techniques for reasoning about relational properties of probabilistic computations, in particular, for verification of cryptographic protocols and differential privacy [2, 3, 1, 6].
There is a simple and natural correspondence between probability theory and quantum theory: probability distributions/density operators (mixed quantum states), marginal distributions/partial traces, and more. This correspondence suggests us to explore the possibility of generalising the coupling techniques for reasoning about quantum systems. We expect that these techniques can help us to extend quantum Hoare logic for proving relational properties between quantum programs and further for verifying quantum cryptographic protocols and differential privacy in quantum computation . But in this paper, we focus on studying quantum couplings themselves.
Strassen theorem  is a fundamental theorem in probability theory that can be used to bound the probability of an event in a distribution by the probability of an event in another distribution coupled with the first. The main technical contribution of this paper is proving an elegant (in our opinion) quantum generalisation of Strassen theorem.
2 Background and Basic Definitions
2.1 Probabilistic Coupling
For convenience of the reader, we first briefly recall the basics of probabilistic coupling, following . Let be a finite or countably infinite set. A sub-distribution over is a mapping such that . In paricular, if , then is called a distribution over . For a sub-distribution over , we define:
The weight of is
The support of is
The probability of an event is
Moreover, let be a joint sub-distribution, i.e. a sub-distribution over Cartesian product . Then its marginals over and are, respectively, defined by
Now we can define the notion of coupling.
Definition 1 (Probabilistic Coupling).
Let be sub-distributions over , respectively. Then a sub-distribution over is called a coupling for if and .
Here are some simple examples of coupling taken from .
Let be the uniform distribution over booleans, i.e.
be the uniform distribution over booleans, i.e.. Then the following are two couplings for :
More generally, let be the uniform distribution over a finite nonempty set , i.e. for every . Then each bijection yields a coupling for :
For any sub-distribution over , the identity coupling for is:
For any distributions over , respectively, the independent or trivial coupling is:
Obviously, coupling for a pair of distributions is not unique. Then the notion of lifting can be introduced to choose a desirable coupling.
Definition 2 (Probabilistic Lifting).
Let be sub-distributions over , respectively, and let be a relation. Then a sub-distribution over is called a witness for the -lifting of if:
is a coupling for ;
Whenever a witness exists, we say that and are related by the -lifting and write .
Let be sub-distributions over , respectively. If there exists a coupling for , then
Let be sub-distributions over the same . Then if and only if .
2.2 Quantum Coupling
With the correspondence of probability distributions/density operators (mixed quantum states) and marginal distributions/partial traces mentioned in the Introduction, we can introduce the notion of quantum coupling. To this end, let us first recall several basic notions from quantum theory; for details, we refer to .
Suppose that is a finite-dimensional Hilbert space. Let be the set of Hermitian matrices in . Let be the set of positive (semidefinite) matrices in , and is the set of partial density operators, , positive (semidefinite) matrices with trace one. A positive operator in is called a partial density operator if its trace , where is an orthonormal basis of .
We define its support:
If is an observable, i.e. Hermitian operator, in , then its expectation in state is Furthermore, let be two Hilbert space. Then partial trace over is a mapping from operators in to operators in defined by
for all and together with linearity. The partial trace over can be defined dually.
Now we are ready to define the concept of coupling.
Definition 3 (Quantum Coupling).
Let and . Then is called a coupling for if and .
Let be a Hilbert space and an orthonormal basis of . Then the uniform density operator on is
where is the dimension of . Indeed, the uniform density operator on is unique and independent with the choice of orthonormal basis. For each unitary operator in , we write , which is also an orthonormal basis of . Then
is a coupling for . In general, for different and , , though they are both the couplings for .
Let be a partial density operator in . Then by the spectral decomposition theorem, can be written as for some orthonormal basis and with . We define:
Then it is to see that is a coupling for . A difference between this example and Example 2 is that can be decomposed with other orthonormal bases, say : In general, , and we can define a different coupling:
Let and be density operators. Then
be density operators. Then tensor productis a coupling for .
The notion of lifting can also be easily generalised into the quantum setting.
Definition 4 (Quantum Lifting).
Let and , and let be a subspace of . Then is called a witness of the lifting if:
is a coupling for ;
The coupling in Example 5 is a witness for the lifting:
where is a subspace of
The coupling in Example 6 is a witness of the lifting , where defined by the orthonormal basis is a subspace of . It is interesting to note that the maximal entangled state is in
The coupling in Example 7 is a witness of the lifting .
As a quantum generalisation of Proposition 1, we have:
Let and . If there exists a coupling for , then .
Let . Then if and only if orthonormal basis s.t. .
Part 1 and Part 2 () are obvious. Here, we prove Part 2 (). If , then there exists a coupling for such that where . Then we have: for some and . Furthermore, for each , we can write: Then it is routine to show that Therefore, it holds that ∎
3 Quantum Strassen Theorem
As mentioned in the Introduction, a fundamental theorem for probabilistic coupling is the following:
Theorem 1 (Strassen Theorem).
Let be sub-distributions over , respectively. Then
where is the image of under : The converse of (1) holds if
In this section, we prove a quantum generalisation of the above Strassen Theorem. For this purpose, for any subspace of , we use and to denote the projections on and (the ortho-complement of ), respectively. We use
to denote the identity matrix of, respectively. is employed to denote the inner product of matrices living in the same space,
Then a quantum Strassen theorem can be stated as follows:
Theorem 2 (Quantum Strassen Theorem).
For any two partial density operators in and in with , and for any subspace of , the following three statements are equivalent:
For all observables (Hermitian operators) in and in satisfying , it holds that
For all positive observables in and in satisfying , it holds that
Suppose is a witness of the lifting . Then for all observables (Hermition operators) in and in , if , then we have:
Let us first define the semidefinite program :Primal problem
To show that the above problems are actually primal and dual, respectively, we only need to check the following equality:
Moreover, the strong duality holds for this semidefinite program as we can check that the primal feasible set are not empty and there exists a Hermitian operator for which :
So, . Now, let us consider the following condition:
(A): For all observable (Hermitian operators) in and in satisfy , then
If condition (A) holds, then . Still remember that . Due to the strong duality, we have . So, which maximizes must satisfy . Consequently, ; in other words, is a witness of . Therefore, On the other hand, condition (A) is equivalent to statement of the theorem. Indeed, this is not difficult to prove as if we replace in condition (A), then
From the above, we can directly derive statement . In summary, we have:
We only need to show that, for any two observables in and in satisfy , there exist two positive observables in and in such that and
Note that and
are Hermitian, so their eigenvalues are real, and we can define. Choose and . Obviously, and are positive observables, and satisfy
Moreover, as , we have
Remark:In the above proof, it is indeed naturally to employing the methods of semidefinite programming. In 
, Hsu deliberately constructs a flow network, and then using the max-flow min-cut theorem to prove the Strassen theorem in the finite case. Essentially, the max-flow min-cut theorem is a special case of the duality theorem for linear programs (LP). Considering the fact that quantum states, quantum operations and so on are all described by matrices, similar to LP, semi-definite programming (SDP) is a powerful and widely used method of convex optimization in quantum theory. Indeed, when all matrices appeared in a SDP are diagonal, then the SDP reduces to LP. In the following section, we will see that in the degenerate case, quantum Strassen theorem also reduces to the classical Strassen theorem.
4 Classical Reduction of Quantum Strassen Theorem
At the first glance, Theorem 1 (Strassen Theorem for Probabilistic Coupling) and Theorem 2 (Quantum Strassen Theorem) are very different. In this section, we show that Theorem 2 is indeed a quantum generalisation of Theorem 1.
To this end, let be a sub-distribution over () and over . And the corresponding degenerate partial density operators (quantum states) are:
in and , respectively. Furthermore, let be a classical relation from to . Then the corresponding (quantum relation) subspace of is defined as
Based on the above definition of the degenerate case, in the rest part of this section, Proposition 3 shows that the left hand side of Eqn.(1) in Theorem 1 is equivalent to the statement in Theorem 2, while Proposition 4 states the equivalence of the right hand side of Eqn.(1) in Theorem 1 and the statement in Theorem 2, concluding that Theorem 1 (Strassen Theorem) is indeed a reduction of Theorem 2 (Quantum Strassen Theorem).
The following proposition indicates that probabilistic lifting is a special case of quantum lifting.
() Suppose that there is a witness of the lifting . We define the partial density operator:
It is easy to check:
So, and ; that is, is a coupling for . Furthermore, we have:
Thus, , and is a witness of the quantum lifting .
() Suppose there is a witness of the quantum lifting . Let us construct the joint sub-distribution :
It is easy to check:
Also, if , then , then
as . Thus, , and is a witness of the lifting . ∎
Two statements are equivalent:
For any , ;
For all positive observables in and in satisfy , then
As , and are diagonal density operators, so we only need to consider those and which are also diagonal. We use the notation and for simplicity. Then it holds that
Now we need a technical lemma:
The following two statements are equivalent:
If , , then
If , , then
where are also diagonal matrices, and , .
For readability, let us first use this lemma to finish the proof of the proposition, but postpone the proof of the lemma itself to the end of this section. As
it is direct to see that statement 2 of the proposition is equivalent to statement 2 of the above lemma. For the statement 1 of the above, we can define the set