Optimal Representation of Quantum Channels

Abstract

This work shows an approach to reduce the dimensionality of matrix representations of quantum channels. It is achieved by finding a base of the cone of positive semidefinite matrices which represent quantum channels. Next, this is implemented in the Julia programming language as a part of the QuantumInformation.jl package.

Introduction

Nowadays the fields of quantum information processing and machine learning are coming together leading to the emergence of quantum machine learning [1, 2]. This area can be broadly divided into three, depending whether the data, algorithms or both are of quantum or classical nature. In this work we are interested in the case of quantum data being processed by a classical algorithm. The natural question arises: how this data should be represented and loaded into our algorithm? To be more precise, we are interested how to represent quantum channels in a succinct manner so that it can be an input into a classical neural network.

The goal of such a network would be to approximate, up to a reasonable error the distance between two channels and . As and are linear mappings transforming matrices into matrices it may not seem obvious how to define the distance between them. Turns out, there exists one notion of distance between channels which has an operational interpretation. The distance between and can be expressed in the terms of so called diamond norm

1

This quantity plays a central role in the problem of quantum operation discrimination which has gained a lot of traction recently. This is due to the fact that this distance provides an upper bound on the probability of discrimination of and .

Consider a following setup. We are given a black box which is said to contain, with equal probability, either or . What is the probability of guessing which of these is in the box if we are allowed to use the box only once? Turns out that this probability p is connected with the distance between and  [3]

2

However the explicit form of the diamond norm contains an optimization over all input matrices X. In principle this can be solved via semidefinite programming, but regrettably this quickly becomes intractable with the growing dimension of the input matrix. That is why it would desirable to have the possibility to train a classical algorithm, like a neural network, on a relatively small set of quantum channels and have the ability to quickly approximate the distance between arbitrary channels utilizing this network.

That is why this paper aims at finding an optimal representation of quantum channels for the purposes of machine learning. By optimal we understand the lowest possible number of real parameters needed to define a quantum channel [4]. Further, we would like this representation to be technically usable so that we could train, for instance, neural networks to approximate functions of this objects. This approach could provide a large speed boost in the problem of quantum channel discrimination [3, 5].

Our work is naturally divided into three parts. In the first part we show the mathematical structures needed to find the optimal representation. This involves dealing with cones of positive semidefinite matrices. The second part we present the example of whereas the last part presents the implementation of this example in the Julia language. This implementation is now a part of the QuantumInformation.jl [6, 7] numerical library available on-line at https://github.com/iitis/QuantumInformation.jl. Surprisingly, despite the complex mathematical structure and quite technical proofs, the implementation is relatively simple and therefore useful.

Mathematical Framework

Quantum Channels

Let , be complex finite-dimensional vector spaces, let be the set of all linear operators transforming vectors from to and denote . Further, consider mappings of the form

3

The set of all such mappings will be denoted and . Quantum channels are such which are trace preserving and completely positive. The former means that

4

The latter is a bit more complicated. Formally this condition can be written as

5

The intuitive explanation is as follows. First, consider a such that and . Such an operator is called a quantum state. We would like our channels not only to transform states into states, but also we would like the ability to perform a channel on only a part of the system. In other words we would like the output of to also be a proper quantum state for an arbitrary space and all . This can only be fulfilled when we introduce the need for completely positivity. We will denote the set of all quantum channels as and .

The mappings may be represented in a number of ways. For our purposes only the Choi-Jamiołkowski isomorphism [8, 9] will be relevant. This representation states that there exists a bijection J between the sets and . This bijection can be explicitly written as

6

is completely positive if and only if ; is trace preserving if and only if . Finally, is Hermiticity preserving if and only if , where denotes the set of all Hermitian matrices in .

Convex Cone Structures

Consider is a real finite-dimensional vector space and is a closed convex cone. We assume that is pointed, i.e. and generating, i.e. for each there exists such that . Such a cone is called a proper cone in the space . The proper cone becomes a partially ordered vector space for each . Let be the space dual to defined by the inner product . Then, we may introduce a partial order in as well with the dual cone

7

The cone is also closed and convex cone. If is generating in space , then is pointed and we may introduce partial order in given by

8

for all .

An interior point of a cone is called an order unit [10] if for each , there exists such that whereas a base of is defined as compact and convex subset such that for every , there exists unique and an element such that The following theorem shows there exists relation between the order unit e and a base of cone .

Theorem 1

The set is the base of (determined by element e) if and only if an element e is an order unit and .

The proof of this theorem is presented in Appendix A.

Base of Hermiticity Preserving Maps

Let us now define the finite-dimensional linear space

9

Due to the Choi–Jamiolkowski isomorphism, the set of all Hermiticity preserving linear maps of a finite-dimensional space is mathematically closely related to the set

10

of all Choi matrices of Hermiticity preserving maps.

In every linear space of Hermitian matrices we can introduce an orthonormal basis . The basis is a collection of matrices. The standard orthonormal basis is denoted by the set

11

If we consider the space of all Choi matrices of Hermiticity preserving maps we receive the dimensional space. To reduce the number of dimensions of we introduce the concept of a cone in this space and the base of cone.

Now we introduce a proper cone in the space as

12

and a subspace such that

13

Fact 1

The set of Choi matrices of quantum channels is the intersection of sets

14

We can also introduce the orthogonal complement of which is given by

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Fact 2

The set is given by

16

The proof of this fact is presented in Appendix B.

We can also consider a proper cone in space given by and a base of the cone . We can prove, using Theorem 1, that the set is the base of cone if and only if for some order unit . The base determined by an order unit E will be denoted as and is given by

17

One can easily see that identity matrix is an order unit in cone . Thus we have the following observation.

Fact 3

For the base is determined by the set of Choi matrices of quantum channels i.e.

18

We are ready to establish the main result of our work.

Theorem 2

The linear space is the smallest linear subspace containing the set of quantum channels with orthonormal basis given by

19

Moreover,

20

The proof of this theorem is presented in Appendix C.

Combining Theorem 2 with Fact 1 we obtain the following corollary.

Corollary 1

Every quantum channel can be uniquely determined by real numbers.

Moreover, there exists extra, single non-zero coefficient which is fixed for all quantum channels . Existence of this coefficient is a consequence of trace preserving condition and it can be calculated via

21

As a conclusion, we reduced the dimension of computational space by ,

Example

In this section we present how one can use the Julia language and QuantumInformation.jl library in order express quantum channels as vectors in the space .

Let us consider and along with quantum channels given by

22

and defined as

23

where denotes the Hadamard product.

First we calculate the Choi matrices of given by

24

Analogously for we have

25

Now we use the function channelbasis. The inputs of this function are the dimensions of spaces and of channels . The function returns an orthonormal basis of . Then, we are able to use the function represent which factor out Choi matrices on basis elements and returns a vector representations of basis coefficients. In our examples we have

26

where denotes vector of zeros of length i. If we want to reverse vector representation process, we can use function combine. The output matrix elements shall be accurate with original Choi matrix elements to or better.

The explicit code of implementation in Julia language is presented in Appendix D.

Conclusion

In this work we find a matrix basis for quantum channels and provide strict mathematical proofs supporting our result. This basis allows us to reduce the dimensionality of the matrix which represents a quantum channel. This, in turn, allows us to speed up computation of a class of functions of these channels, which is applicable in, for instance, the study of quantum channel discrimination. Our analytical results are accompanied by functions written in the Julia language which decompose a given quantum channel in our basis. This implementation is now a part of the QuantumInformation.jl package [6, 7].

A Proof of Theorem 1

Proof

“” Consider that is a base of . An element if and only if there exists such that the ball , which is equivalent above condition

27

By using the fact that in finite-dimensional spaces all norms are equivalent, we use the definition of induced norm given by

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Then, we have

29

Assume that , and . Then, we have

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If , then . Hence That entails that . By using the assumption we have , which implies that .

“” Now consider that is an order unit. It easy to see that is a convex set. First we prove that . Let and . If , then . It suffices to show that . We will show this fact by contradiction. Assume that and let . By the Hahn–Banach theorem [11], there exists such that . Then,

31

It implies that , which is contradiction with the assumption . Therefore, .

Let us see that if and , then each element can be written as . To prove that is compact we note that is a finite-dimensional space. Then the set is compact if and only if is closed and bounded. To prove that is closed, we take any sequence such that . By the inner product continuity, we get

32

It implies that . Therefore is closed. To prove that is bounded we show there exists such that for every . Let us take a compact sphere and closed cone . Then is also compact. Notice the function given by , where e is an order unit. By the Weierstrass theorem, a function f attains infimum and supremum. Therefore there exists such that . Consider by contradiction that . We have , where , which is a contradiction with the assumption . Thus there exists such that for every , hence Taking , we get thesis.

B Proof of Fact 2

Proof

It is clear that . Consider a linear space which is dimensional. Let . The condition in the space is equivalent to

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for all . This homogeneous system of linear equations is linearly independent. By rank–nullity theorem [12], we have

34

Therefore, . To complete the proof, note that

35

C Proof of Theorem 2

Proof

According to Fact 3 the set is the base of a proper cone . That means

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Now we fix an orthonormal basis of the space . Let it be given as the collection

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By using Fact 2, if we take , than there exists , such that . Let us set up the basis of

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Bearing in mind the relation , we conclude that basis of can be chosen as , namely

39

which completes the proof.

D Julia implementation

Here, we present the code structure for the basis representation of Choi matrix of a qubit unitary channel given by Eq. (22).