Additivity rates and PPT property for
random quantum channels
Inspired by Montanaro , we introduce the concept of additivity rates of a quantum channel , which give the first order (linear) term of the minimum output -Rényi entropies of as functions of . We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the -Rényi entropy for all .
Key words and phrases:
2000 Mathematics Subject Classification:
- 1 Introduction
- 2 Preliminaries
- 3 Additivity rates for quantum channels
- 4 Additive bounds for the Rényi entropies via (partial) traces and transpositions
- 5 Partially transposed random Choi matrices and their norm
- 6 Other bounds for random quantum channels
- 7 Minimum output entropies for a single random quantum channel
- 8 Additivity rates of random quantum channels
- 9 Classical capacity for random quantum channels
- 10 PPT properties for random quantum channels
In this paper, we focus on three questions related to additivity properties of quantum channels. First, we introduce the concept of additivity rates by which we can bound additivity violations for tensor powers of channels. Then, we use these results to upper bound the classical capacity of quantum channels. Finally, we prove the existence of PPT quantum channels violating the additivity of minimum output -Rényi entropy. In the following three subsections, we introduce the above questions and present our main results.
1.1. Additivity rates of quantum channels
One of the most important conjectures in quantum information theory had been the additivity of -Rényi entropy: for any (quantum) channels ,
for . Here, is defined for a quantum channel by
where runs over all the quantum states, and the -Rényi entropy is defined by
These conjectures were disproved by Hayden and Winter for  and by Hastings for . The case , and close to , was disproved in . Importantly, the violation in the case implies that we can increase the classical capacity of some quantum channels by using entangled inputs . Then, an important question comes to our mind: how much one can increase the classical capacity by using entanglement over as many quantum states as possible. Although this question on classical capacity should be most important, it is difficult to treat it directly. On the other hand, approaches via Rényi entropy only involve eigenvalues of matrices and this fact enables us to use random matrix and free probability to investigate the generic behavior of random quantum channels on this issue. In this paper, for natural class of random quantum channels, we bound additivity violation of Rényi entropy.
In fact, Montanaro  investigated on the limit of additivity violation for and extended the result to by using the monotonicity of the Schatten -norms in . In our paper, we study this problem first for and then extend it to . His paper and ours both depend on estimate of norm of random matrices. However, those two random matrices are different and give different estimates. Detailed discussions on this matter are made in Section 6.4 and Section 8.2.
Finally, our informal main theorem on limitation of additivity violation can be stated as follows.
Consider a sequence of random quantum channels , defined via random embeddings of into , where is a fixed parameter and for a fixed . Then, almost surely as , for all , there exist constants such that, for all ,
The constants satisfy the following relations
When is a constant,
When is large and with ,
The above statements hold for the complementary channels , where the roles of and are swapped.
In the result above, the larger the constant is, the more restrictive the additivity violation is. A precise definition of additivity rates is given in Definition 3.1 and more detailed estimates on are made in Theorem 8.4. Also, our model of random quantum channels together with the idea of complementarity is described in detail at the beginning of Section 2.1.
1.2. Range of capacity
In [36, 48], the Holevo capacity of quantum channels, denoted by , was proven to be the capacity of transmitting classical information without entangled inputs. Here, is defined for quantum channels :
where the are all possible ensembles with and being a probability distribution and quantum states, respectively. It is not difficult to see that
Here, ; note that this quantity is trivially additive, . The equality is saturated, for example, if the channel has covariant property , and it is also the case in our setting for a similar reason, which will be discussed later. Since the classical capacity, denoted by is obtained by regularizing Holevo capacity [36, 48], we can show the following estimate.
Consider a sequence of random quantum channels , defined via random embeddings of into , where is a fixed parameter and for a fixed . Then, almost surely in the regime , the classical capacity is asymptotically bounded as follows:
When is a constant, we have
When and , we have, for some constant
When and , the classical capacity is almost surely bounded by a constant.
1.3. PPT property and additivity violation
Another topic treated in this current paper is the positive partial transpose property (PPT) for quantum channels; a quantum channel is called PPT iff the partial transposition of its Choi matrix is positive semi-definite. The importance of PPT channels stems from their recent use in the proofs of superactivation for the quantum capacity, see . Hence, it is interesting to investigate typical PPT/non-PPT property for random quantum channels. Also, we show that there exist PPT channels which violate additivity of Rényi entropy. This result is interesting because all entanglement-breaking channels are proven to be additive [42, 50]. Note that the set of entanglement-breaking channels is contained by the set of PPT channels but for qubit channels, these sets are the same. The above two problems are investigated in Section 10, and we obtain the following results.
First, in Section 10.1 we have
Consider a sequence of random quantum channels of parameters , and let
If then, almost surely as , the sequence has the PPT property, whereas if , then, almost surely, the sequence does not have the PPT property. We say that the value is a threshold for the PPT property of random quantum channels.
Second, we have
Consider a sequence of random quantum channels , defined via random embeddings of into and let . Suppose one of the following two procedures are made:
fix and take large enough and , or
fix and take large enough and
then, typically are PPT and violate additivity:
1.4. Structure of the paper
More precisely, after recalling some basic definitions and results in Section 2, we introduce in Section 3 one of the main topics of this paper: additivity rates of quantum channels. Then, in Section 4, we introduce lower bounds for minimum output Rényi entropies, which are additive with respect to tensor products. These results can be used to lower bound the additivity rates. In Sections 5 and 6 we study these bounds for random quantum channels. After recalling some known results about the minimum output entropies of random quantum channels in Section 7, we give lower bounds for the additivity rates of random quantum channels in Section 8, limiting the possible violations of the additivity of the minimum output entropies of these channels. Based on previous results, we present in Section 9 upper bounds for the classical capacitiy of (random) quantum channels, and in Section 10 examples of random channels which are PPT and violate the additivity of the minimum Rényi output entropies.
In this section, we go through basic definitions and knowledge, which are needed through this current paper. We give definitions on quantum states, channels and entropy in Section 2.1, and then make quick overviews on graphical Weingarten calculus and free probability in Section 2.2 and Sec 2.3, respectively, as much as we need.
Let us start by introducing some notation. In this paper, the operator denotes the usual, unnormalized trace. The reader may choose to denote the logarithm in basis or , depending on her/his background. Finally, we use the following asymptotic notations for sequences:
2.1. Quantum states and channels
In this paper we will consider quantum channels , defined via the Stinespring dilation 
for an isometry . In this case, the dimensions of input, output and environment spaces are , and , respectively. If we swap the roles of and , we get another channel , called the complementary channel [38, 43]. Outputs of this channel share non-zero eigenvalues with those of the channel as long as inputs are pure states. Hence, our results on entropy bounds are also shared by and . For such maps and we know that and are positive for any . This property is called completely positivity. Later, we shall consider random quantum channels obtained by choosing the isometry randomly. The probability distribution of the random variable will be the uniform one on the set of isometries, obtained by truncating a Haar-distributed unitary matrix : will consist of the first columns of a random Haar unitary matrix. For now, let us introduce a key concept in this paper, the Choi matrix of a quantum channel . To a channel , we associate its Choi matrix , defined by
where is the (unnormalized) maximally entangled state
It is a classical result that a linear map is completely positive if and only if its Choi matrix is positive semidefinite.
Finally, we shall denote by the set of -dimensional quantum states
Let us now introduce the entropic quantities we are interested in. The Shannon entropy of a probability vector , , is defined by
where we put . This quantity is extended, via functional calculus, to quantum states, where it is known as the von Neumann entropy. The Rényi entropies are a one-parameter generalizations of these quantities. They are defined for any , as follows:
The same quantities are defined for quantum states, and satisfy . In what follows, we shall use the following well-known result :
For a fixed probability vector (resp. quantum state ), the function
(resp. ) is non-increasing in . In a similar manner, for a fixed quantum channel , the function is decreasing in .
Recall from the introduction that the minimum output -Rényi entropy of a quantum channel is defined by
for an arbitrary entropy parameter . The functionals are sub-additive, in the sense that for any quantum channels , we have
For a pair of quantum channels such that has no pure outputs, define the relative violation of minimum output -entropy of the pair by
With this notation, we call the pair -additive iff. .
2.2. The graphical Weingarten integration formula
The model of random quantum channels we are interested in involves random isometries, which can be seen as truncated random Haar unitary matrices. Since our approach to understanding statistics of such channels is the moment method, we shall compute integrals of polynomials in the entries of unitary matrices. The main result here is the Weingarten formula, which was introduced by Weingarten  in the physics literature and rigorously developed by Collins , and Collins and Śniady .
Let be a positive integer and , , , be -tuples of positive integers from . Then, the following integral over the Haar measure of can be evaluated as
where the function is called the Weingarten function (see the next definition). If then
For a permutation , denotes the number of cycles of , and is the length of , i.e. the minimal number of transpositions that multiply to . Note that the length function defines a distance on , via . Let us recall the definition of the unitary Weingarten function.
The unitary Weingarten function is a combinatorial function which depends on a dimension parameter and on a permutation in the symmetric group . It is the inverse of the function with respect to the following convolution operation:
In the large limit ( is being kept fixed), it has the following asymptotics
where the Möbius function is multiplicative on the cycles of and its value on a -cycle is
where are the Catalan numbers. Note that we omit the dimension in the Weingarten function when there is no confusion and write for . Also, we use the notation .
When applying the above integration formula, especially in the cases where the degree of the polynomial to be integrated is high, one has to deal often with sums indexed by a large set of indices. Computing such sums is a tedious task, so we use here a graphical formulation of the Weingarten formula, introduced in . Here we sketch the main ideas, referring the reader to original work  for the technical details. This method has been used recently in relation to channel capacities [21, 14, 15, 30], entanglement theory [4, 22] and condensed matter physics .
The Weingarten graphical calculus builds up on the tensor diagrams introduced by theoretical physicists and adds to it the ability to perform averages over diagrams containing Haar unitary matrices. In the graphical formalism, tensors (vectors, linear forms, matrices, etc.) are represented by boxes, see Figure 1, left diagram. To each box, one attaches labels of different shapes, corresponding to vector spaces. The labels can be filled (black) or empty (white) corresponding to spaces or their duals: a -tensor will be represented by a box with black labels and white labels attached. The example in Figure 1, left corresponds to a (square) matrix .
Besides boxes, our diagrams contain wires, which connect the labels attached to boxes. Each wire corresponds to a tensor contraction between a vector space and its dual (). See Figure 1 for the example of the partial trace. A diagram is simply a collection of such boxes and wires and corresponds to an element in a tensor product space (which might be degenerate, i.e. the scalars ).
Let us now describe briefly how one computes expectation values of such diagrams containing boxes corresponding to Haar-distributed random unitary matrices. The idea in  was to implement in the graphical formalism the Weingarten formula in Theorem 2.2. Each pair of permutations in (2.13) will be used to eliminate and boxes and wires will be added between the black, resp. white, labels of the box with index and the black, resp. white, labels of the box with index , resp. . In this way, for each pair of permutations, one obtains a new diagram , called a removal of the original diagram corresponding to . The graphical Weingarten formula is described in the following theorem .
If is a diagram containing boxes corresponding to a Haar-distributed random unitary matrix , the expectation value of with respect to can be decomposed as a sum of removal diagrams , weighted by Weingarten functions:
Since the above Weingarten formula is written as a sum over permutations we review next some additional properties of the symmetric group endowed with the distance . The function has nice properties, for example, and ; we refer the readers to  for more details. For three permutations the triangle inequality holds:
When the bound above is saturated, we say that is on a geodesic between and , and write . When is the full cycle permutation, , permutations lying on the geodesic between and are simply called geodesics permutations. In [47, Proposition 23.23], it is shown that geodesic permutations are in bijection with non-crossing partitions. Recall that a partition of is said to be non-crossing if
where denotes the equivalence relation on induced by . We denote by the set of non-crossing partitions on elements. Moreover, the isomorphism between geodesic permutations and non-crossing partitions respects many combinatorial properties of the objects, such as the number of cycles (resp. blocks); see [47, Section 23] for more details.
2.3. Some elements of free probability
An excellent reference for the theory of free probability is ; we recall now only some basic facts from this theory needed in the current paper.
A probability space is a unital -algebra equipped with a state , which gives a norm; . Such a probability space is denoted by .
The convergence of the eigenvalues of random matrices can be stated in the language of probability spaces as follows. Note that we define two types of convergence: the convergence in distribution (which is the convergence of all moments) and the strong convergence (which implies, in particular, the convergence of the extreme eigenvalues of the matrices). In this paper, we are interested in the operator norms of random matrices, and the usual convergence in distribution does not guarantee the convergence of the norms in the case when the size of the matrices grows; hence we shall make use of the strong convergence.
Suppose we have -probability spaces: and with , where and are faithful traces. For -tuple elements in and in ,
we say converges to in distribution if
we say converges to strongly in distribution if in addition
Here, is any polynomial in non-commuting variables.
The strong asymptotic freeness of random unitary matrices and deterministic matrices has been proven by Collins and Male:
Proposition 2.6 ().
Suppose we have -probability spaces: and with . Here, is a faithful trace and is the usual normalized trace on the matrix space . Take
a -tuple of free Haar unitary elements in , and
a -tuple of i.i.d. Haar-distributed unitary matrices in .
Suppose we are given
a -tuple of elements free from in , and
a -tuple of matrices independent from in .
such that converges to strongly in distribution. Then, almost surely converges to strongly in distribution.
The following useful statement was proven by Male:
Proposition 2.7 (Proposition 7.3 in ).
Suppose we have -probability spaces: and with , where and are faithful traces. Take
a -tuple of self-adjoint elements in , and
a -tuple of self-adjoint elements in .
If we assume that converges to strongly in distribution, then we have strong convergence in the following sense: for any polynomial in non-commuting variables with coefficients in ,
Note that in the above result, one can drop the self-adjointness assumption by considering real and imaginary parts of the operators involved.
We prove next a simple lemma about push-forwards of free additive convolution powers of probability measures and we recall a well-known result about the free multiplicative convolution product of Bernoulli distributions . Recall that given two free elements having distributions , the distributions of and respectively are denoted by , respectively , and they are called the free additive (resp. multiplicative) convolutions of and (for the latter, we require ). We denote by the push-forward of a measure by a measurable function : if the random variable has distribution , then has distribution .
Let be a compactly supported probability measure on so that, for any , is well-defined. Then, we have, for any
First, let be an element in the -algebra which gives the probability measure so that induces the probability measure . Then, first by using multi-linear property of cummulant,
Also, shift does not change cummulants except for the case when : . Therefore,
This completes the proof. ∎
The free multiplicative convolution of two Bernoulli distributions (with ) is given by
where the bounds of the a.c. part of the support are given by
Equivalently, for any ,
where . Note that .
3. Additivity rates for quantum channels
In this section, we discuss how the functional behaves with respect to tensor products. The ideas and results developed here will be applied to random quantum channels later. Our inspiration comes from , where a multiplicative version of the additivity rates was established.
3.1. Definition and basic properties
First, as is explained in Section 1.1, because of non-additivity properties of quantum channels, we do not know how behaves as grows. So, we introduce a notion which quantifies in terms of :
For a quantum channel and an entropy parameter , define the -additivity rate of by
Different characterizations as well as some basic properties of -additivity rates can readily be obtained from basic properties of the entropy functionals.
The -additivity rate of a quantum channel can be characterized in the following equivalent ways:
The statements follow from Fekete’s sub-additive lemma [52, Lemma 1.2.1] and (2.11): the in (3.1) is actually a limit and it is equal to the infimum of the sequence . The zero entropy case follows from the fact that if the channel has zero minimum output entropy for some , then the same holds for all tensor powers because . Moreover, such a channel is additive with any channel (see [28, Lemma 1]). ∎
The -additivity rate functionals have the following set of properties:
The additivity relation holds for all if and only if .
Monotonicity with respect to tensor powers:
for all integer tensor powers .
Convex-like behaviour with respect to tensor products:
where is the relative violation of the minimum -output entropy (2.12) and ; if , just put .
Additivity violations yield upper bounds:
The first property follows directly from the definition. For the second one, in the case when , write
The last two statements follow from the definition of the relative violation . ∎
3.2. Examples: the Werner-Holevo and the antisymmetric channels
In Proposition 3.3, we have seen that any -additive channel has unit additivity rate . We discuss next some examples of non-additive channels. Below, we shall denote with the transposition of a matrix .
The Werner-Holevo channel ,
is the first known example of a quantum channel that violates the additivity of the minimum -output Rényi entropy. In , it has been shown that violates the additivity for any value . From  we have explicitly
from which we can infer the following upper bounds for additivity rates (see (3.7)):
4. Additive bounds for the Rényi entropies via (partial) traces and transpositions
In this section we introduce several additive bounds for the Rényi entropies of quantum channels (we focus on ) that we obtain by considering the operator norm of the vectorized version of the isometry defining the channel, after applying one or several traces or transpositions. We perform an exhaustive study of this method, concluding that the method yields 5 non-trivial bounds, including the one studied by Montanaro . The key point is that the bounds we are providing are additive with respect to tensor powers of channels, so they can be used to bound the additivity rates defined in the previous sections. Interesting bounds for the classical capacity of quantum channels can be obtained from these bounds.
Recall that the , resp. -minimum output Rényi entropies of a quantum channel are
where runs over all the input quantum states. In what follows, we shall write for the transposition map, which is an involution on matrix algebras. Moreover, for bi-partite matrices , we write for the partial transposition of with respect to the second subsystem,
Equivalently, the partial transposition operation can be defined on simple tensors by .
4.1. Quantities arising from vectorized isometries
The starting point of our study is the vectorization of the isometry defining the channel as in (2.3). To this isometry we associate its vectorization (which is a tripartite tensor) by the relation
where , , are orthonormal bases of respectively , , . The Choi matrix of the channel (see (2.4)) is related to the third order tensor by the partial trace operation:
For a graphical representation of the vectorization and its relation to the Choi matrix , see Figure 2. Note also that .
We shall now apply different operations to the orthogonal projection on and take the operator norm of the resulting matrix, in order to obtain a scalar quantity ; in the next subsection, we show that some of these quantities are (additive) bounds for the or the minimum output Rényi entropies of quantum channels. We are interested in three operations: the identity , the trace , and the transposition . These operators will act on the three tensor factors of , and we shall denote by the quantity
where . As an illustration, in the case of the Choi matrix, where we apply the trace map on the first factor and the identity on the other two, we have .
We gather in Table 1 the possibilities we obtain by applying on each of the tensor factors the maps above. We obtain 5 different bounds (, , , , and ) which appear several times in the list, as well as some trivial, constant bounds which do not depend on the channel.