Entanglement cost and entangling power of bipartite unitary and permutation operators

Entanglement cost and entangling power of bipartite unitary and permutation operators

Lin Chen School of Mathematics and Systems Science, Beihang University, Beijing 100191, China International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China    Li Yu yupapers@sina.com National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
July 15, 2019
Abstract

It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank , and showed that one of the protocol uses ebits of entanglement and bits of classical communication, while these two types of costs for the other protocol scale as but the actual values are smaller for all . Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively ebit and some value between and ebits.

pacs:
03.67.Ac, 03.67.Lx, 03.65.Ud, 03.67.Mn

I Introduction

The implementation of unitary operations is a key task in quantum information processing. Bipartite unitaries are a particularly important class to study, because they are the base case for studying multipartite unitaries. Many tasks in quantum communication, games and cryptography are restricted to two parties. The evaluation of entanglement cost and/or classical resources for implementing unitary operations belong to a type of communication cost problems in quantum information theory. It has applications in the study of quantum networks and distributed quantum computation, see Akibue15 (); ick16 () for recent progress on implementing nonlocal unitaries or isometries on multiple qubits, using shared entanglement in a network or using a limited set of basic gates.

Any bipartite unitary is the product of controlled unitaries bry02 (); blb05 (). The controlled unitary can be implemented with local operations and classical communication (LOCC) and a maximally entangled state ygc10 (). The entanglement cost scales with the logarithm of the number of terms of control. The number can be as large as the dimension of the controlling system. Bipartite unitaries of Schmidt rank not greater than three are equivalent to controlled unitaries under local unitaries cy13 (); cy14 (); cy14ap (). Every Schmidt-rank-two bipartite unitary can be implemented using one ebit and LOCC cy13 (), but the best upper bound for the entanglement cost of Schmidt-rank-three unitaries appears to depend on the dimensions of the Hilbert spaces: an upper bound on system is ebits for cy14ap (). In this paper we show that all Schmidt-rank-three bipartite unitaries can be implemented using ebits, where is the controlling side of the unitary. This is presented in Theorem 10 based on a standard form constructed in Eq. (9). We present a protocol for implementing some bipartite unitaries using multiple levels of control, and apply it to Schmidt-rank-three unitaries.

Reducing the entanglement cost for implementing nonlocal unitary gates is a key problem in computation or communication tasks on networks, because entanglement is often imperfect and costly to produce. A protocol that uses less entanglement would have less error in the implemented unitary gate, giving rise to less error in the final outcome of the computation or communication task. Some tasks may involve multipartite unitaries or non-unitary operations, and studying the entanglement cost of bipartite unitaries may help the study of the entanglement cost of those operations. The classical communication cost of the protocols in this paper is linear in the entanglement cost. Thus our protocols have less classical communication cost than the previous protocols. This is beneficial since classical communication is subject to noise and security concerns.

It is known that there is a dimension-independent upper bound for the entanglement cost of bipartite permutation unitaries with the help of a one-qubit ancilla on one side cy15 (). The ancilla can be dropped from this statement at the cost of using more entanglement, since it can be prepared from another shared entangled pair of qubits. We construct a standard form of bipartite complex permutation unitaries of Schmidt rank , when a “big row” of the unitary contains at least nonzero blocks. (The big row is defined in Sec III.) We further investigate the maximum number of distinct nonzero diagonal blocks of a controlled permutation unitary of Schmidt rank . The above two results give upper bounds of entanglement cost for implementing the corresponding types of unitaries. This is presented in Lemmas 13 and 14. When the Schmidt rank is not greater than four, we give tighter upper bounds of entanglement cost in Lemmas 15 and 16, and Corollary 23. In particular, any Schmidt-rank-three bipartite permutation unitary needs only ebits to implement. We give a protocol that implements any bipartite permutation unitary of Schmidt rank using ebits of entanglement and bits of classical communication. Then we present another protocol for the same task with the costs only scaling as , but the actual values are larger for all , as discussed below Theorem 22. These results give upper bounds for the number of nonlocal CNOT gates for implementing a bipartite classical reversible map using a classical circuit under two different conditions (A nonlocal CNOT gate is a CNOT gate that acts across the two parties, as opposed to acting locally on the bits within each party). The number is larger in the case that ancillas are required to be restored to the initial value, compared to the opposite case, and both results are under the assumption that the initial values of the ancillas are known. These results are an exponential improvement over the corresponding results in cy15 (). An example of a Schmidt-rank-four permutation unitary is given in Sec. V.3 with its entanglement cost analyzed. As a byproduct, we point out that the expression of bipartite complex permutation unitaries in (13) is further evidence supporting a recent conjecture on the ranks and marginals of multipartite states chl14 ().

Classical reversible circuits may have lower energy cost compared to the circuits that involve erasures Bennett73 (). The current paper touches upon the topic of classical reversible circuits, not only because our main result applies to it, but also we find that the design for the classical reversible circuits could provide hints for designing better quantum LOCC protocols or quantum unitary circuits.

The results so far are for the upper bound of entanglement cost for implementing bipartite unitaries. Another interesting topic is finding lower bounds for this quantity, such as the entangling power defined in (27). Any Schmidt-rank- unitary can have entangling power at most ebits, see the beginning part of Sec. V.4. In the case of , it is much smaller than the upper bound in this paper when and are large. Recently, Soeda et al stm11 () proved that ebit of entanglement is needed for implementing any 2-qubit controlled unitary by LOCC when the resource state is of Schmidt rank two. Stahlke et al sg11 () proved that if the Schmidt rank of the resource state is equal to the Schmidt rank of the bipartite unitary, and the unitary can be implemented by the state using LOCC or separable operations, then the resource state has equal nonzero Schmidt coefficients. In Example 12 we present a class of Schmidt-rank-three unitaries for which we do not know of a protocol with constant entanglement cost. In fact it is an open problem whether there is a constant upper bound for the entanglement cost of all Schmidt-rank-three bipartite unitaries.

Next, we show that the entangling power of any Schmidt-rank-two bipartite permutation unitary is exactly 1 ebit by Lemma 26. The counterpart of Schmidt-rank-three permutation unitary is some value between and ebits, as shown in Proposition 27. Again, there is a curious gap between the best known entanglement cost and the entangling power, similar to the case of general Schmidt-rank-three unitaries.

The rest of this paper is organized as follows. In Sec. II we briefly introduce the appendix. In Sec. III we introduce the notations and preliminary lemmas used in the paper. In Sec. IV we present the main result on Schmidt-rank-three bipartite unitary operators. In Sec. V we study bipartite complex permutation unitaries. We first present some preliminary lemmas, and then investigate the entanglement cost of bipartite permutation unitaries of Schmidt rank up to three in Sec. V.1, and study the protocol and entanglement cost for general bipartite permutation unitaries in Sec. V.2. An example is given in Sec. V.3, and the entangling power of bipartite permutation unitaries is studied in Sec. V.4. Finally we conclude in Sec. VI.

Ii Summary of technical results

To enhance readability we briefly summarize the results of the current work and their relationships in this section. We have introduced Theorem 10 in the introduction, which reduces the entanglement cost to about half of the previous upper bound in cy14ap () for large classes of bipartite Schmidt-rank-three unitaries. To study this theorem, we introduce Lemma 9 as a hard case among the possible forms of bipartite unitaries of Schmidt rank three. The proof of Theorem 10 makes use of Protocols 7 and 8, which are respectively a new two-level controlled unitary protocol, and a protocol from ygc10 () for implementing unitaries with group-type expansion.

We study some basic properties of the real or complex bipartite permutation unitaries in terms of the Schmidt rank in Lemmas 13 and 14. The results are used throughout Sec. V. In Lemmas 15 and 16 we investigate the structure and entanglement cost for (complex) permutation unitaries of Schmidt rank two or three. In Theorem 22 we show that any bipartite permutation unitary of Schmidt rank can be implemented using local operations with the help of ebits of entanglement and twice as many bits of classical communication, where is the Bell number defined before Lemma 19. The two terms in the result arise from Protocol 18 and Protocol 21, respectively. This significantly improves over the result in Theorem 22 of cy15 (), which states that such unitary can be implemented using LOCC with ebits. In Theorem 24, we adapt the two methods of implementing bipartite permutation unitaries in the proof of Theorem 22 to the decomposition of classical bipartite reversible circuits into local gates and nonlocal CNOT gates. In Proposition 27, we prove that the entangling power [defined in Eq. (27)] of bipartite permutation unitaries of Schmidt rank three is in the range of ebits.

Iii Preliminaries

In this section we introduce the notations and preliminary lemmas used in the paper. Let be the usual Pauli matrices. Denote the computational-basis states of the bipartite Hilbert space by , . Let and be the identity operators on the spaces and , respectively. We also denote and , respectively, as the identity and zero matrix of order . The bipartite unitary gate acting on has Schmidt rank if there is an expansion where the matrices are linearly independent, and the matrices are also linearly independent. An equivalent definition named as the operator-Schmidt rank has been presented in Nielsen03 (); Tyson03 (). The above expansion is called the Schmidt decomposition. We name the space of as the space spanned by all that appear in a Schmidt decomposition of . It is well defined in the sense that the space is independent of the specific choice of the Schmidt decomposition.

Next, is a controlled unitary gate, if is equivalent to or via local unitaries. To be specific, is a controlled unitary from or side, respectively. In particular, is controlled in the computational basis from side if . Bipartite unitary gates of Schmidt rank two or three are equivalent to controlled unitaries via local unitaries cy13 (); cy14 (); cy14ap (). We shall denote as the ordinary direct sum of two matrices and , and denote as the direct sum of and from the side. The latter is called the -direct sum, and and respectively act on two subspaces and such that . A permutation matrix (or called “permutation unitary” or “real permutation matrix”) is a unitary matrix containing elements and only. The partial permutation matrix is a matrix with elements being and only, satisfying that any row sum or column sum is not greater than . So the partial permutation matrix may be not unitary. A bipartite controlled-permutation matrix is a permutation matrix controlled in the computational basis of one system, i.e., , where the projectors , is a permutation unitary, and each is a term of . A complex permutation matrix is a unitary matrix with exactly one nonzero element in each row and column. A “big row” of the unitary matrix refers to a submatrix given by , for some . Similarly, a “big column” of refers to a submatrix given by , for some . A “block” of refers to a submatrix given by , for some , and when , the block is called a “diagonal block.”

In all the protocols in this paper, the computational basis starts from instead of . For an -dimensional system, we respectively define the Fourier gate , and the gate usually as but sometimes generalizing the to a high-rank projector, see Protocol 4. The basis is the computational basis. The -information means the information about which computational basis state that the state of the quantum system is in.

In this paper, the “entanglement cost” of a bipartite unitary is defined as

(1)

where is any one-shot exact deterministic LOCC protocol to implement , and is the amount of initial entanglement needed in the protocol. “One-shot” means that only one copy of the unitary is implemented, while the word “exact” excludes the case that some other unitary that might approximate the given unitary is implemented, and “deterministic” means that the unitary is implemented with no chance of failure. The Schmidt rank of initially entangled state and the dimension of ancillary space are finite in each protocol , and there is no constant upper bound for these quantities. In the case that the resource entangled state is mixed, we suggest to use the entanglement of formation pv07 () as the entanglement measure, although we do not discuss the mixed entangled state in this paper. If there is entanglement left after the protocol, subtraction of the latter from the cost would lead to definitions of assisted entanglement cost. It is beyond the scope of this paper.

The unit for entanglement is “ebit.” The entanglement contained in a maximally entangled pure state of Schmidt rank is regarded as ebits. Also, to simplify the notation, every bit of classical communication used in a protocol is called a “c-bit.” If the classical message is a signal among equally possible signals, the amount of classical communication is regarded as c-bits.

iii.1 Linear algebra

Here we present a few preliminary results of linear algebra used throughout our paper.

Lemma 1

Let be a diagonal unitary matrix. The following four statements are equivalent.
(i) has at least three distinct eigenvalues;
(ii) the identity, and are linearly independent;
(iii) any unitary in the linear span of the identity and is proportional to one of them;
(iv) any multiple of unitary in the linear span of the identity and is proportional to one of them.

Proof.

. Let be the three distinct eigenvalues of . Since all have modulus one, the matrix is the product of the diagonal matrix and a Vandermonde matrix with columns permuted, the latter has determinant . Since are distinct, is invertible. Since is a submatrix of the matrix whose columns are the diagonal vectors of the identity, and , the latter are linearly independent. We have proved .

. Let the unitary be where are complex numbers. We have , hence . Then follows from , because of .

Finally the relations , and are trivial. This completes the proof.    

In the following lemma, a matrix is said to be “block diagonal” iff there is a permutation matrix such that , where and are square matrices. We regard a matrix as being of order .

Lemma 2

Suppose is a unitary matrix of order at least two, and there is a nonzero diagonal matrix such that there is a nontrivial linear combination of and that is unitary, and we denote it as . Then is block diagonal, where , and is an unitary matrix.

Proof.

By assumption, for the given unitary matrix , where , there exists a nonzero complex number and a nonzero diagonal matrix such that is proportional to a unitary matrix of order with , where . This differs from the in the assertion by a constant factor, hence it suffices to prove the assertion for the current . Suppose , and the matrix elements of are , . The rows of are mutually orthogonal. From that the ’th and ’th rows of are orthogonal, where , we have , hence

(2)

Therefore, for any , it must be that those () that are equal to and those () that are equal to satisfy that their row and column coordinates determine a rectangular block in consisting of elements , and any element of outside of this block that are in the same rows or the same columns of this block must be zero. The last statement is due to the following reason: Suppose such a rectangular block contains , then an element where satisfies is in the row labeled by and outside of the rectangular block containing ; and from (2), we have . Now we consider two cases:

The first case is that there exist such that . In this case, the contains some rectangular blocks that do not overlap in the rows and columns that they occupy. Since is unitary, these rectangular blocks must be square blocks. Hence, is block-diagonal after suitable permutation matrices are multiplied before and after it. From the form of , this implies that is block diagonal in the sense defined before the lemma. Thus the assertion holds with being the identity matrix .

The second case is that . Then it must be that , since otherwise it can be deduced from (2) that there would be a column of that is zero, violating that is unitary. Since is unitary, there is an diagonal matrix and an unitary matrix such that , then

(3)

where . Since , , and are all diagonal, the matrix is the direct sum of matrices up to a similarity transform by a permutation matrix. The rows and columns of the ’th matrix correspond to the ’th and the ’th rows, and the ’th and the ’th columns of the original matrix, respectively. This completes the proof.    

Lemma 3

Any real linear combination of the three matrices , , and is proportional to a unitary matrix.

Proof.

Let where are real numbers. By direct computation one can show that is proportional to a unitary matrix. This completes the proof.    

Iv Tighter upper bound for entanglement cost of implementing Schmidt-rank-3 unitaries

On the problem of exact implementation of bipartite nonlocal unitaries using LOCC and shared entanglement, we use or discuss the following three known protocols. (1) The two-way teleportation protocol, i.e., teleporting the system of one party to the other party, performing the unitary there, and teleporting the system back to the original party. (2) The protocol for implementing controlled unitaries in Sec. III of ygc10 (), which is briefly reviewed as Protocol 4 below, and it will be called “the basic controlled-unitary protocol.” A simple extension of it is Protocol 5, and the latter is the basis for the two-level controlled Protocols 6 and 7. (3) The group-type protocol in Sec. IV of ygc10 (), which is briefly reviewed as Protocol 8 below. Protocol 6 is used in Sec. V, and Protocols 7 and 8 are used in the proof of Theorem 10 (ii).

Protocol 4

(The basic controlled unitary protocol.)

The unitary to be implemented by two parties, Alice and Bob, is

(4)

where are mutually orthogonal projectors on , and are unitary operators on . The may be of rank greater than , meaning that the dimension of may be larger than .

A figure for this protocol is Fig. 5 of ygc10 (). This figure was originally for the case that are all rank-one, but with suitable interpretation of the gates in the circuit (see Sec. III C of ygc10 ()), it works for the general case of higher rank . For the protocols in this section only, the gate on a -dimensional Hilbert space is defined as

(5)

The steps of the protocol are as follows.

0. The two parties initially share the following entangled state on ancillary systems and , which are with Alice and Bob, respectively:

(6)

1. Alice performs a controlled- gate on systems and , with as the control. (The means to the power .) Then Alice performs a measurement on in the standard basis, and sends the result to Bob.

2. Bob applies the gate to . This is followed by a controlled gate on and , with as the control. Then Bob does a Fourier gate on (defined in Sec III), and measures in the standard basis. The outcome is sent to Alice.

3. Alice carries out a correction on , where the is defined as (c.f. Sec. III C of ygc10 ()), and this definition of reduces to that in Sec III in the case that all are rank-one. This completes the protocol.

The resource consumption of the protocol is ebits and c-bits.

Protocol 5

(The extension of the basic controlled unitary protocol to the case that some projectors in (4) are replaced with zero operators.)

If the unitary to be implemented by Alice and Bob is given by (4), but only some are projectors, and some others are zero operators (the output is zero for any input), then the steps of Protocol 4 can still be carried out. Note that the controlled- gate in step 1 and the gate in step 3 could be defined using the same expression as before but with the understood as being projectors or zero operators. The protocol still uses ebits and c-bits. Suppose there are operators among the that are nonzero; then the same unitary could be carried out with only ebits and c-bits using Protocol 4. Nonetheless, the less efficient protocol turns out to be useful in Protocols 6 and 7 below.

Next, we introduce a recursive-control protocol for implementing some bipartite unitaries with LOCC and initial entanglement.

Protocol 6

(Protocol for implementing a bipartite unitary with two levels of control — The special case that the lower-level controlled unitaries are controlled from a fixed side.)

The bipartite unitary to be implemented on is of the following form:

(7)

where , and , and are orthogonal projectors on , and

(8)

are controlled unitaries with local unitaries on . The are projectors on and are orthogonal among different for the same . Let . By introducing some zero operators to the set of and calling the new operators , we may write all using terms:

(9)

where are still local unitaries and some of them are not present in Eq. (8).

The idea of the protocol can be roughly summarized as follows. The higher level of the protocol is “ controls ,” and the lower level is “ controls .” The steps are as follows.

0. Alice and Bob share a maximally entangled state of Schmidt rank on , and another maximally entangled state of Schmidt rank on . The subsystems and are on Alice’s side, while and are on Bob’s side.

1. They perform the first half of the basic controlled-unitary protocol (Protocol 4) on and , until the gate in the protocol is done [the is defined in Eq. (5)]. Now they share a maximally entangled state .

2. They perform Protocol 5 to implement using their information about stored in the entangled state above, with the help of a maximally entangled state of the form . More specifically, in the lower-level protocol, every unitary gate is controlled by the state on Alice’s side or the on Bob’s side. If there are measurements not in the standard basis in the lower-level protocol, we decompose it as a unitary followed by a measurement in the standard basis, so that all measurements are in the same basis and thus need not be controlled by information about .

3. They have effectively performed the gate from the protocol in Sec. III of ygc10 (), which is the gate in the higher-level of the current protocol. Next, the subsystem is measured in the Fourier basis, and a local unitary correction, i.e., the integer powers of the generalized gate defined in the basic controlled-unitary protocol is done on . Note that is not being measured, since it is a “data” system and not an ancilla.

The whole protocol uses ebits and c-bits. Note that in step 2, the measurement outcomes in the lower-level protocol are the same for different controlling states labelled by . This is acceptable, since the Protocol 5 (used as the lower-level protocol here) works under any measurement outcome anyway.

Protocol 7

(Protocol for implementing a bipartite unitary with two levels of control — The general case that the lower level unitaries are controlled from different sides.)

In Protocol 6, the lower level unitaries are all controlled from the same side (and opposite to the direction of control in the higher level, since the case of same direction is trivial in that the unitary is then a one-level controlled unitary). Here we consider a generalization: the lower-level unitaries can be controlled from different sides. Formally, the target unitary is of the following form:

(10)

where , and , and are orthogonal projectors on . For each , there exists an integer , such that at least one of the following two equations hold:

(11)
(12)

where and are local unitaries on and , respectively. The are projectors on and are orthogonal among different for the same . The are projectors on and are orthogonal among different for the same . Let . By introducing some zero operators to the set of and , and calling the new operators or , we have that for each , at least one of the following two equations hold:

(13)
(14)

where and are local unitaries on and , respectively, and some of them are not present in Eq. (8).

The steps of the protocol are modified from Protocol 6 as follows: The first two steps are the same as the Steps 0 and 1 of Protocol 6, after which both sides have a copy of the computational-basis information of the higher-level controlling state. And since the form of the overall unitary is known, each party knows whether he or she is to act as the controlling party in the lower-level protocol, depending on the higher-level controlling state. So in the modified Step 2 of the protocol, each party does what is supposed to be done locally in the lower-level controlled-unitary protocol, with each unitary gate being controlled by the local higher-level controlling state labeled by , but the measurements are all in the standard basis and thus need not be controlled (if there are measurements not in the standard basis, we decompose it as a unitary followed by a measurement in the standard basis). There are two stages of classical communication (in opposite directions) in Step 2, and for each such communication stage, the party that is supposed to send classical messages does exactly the same operations as before, but the opposite party measures in the computational basis on an extra ancilla initially in the state, and sends the outcome to the other party. The choice of measuring a useful system or a dummy ancilla introduced above is determined by the higher level controlling state labeled by . However, for actual implementation, the actual measurement should be on a fixed system. This can be resolved by a controlled-swap gate controlled by , which conditionally swaps the system to be measured into a fixed system before doing the measurement. The final step is similar to Step 3 of Protocol 6.

The whole protocol requires the same amount of entanglement as in Protocol 6, but generally requires more classical communication, since the correct and dummy messages are sent in both directions simultaneously in the two stages of classical communication in Step 2, so we allow twice as much classical communication in the lower-level protocol. Thus the overall protocol uses ebits and c-bits. A dummy message is the measurement outcome of a system which was originally (before the controlled-swap gate mentioned in the previous paragraph) an ancilla in a fixed initial state. Note that the dummy classical message is only dummy for some of the higher-level controlling states labeled by , but is the correct message for some others. Such message, even if “correct”, does not carry any information about the input state for the overall unitary, by the design of the basic controlled-unitary protocol. The rationale behind the above technique is as follows: The choice of which lower-level unitary is being implemented should be indistinguishable from an outside observer, since the information about the higher-level controlling state should not be leaked to the outside observer, which is necessary for implementing a unitary operation. The reason is in Theorem 1 of ygc10 (), which says that implementing a unitary successfully is equivalent to that no information about the input state of the unitary is leaked to an “environment” system (the tensor product of the environment system and the output system of the unitary is the entire output system of the protocol).

Protocol 8

(Protocol for implementing a bipartite unitary given its group-type expansion.)

This protocol is illustrated in Fig. 8 in ygc10 () (except for changes in symbols in the description below), and it implements bipartite unitaries of the form

(15)

where the are unitaries acting on , and they form a projective unitary representation of a finite group , and are arbitrary operators acting on but they satisfy that is unitary. This protocol uses a maximally entangled resource state of Schmidt rank (the order of ). Thus the entanglement cost is ebits. The classical communication cost is c-bits. For any unitary , we may expand it in the form (15) by letting be the generalized Pauli group (ignoring overall phases) which is of order , since the generalized Pauli matrices form a basis for the space of matrices.

We abbreviate the steps of the protocol here. For our purposes, a good property of the protocol to be utilized for the proof of Theorem 10 is that when is the -direct sum of some unitaries, it is often the case that there is a relatively small group (by “small” we mean smaller than ) such that can be expanded in the form (15). This is because of the following reason: Each component in the -direct sum form of is also expandable using the form (15); thus, its size divided by is the dimension of a (projective) unitary representation of the group , where the representation is obtained by restricting to the relevant subspace of , for all . Denote the dimension of such a projective representation as , , where is the total number of components in the -direct sum form of . Assume that there is a group that has inequivalent irreducible projective unitary representations of sizes , , where , and the with (in the case ) are arbitrary positive integers (this is, of course, a big assumption and does not hold for most bipartite unitaries, but note that we may regard several blocks in an -direct sum form of as one block to increase the chance that such a group exists, which is a technique used in the proof of Theorem 10), then we may do the following steps: Arbitrarily choose a factor system (see the definition in ygc10 ()) from the set of factor systems of that admit inequivalent irreducible projective unitary representations of sizes , (the existence of such a factor system is guaranteed by the assumption above). Then choose a projective unitary representation of that contains all inequivalent irreducible projective unitary representations belonging to this factor system. This would be a linearly independent set of matrices according to (ygc10, , Theorem 4), and they are of a simultaneous block diagonal form. We then remove some diagonal blocks from all these matrices so that the remaining blocks are of sizes , . Then the resulting matrices would be generally linearly dependent, and from the construction, the resulting set forms a (possibly overcomplete) basis for the space of matrices with the same block structure. Thus, this set of unitary matrices can be used to expand the bipartite unitary in the form of (15).

In our application in the proof of Theorem 10 in this paper, we choose the type of group directly and figure out its suitable size. A different problem has been discussed in Cohen10 (), which is trying to find the smallest group when the matrix of is known. However, there is some similarity: Our reason for choosing the dihedral groups as the type of group in the proof of Theorem 10 is based on the -direct-sum form of that we proved. The algorithm for choosing the group in Cohen10 () also is based on finding the -direct-sum form of (which corresponds to the block diagonal structure of the operators on that are used to expand ).

The protocols with two levels of control can be generalized to protocols with multiple levels of control. Some other generalizations are possible (but not used in this paper): The lower-level operators in the target unitary of the form (7) need not be a controlled unitary, but could be unitaries with group-type expansion in Protocol 8, and thus the inner level of the protocol becomes Protocol 8.

For studying Theorem 10, we introduce the preliminary lemma below. We note that the simplest type of Schmidt-rank-three bipartite unitaries, which are controlled unitaries with three terms, are generally not included in Lemma 9, due to the restrictions on the coefficients and the matrices and below.

Lemma 9

Suppose there are three linearly independent unitary matrices , and , where is the identity matrix, and is diagonal, and is not diagonal, and are not simultaneously diagonalizable under a unitary similarity transform; and distinct triplets , where , are real and nonnegative, and are nonzero complex numbers, such that

(16)

is a bipartite unitary of Schmidt rank on a space .

Then up to local unitaries, there is a decomposition of with the following direct sum structure on : , , and , satisfying that each

(17)

is a unitary on the subspace with , and that is diagonal, and for each , , ; is a non-scalar unitary whose non-diagonal entries are equal and positive.

The proof of this lemma is given in Appendix A. Lemma 9 leads to the following result, where assertion (i) is a structure theorem for Schmidt-rank-3 bipartite unitaries. Note that the assumption of the result implies and .

Theorem 10

Assume that is a Schmidt-rank-3 bipartite unitary controlled from the A side. Then the following assertions hold.
(i) Either is the -direct sum of at most three unitaries of Schmidt rank at most , or is locally equivalent to a -direct sum of controlled unitaries of Schmidt rank at most . Each of the controlled unitaries is on a or space controlled in the computational basis of .
(ii) can be implemented by local operations and

(18)

ebits of entanglement and

(19)

c-bits.

The proof of this theorem is given in Appendix B. Given that the side is the control, the result in cy14ap () gives an entanglement cost upper bound of ebits. This old upper bound is always not less than the new upper bound in (18). When are both large and is about , the new upper bound in (18) is about ebits, which is about half of the old upper bound which is about ebits.

We show two classes of examples. The first shows that for some , the entanglement cost can be much less than the upper bound in Theorem 10(ii).

Example 11

Consider a Schmidt-rank-three unitary of the form (41). Let be of dimension , and , , , where () are some different real numbers. Then , and . Actually, by conjugation using a local diagonal unitary on , we can transform into while keeping and unchanged. The other with are given by , where and are real. The space of is spanned by a projective representation of an Abelian group of order (the Klein-four group), hence Protocol 8 implements using 2 ebits of entanglement and LOCC. This is much less than the upper bound in Theorem 10(ii) when and are large.

The second class of examples is still for unitary of the form in Lemma 9, but with essentially different blocks in different subspaces of . It suggests that there might not be an easy improvement to the upper bound in Theorem 10(ii) for general Schmidt-rank-three bipartite unitaries.

Example 12

We use the notations in the proof of Lemma 9, but assume that the unitary is without the diagonal part, i.e. the subspace is a null space. Assume the diagonal elements of the matrices and are and , respectively, where is a variable dependent on , and is a positive constant less than , e.g. , and the sign factor for the real part is either or . Suppose the diagonal elements with the positive appear first in each and , and denote such elements as and , respectively. Then , , and are , , and , respectively, which is useful for checking the result below. Since , the two off-diagonal elements of are chosen to be equal real numbers such that is unitary. Let the satisfy that , and , , for , where is an arbitrary positive integer, and is a real positive number independent of but dependent on . Note that is independent of . The diagonal part of Eq. (A) can be written as

(20)

for . Here we have used that is real, and and are pure imaginary, and that , are unitary, and we denote , . It is easily verified that there are an infinite number of solutions of and for (12) when is fixed, and by choosing some sufficient but finite number of them to be used in the matrix , the would have Schmidt rank three. The is unitary because each block in each controlled operator on the side is unitary, and the latter follows from Lemma 3 and our choice of the and , and that is real, and and are pure imaginary. The statement about the number of solutions above implies that the dimensions and are arbitrarily large, and we do not know of any simple protocol that implements this class of unitaries with a constant number of ebits and LOCC. This suggests there might not be an easy improvement to the upper bound in Theorem 10(ii).

V Entanglement cost and entangling power of bipartite permutation unitaries

This section is motivated by the following question. What is the entanglement cost for Schmidt-rank-three bipartite permutation unitaries? The result in Theorem 22 of cy15 () gives an upper bound of 24 ebits, with the help of a one-qubit ancilla on one side. Other motivations to study the permutation unitaries are in the first paragraph of Sec. V.3, and also in cy15 (). We shall first develop some preliminary results about bipartite (complex) permutation unitaries of general Schmidt rank, and then derive the improved upper bounds for the entanglement cost for bipartite permutation unitaries of small Schmidt rank in Sec. V.1. The case of general Schmidt rank is studied in Sec. V.2. We give an example in Sec. V.3, and study the entangling power of bipartite permutation unitaries of Schmidt rank up to three in Sec. V.4.

Lemma 13

Let be a complex bipartite permutation matrix of Schmidt rank . Then the following assertions hold.
(i) The nonzero blocks in any big row or big column of are linearly independent. The number of them is at least and at most .
(ii) Suppose a big row of contains linearly independent blocks. Then up to local complex permutation matrices the first blocks in the big row are orthogonal projectors, whose sum is the identity matrix.

A similar statement holds when all “row” are replaced with “column”.
(iii) Under the assumption in (ii), up to local complex permutation matrices is a complex -term controlled-permutation unitary from the side. The projectors in the terms are exactly the projectors in (ii). Such unitary can be implemented using ebits and LOCC.
(iv) If is a real permutation unitary, then (ii) and (iii) hold with all occurrences of the word “complex” removed.
(v) Suppose a big row of contains linearly independent blocks. Then up to local complex permutation matrices the first blocks in the big row are orthogonal projectors, whose sum is the identity matrix.

A similar statement holds when all “row” are replaced with “column”.
(vi) Under the assumption in (v), assume that the projectors and their orders are respectively and for . Up to local complex permutation matrices, we have

where , , and are all complex permutation matrices on their respective subspaces. is of size , and the pair of matrices and () are orthogonal in both the input and output spaces. Furthermore, is a complex permutation matrix of Schmidt rank at most two on the bipartite Hilbert space . The space of contains .

If , then can be implemented using ebits and LOCC. If , then can be implemented using ebits and LOCC. If , then can be implemented using ebits and LOCC.
(vii) In (vi), if is a real permutation unitary, and , then under local permutations, either can be written in the case of the form of (13), or is a controlled-permutation unitary controlled from the side with at most terms, thus can be implemented using ebits of entanglement.

The proof of this lemma is given in Appendix C. The partial transpose has been used to study the separability problem in entanglement theory peres1996 (); hhh96 (). Recently it has been used to study the ranks of marginals of multipartite quantum states chl14 (), in terms of the following conjectured inequality

(22)

where (resp. ) are matrices of the same size and denotes the transpose. In previous works we have presented a few bipartite unitaries satisfying the inequality cy14 (); cy14ap (). One can verify that the partial transpose of the complex permutation unitaries in (ii) and (13) are still unitary matrices. When considered as one of the bracket expressions in the lhs or rhs of (22), they both satisfy (22). They provide further evidence supporting the conjecture. We do not know whether all bipartite permutation matrices or complex permutation matrices satisfy (22).

Next we describe some simple properties about the blocks in bipartite permutation matrices. Let denote the maximum possible number of distinct diagonal blocks in a Schmidt-rank- bipartite controlled-permutation unitary. Let denote the maximum possible number of distinct permutation matrices in the -space of a Schmidt-rank- bipartite permutation unitary. Let denote the maximum possible number of distinct nonzero partial permutation matrices in the -space of a Schmidt-rank- bipartite permutation unitary. Using these definitions we state the following lemma.

Lemma 14

(i) is equal to the maximum number of distinct permutation matrices in the linear span of arbitrary permutation matrices of the same size.
(ii) .
(iii) The entanglement cost of any Schmidt-rank- controlled-permutation unitary is not more than ebits.
(iv) is not greater than the maximum number of distinct permutation matrices in the linear span of arbitrary partial permutation matrices of the same size.
(v) .
(vi) , and the maximum in the definition of is achieved only when the bipartite permutation unitary is equivalent to a controlled unitary from the side under local permutation unitaries.

The proof of this lemma is given in Appendix D. Evidently the and would be unaffected if we replace by in their definition.

v.1 Entanglement cost of bipartite permutation unitaries of Schmidt rank two or three

We have studied the properties of the complex bipartite permutation unitaries in terms of the Schmidt rank in Lemmas 13 and 14. In this subsection we study the bipartite permutation unitaries of Schmidt rank two or three. They are locally equivalent to controlled unitaries cy13 (); cy14 (); cy14ap (). So they can be implemented using the basic controlled-unitary protocol by directly using the controlled form, however this might require more than minimal amount of entanglement. The Lemma 15 (i) below, together with Lemma 26, imply that the entanglement cost by directly using the controlled form is minimal for the case of Schmidt rank two.

Lemma 15

(i) Any Schmidt-rank-two bipartite permutation unitary is equivalent to a two-term controlled-permutation unitary under local permutation unitaries.
(ii) Any Schmidt-rank-two bipartite complex permutation unitary is equivalent to a two-term controlled-complex-permutation unitary under local complex permutation unitaries.

Proof.

Let us prove (ii) first. Denote the complex unitary as . Its standard matrix form, also denoted by , is a matrix. If there is a big row or column of containing two nonzero blocks, then the assertion follows from Lemma 13(ii)(iii). It suffices to consider the case that there is exactly one nonzero block in any big row or column of . Up to local permutation matrices on we may assume that is a block-diagonal complex permutation matrix, and the first two diagonal blocks are linearly independent. Up to a local complex permutation matrix on , we may assume . If all diagonal blocks of are proportional to or , then the assertion follows. If there is a diagonal block which is not proportional to any one of , then has to be diagonal and if has only two distinct diagonal entries, then is equivalent to a controlled complex permutation unitary from the side with two terms, up to local permutation unitaries. Thus we only need to consider the remaining case, i.e., that is diagonal and has at least three distinct diagonal entries. However in this case cannot be unitary by Lemma 1. This completes the proof of (ii).

The proof for (i) is similar. If there is a big row or column of containing two nonzero blocks, the assertion follows from Lemma 13(iv). In the remaining case, the result follows from Lemma 14(ii).    

Now we investigate the structure and entanglement cost for complex permutation unitaries of Schmidt rank three. In particular, the real counterpart is completely characterized in (i).

Lemma 16

(i) Up to local permutation unitaries, any Schmidt-rank-three bipartite permutation unitary is either equivalent to a three-term or four-term controlled-permutation unitary, or equivalent to the direct sum of a product permutation unitary and a two-term controlled-permutation unitary. Therefore such unitary can be implemented using ebits and c-bits.
(ii) Any Schmidt-rank-three bipartite complex permutation unitary that is not equivalent to a diagonal unitary under local permutation unitaries can be implemented using ebits and LOCC.
(iii) Any diagonal Schmidt-rank-three bipartite complex permutation unitary, whose diagonal blocks contain the identity matrix and a diagonal matrix of exactly two distinct diagonal elements, can be implemented using ebits and LOCC.

The proof of this lemma is given in Appendix E. An example for “the direct sum of a product permutation unitary and a two-term controlled-permutation unitary” is given by the following unitary on system:

v.2 Entanglement cost of bipartite permutation unitaries of general Schmidt rank

The following Protocol 18 implements bipartite permutation unitary of arbitrary Schmidt rank . The computational basis for each system starts with . The entanglement and classical communication cost of the protocol in terms of is analyzed in Theorem 22. Before introducing the protocol, we define the so-called effective input and output dimensions for . An example unitary illustrating these definitions is in Example 25 in Sec. V.3.

Definition 17

(i). The effective input dimension of is the number of types of input states of . A type of input states of (or “an input type of ”) is a subspace of spanned by computational basis states, so that any two big columns of corresponding to two computational basis states in the subspace have the same collection of blocks in them, ignoring the positions and the relative order of the nonzero blocks in the big column.

(2). The effective output dimension of relative to an input computational basis state of is the number of nonzero blocks in the big column of corresponding to the input computational basis state of . And the labels for each effective dimension for a given input computational state of is determined by the order in which the nonzero block appears in the big column. The output computational basis state of corresponding to the big row with a nonzero block in the given big column is called an output type of relative to the input of , abbreviated as “a relative output type of ”.

(3). The effective output dimension of is the number of output types of , where an output type of is a subspace of spanned by computational basis states, so that each computational basis state in such subspace has the same combination of being in or not in the output space of the partial permutation operators in the space of . It turns out that for this definition of the output type of , it suffices to consider a linearly independent set of partial permutation operators in the space of , which form a basis for the space of , and we call such revised definition the simplified definition. Such a basis of partial permutation operators do exist, and they can be selected from the blocks in the matrix . Any other partial permutation operator in the space of is a linear combination of these basis operators. Suppose the simplified definition is inequivalent to the original definition. Then there are two computational basis states in the output space so that they are simultaneously in or not in the output space of any of the basis operators, while one and only one of them is in the output space of another partial permutation operator in the space of