
TL;DR
This paper introduces unique basic matrices for normal matrices, enabling new representations and algorithms for quantum gates, idempotents, and pseudo-inverses, with broad applications in matrix theory and quantum computing.
Contribution
It develops a framework of unique basic matrices for normal matrices and derives algorithms and formulas for quantum gates, idempotents, and pseudo-inverses.
Findings
Unique basic matrices for normal matrices are established.
An efficient algorithm for expressing idempotents as sums of rank 1 idempotents is presented.
Formulas for symmetric unitary matrices and pseudo-inverses are derived.
Abstract
Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) is a symmetric unitary matrix if and only if it has the form for a symmetric idempotent , (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.
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Unique builders for classes of matrices 111MSC 2010 Classification: 15A23, 81P45, 94A15
Ted Hurley222National Universiy of Ireland Galway, email: [email protected]
Abstract
Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) is a symmetric unitary matrix if and only if it has the form for a symmetric idempotent , (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.
1 Introduction
Basic matrices are defined and it is shown that any normal matrix is the product of basic commuting matrices and that the product is unique apart from the order. A matrix is normal when where denote the complex conjugate transposed of . The class of normal matrices include the classes of unitary matrices (), and Hermitian, also called self-adjoint, matrices (). These occur in many applications: for example quantum logic gates are represented by unitary matrices and their properties and applications depend ultimately on the structure of unitary matrices.
Each basic matrix itself is a product of minimal basic matrices with the same eigenvalue. A basic matrix is expressed in terms of a symmetric idempotent matrix; an idempotent occurring in the expression as a product of basic matrices has the property that acts on by and the eigenvalue occurs with multiplicity equal to the rank of .
An efficient algorithm is given for expressing a rank idempotent matrix as a sum of rank (pure) orthogonal idempotents with increasing initial zeros and such an expression is unique. From this a unique expression for a mixed matrix as a sum of rank idempotents of this type is derived.
A quantum logic gate is represented by a unitary matrix and these gates are the basis for quantum information theory, see for example [3]. Indeed a quantum logic gate is often stated to be a unitary matrix itself. A quantum logic gate is thus a unique, apart from order, product of basic logic gates and these basic logic gates are building blocks for quantum logic gates in general.
Examples are readily available and applications are given throughout: It is shown that a unitary matrix is symmetric if and only if it has the form for a symmetric idempotent . An easy formula for the roots and powers of the matrices follows directly from its expression as a product of basic matrices; powers and roots are given explicitly in terms of basic matrices. A formula for the pseudo inverse is immediate.
Expressions for well-known quantum gates (such as Pauli, Hadamard gates) as products of unique basic matrices are explicitly derived in section 4.
Comparisons may be made with the famous 1D factorization theorem of Belevich and Vaidyanathan which derives building blocks for 1D paraunitary matrices, [5] pp. 302-322. The basic matrices derived here for building normal matrices are influenced by methods in [2] for constructing generators/builders for multidimensional paraunitary matrices333A paraunitary matrix is a matrix such that where and . , and in particular for constructing paraunitary non-separable (entangled) matrices.
A summary of the main results is given in Section 1.1. Further notation from Section 1.2 may be consulted as required.
1.1 Summary
Let be a normal matrix. Then is a product the are distinct and is an orthogonal symmetric set of idempotents. Each is termed a basic matrix and the product is unique apart from the order of these commuting basic matrices.
acts on the idempotents by and so is an eigenvalue of occurring to multiplicity equal to . When is not a complete set then completes the set and occurs as an eigenvalue of with multiplicity equal to where is the size of the matrices.
A basic matrix is the product of minimal basic matrices where the are mutually orthogonal, have rank and each minimal basic has the same eigenvalue as the original.
The idempotent matrix is of the form for mutually orthogonal unit column vectors . Given an idempotent matrix of rank an efficient algorithm is given for expressing as a sum of such rank idempotents with increasing initial zeros so that the expression obtained is unique. Mixed density matrices have a particularly useful form when viewed as products of basic matrices. A mixed density matrix is defined as a convex sum of pure density matrices though not in a unique way. By writing the mixed matrix as a product of basic matrices, uniqueness is obtained and this gives a unique perspective on mixed matrices. Further each basic matrix is a unique product of ordered rank basics matrices giving a unique expression for a mixed density matrix. See Section 3.1.
Unitary matrices are normal matrices with eigenvalues of the form . Quantum logic gates are represented by unitary matrices; indeed a quantum logic gate is often defined to be a unitary matrix. The expression of a unitary matrix as a unique product, except for order, of basic matrices then expresses a quantum logic gate as a unique product of basic quantum logic gates. Each basic quantum logic gate is a product of minimal quantum logic gates with the same eigenvalue.
A number of easy consequences are derived. A formula for the pseudo inverse of the matrix follows directly from the expression of the matrix as a product of basic matrices. It is shown that is a symmetric unitary matrix if and only if for a idempotent ; thus building symmetric unitary matrices from idempotent matrices is straight forward. Many of the mostly used quantum logic gates are symmetric. In Section 3.2 a useful direct formula for writing down all the roots of a normal matrix is derived giving in particular a useful direct formula for roots of a unitary matrix/logic gate. Section 4 finds expressions for common logic gates as products of basic logic gates. Section 5 discusses techniques for building normal matrices including unitary matrices from basic matrices.
There are easily constructed interesting examples; those displayed are of small size but the constructions can be efficiently applied to large size matrices.
1.2 Additional notation
Necessary background on algebra may be found in many algebra and linear algebra books. Background on quantum information theory may be found in [3].
denotes a ring with identity ; the suffix R may be omitted when a particular ring is understood. A mapping in which is said to be an involution on if and only if (i) , (ii) , and (iii) . Let be a ring with involution ∗. An element is said to be symmetric, with respect to ∗, if . An idempotent in is an element such that . are said to be orthogonal in if . The set is said to be a complete set of orthogonal idempotents in if each element is an idempotent, the are mutually orthogonal and . The set is further said to be symmetric if each is symmetric (with respect to ∗).
Here we work in the field of complex numbers although many of the results work over other systems but are not included. For , denotes the complex conjugate of and then denotes the complex conjugate transposed of for or . Now denotes the identity matrix; the suffix n will be omitted when the size is clear. An matrix is said to be a normal matrix in if and only if . An matrix is unitary if and only if . An matrix is Hermitian (or self-adjoint) if and only if . Thus unitary and Hermitian matrices are normal matrices. Unitary matrices by their definition are invertible but an Hermitian matrix may not be invertible.
Column vectors are orthogonal if ; matrices are orthogonal if ; in all cases here orthogonality will refer to symmetric matrices. For a column vector , is a symmetric matrix and the matrix is necessarily of rank . When is a unit vector () then is an idempotent matrix as then . Suppose are orthogonal vectors. Then the matrices and are orthogonal matrices as since .
2 Basic Matrices
A matrix is normal if and only if there exists a unitary matrix such that where is a diagonal matrix. Normal matrices include Hermitian matrices () and unitary matrices (); Hermitian matrices include real symmetric matrices. Eigenvalues of a unitary matrix have the form and an Hermitian matrix has real eigenvalues.
Proposition 2.1
Let be an matrix. There exists a unitary matrix such that , where is diagonal, if and only if for an orthogonal set of unit vectors .
This can be restated as follows:
Proposition 2.2
Let be an matrix. Then is a normal matrix if and only if for an orthogonal set of unit vectors .
The consist of the columns of and the are the diagonal entries of in Proposition 2.1 and are eigenvalues of . Now is an orthonormal basis for . Define . Then is an orthogonal symmetric set of idempotents. The set is also a complete set of idempotents for if not then and is orthogonal to each of the ; this would give a linearly independent set of vectors in the -dimensional space .
Now denotes the trace of the matrix which is the sum of its diagonal elements. One nice property of an idempotent matrix is that its rank is the same as its trace, see for example [1].
Lemma 2.1
Let be orthogonal symmetric idempotent matrices. Then
* is a symmetric idempotent matrix.* 2. 2.
If , then is a symmetric idempotent matrix orthogonal to both and . 3. 3.
. 4. 4.
.
Proof: The proofs of the first two items are straight forward. Proof of 3: It is known that for an idempotent matrix , see for example [1]. Now is an idempotent and so . The proof of 4 is similar.
Lemma 2.1 may be generalised as follows:
Lemma 2.2
Let be a set of orthogonal symmetric idempotent matrices. Then
- •
* is an idempotent symmetric matrix.*
- •
If then is a symmetric idempotent orthogonal to each for .
- •
.
- •
.
Lemma 2.3
Let be mutually orthogonal symmetric idempotent matrices. Then .
The following follows directly from Lemma 2.3.
Proposition 2.3
Let be an matrix with where is a complete orthogonal set of symmetric idempotents. Then .
A basic matrix is one of the form where is an idempotent. This has eigenvalue occurring to multiplicity equal to and has eigenvalue occurring to multiplicity equal to , where is the size of . When , it has inverse .
Lemma 2.4
Let be orthogonal symmetric idempotent matrices. Then
Lemma 2.4 enables basic matrices with orthogonal idempotents and the same eigenvalue to be collected together.
The following is a consequence of Lemma 2.4 and Proposition 2.3.
Proposition 2.4
Let be an normal matrix. Then , the are distinct and is a complete orthogonal set of idempotents. Moreover the eigenvalues of are and occurs with multiplicity equal to .
It is shown below, Proposition 2.12, that the expression for in Proposition 2.4 is unique apart from the order of the commuting basic matrices. Note that it is required for this uniqueness that the are distinct; the equal have been gathered into one basic matrix by Lemma 2.4.
Each idempotent is a sum of rank idempotents but not in a unique manner. Algorithm 3.1 and Theorem 3.1 below give a method of expressing an idempotent as the sum of rank (pure) idempotents which have increasing initial zeros thus giving a unique expression for as a sum of such pure idempotents.
One or more of the in Proposition 2.4 may be [math]. Some of the may be also and then giving that where the are distinct and . Here then the eigenvalue is in ‘disguise’ and if so, it occurs with multiplicity equal to .
Now acts on its idempotents by , and when .
In particular Proposition 2.4 can be applied to unitary matrices. The eigenvalues of a unitary matrix are of the form .
Proposition 2.5
Let be an unitary matrix. Then , the are distinct and is an orthogonal set of idempotents. Moreover the eigenvalues of are and occurs with multiplicity . Now eigenvalue occurs with multiplicity which may be [math]. Moreover a matrix of the form is a unitary matrix for any orthogonal set of symmetric idempotents .
A basic unitary matrix is one of the form .
In Proposition 2.4, may be singular in which case the eigenvalue [math] appears to a certain multiplicity. In this case , where the are distinct and and is an orthogonal set of idempotents. (The eigenvalue is hidden when is not complete.) It is easy to check that is the pseudo inverse of .
Proposition 2.6
Let , where the are distinct and and is an orthogonal set of idempotents. Then is the pseudo inverse of .
Proposition 2.7
Let be a normal matrix. Then if and only for a symmetric idempotent .
Proof: Suppose . Then the eigenvalues of are . Thus by Proposition 2.4, for symmetric orthogonal idempotents . If then clearly and can be diagonalised by a unitary matrix.
The following is a corollary:
Proposition 2.8
* is a symmetric () unitary matrix if and only if for an idempotent .*
Notice in Proposition 2.7 and in Proposition 2.8, the eigenvalue occurs with multiplicity equal to and the eigenvalue occurs with multiplicity equal to .
Corollary 2.1
Let and for an matrix . Then for a symmetric idempotent . In particular this applies to a symmetric Hadamard matrix.
2.1 Examples
. This has eigenvalues and corresponding eigenvectors (1,1)\text{{}^{\text{T}}},(1,-1)\text{{}^{\text{T}}}. Then where . Using this formula, roots of may be obtained directly – see Section 3.2. 2. 2.
A commonly referenced matrix is the following real orthogonal/unitary matrix . This has eigenvalues . Then , which may be checked independently. The are symmetric idempotents. It is seen that which gives as a product of basic unitary matrices, which are defined in Section 2.2 below.
2.2 Uniqueness
- Definition
A basic matrix is one of the form for a symmetric idempotent matrix .
The idempotent of is said to be the idempotent involved in and is the eigenvalue involved in .
- Definition
Say a basic matrix is a simple basic unitary matrix if the idempotent has .
Proposition 2.9
A symmetric idempotent has if and only if for a unit column vector .
Proof: If has the form for a unit column vector , then clearly is a symmetric idempotent of . On the other hand if is a symmetric idempotent of then is of the form for a unit column vector . This is shown in for example [2] Proposition 4.5.
A minimal basic is of the form for where is a unit column vector and has rank and .
Note
Let and be basic matrices with the same idempotent. Then . A product of a minimal basic with another minimal basic with the same idempotent is another minimal basic. The minimality is expressed in terms of the rank of the idempotent involved.
Proposition 2.10
A symmetric idempotent has if and only if for unit column mutually orthogonal vectors .
Proof: When the result follows from Proposition 2.9. If the first column of is zero then the first row of is zero and the result follows by induction. Suppose then the first column of is non-zero and define which is a unit column vector. Then is an idempotent of rank . Let . Then is an idempotent symmetric matrix orthogonal to which has first column, and hence first row, consisting of zeros. Let be the matrix with first column and first row of omitted. Since are orthogonal it follows that and so has . Thus is rank symmetric idempotent matrix. The result then follows by induction.
Thus a basic unitary matrix is of the form where for unit column mutually orthogonal vectors . When the basic unitary matrix is what is termed a simple basic unitary matrix and its idempotent is of the form .
Proposition 2.11
Suppose the space is generated by the unit orthogonal columns vectors and by the unit orthogonal column vectors . Then
.
Proof: Now
[TABLE]
for some .
In other words
[TABLE]
Denote the matrix on the right (involving ) by .
Then
[TABLE]
Then by (1),
[TABLE]
Thus and hence . Therefore by (2)
and hence .
Suppose now is a product of basic matrices. Implicit in this is that if then is a non-zero idempotent with eigenvalue of multiplicity .
By Lemma 2.4 a product basic matrices using orthogonal idempotents with the same eigenvalues can be collected together into a basic matrix.
Suppose where the are mutually orthogonal idempotents and the are all different. We now show that except for the order such an expression is unique. Each in the product has the form where is the rank of and is also the multiplicity of the corresponding eigenvalue .
Proposition 2.12
Suppose where the are orthogonal idempotents and the are distinct and where the are orthogonal idempotents and the are distinct. Then and by reordering .
Proof: Now has eigenvalues occurring with multiplicity and eigenmatrix and looking at it another way has eigenvalues occurring with multiplicity . Thus must equal for some . We may assume this by reordering. Then . Now for orthogonal unit , () and for orthogonal unit vectors . Now by Proposition 2.11, . Similarly by reordering if necessary in general for and of necessity . 444Induction is more complicated because we cannot assume is invertible.
3 Pure and mixed idempotents
In order to obtain a normal matrix as a unique product of basic matrices it is necessary to collect the basics with the same eigenvalue, see Propositions 2.4 and 2.12. An idempotent is not a unique sum of idempotents of rank . An idempotent of rank is often described as a pure idempotent by analogy with the mixed matrix in quantum theory, see Section 3.1 below.
It is now shown that an idempotent matrix of rank may be written as the sum of special idempotents of rank in a unique way and an efficient algorithm is given.
Let be an idempotent matrix of rank . Then is an idempotent matrix of rank . Also has eigenvalues . Now and so the multiplicity of the eigenvalue is since has rank . Also and so the multiplicity of the eigenvalue [math] is as has rank . Hence the multiplicity of the eigenvalue of is exactly and the multiplicity of the eigenvalue [math] of is exactly .
The following gives as a sum of rank idempotents from the column space of .
Proposition 3.1
Let be an orthonormal basis for the column space of . Define for . Then .
Proof: The space of eigenvectors of corresponding to the eigenvalue is the solution space of which is the solution space of and this has dimension . Now and so the columns of form a basis for this solution space. Let be an orthonormal basis for this solution space which is an orthonormal basis for the columns of .
The space of eigenvectors of corresponding to the eigenvalue is the solution space of which is the solution space of and this has dimension . Now and so the columns of form a basis for this solution space. Let be an orthonormal basis for this solution space which is an orthonormal basis for the column space of .
Now vectors corresponding to distinct eigenvalues of the symmetric matrix are othogonal and so is an orthonormal basis for . Let . Then is a unitary matrix and
where the occurs times and the [math] occurs times on the diagonal.
Then as required.
This expression for is not unique as any orthonormal basis for the column space of may be used.
An efficient algorithm for finding as a sum of idempotents of rank is then formulated as follows:
Algorithm 3.1
Let be the first non-zero column of which occurs at column so that columns consist of zeros. 2. 2.
Define so is a unit column vector. 3. 3.
Define . Then is rank symmetric idempotent. 4. 4.
Let . If then and has rank and we are finished. If then is a nonzero idempotent with columns all zero vectors. Now let be replaced by and go back to item 1.
To apply the algorithm it is not necessary to know the rank of the idempotent beforehand and this comes out at the end. The algorithm is efficient and at each step just involves forming an idempotent from the first non-zero column of a matrix and subtracting this from the matrix.
Example 1
. Then is a rank idempotent. The first column of is non-zero so start there and let
Then . Now is an idempotent of rank and .
Example 2
Let .
Apply the algorithm and get in turn idempotents of rank and where
, , .
is formed using the first (non-zero) column of and notice that the number of initial zeros increase from to to .
For a mixed matrix with , as in Section 3.1. Now as shown where the are distinct, each is a basic idempotent and this expression for is unique apart from the order of the . Each is a sum of of rank idempotents in a non-unique manner and such a sum may be obtained from Proposition 3.1 in theory and by Algorithm 3.1 in an efficient algorithm.
Let u=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\text{{}^{\text{T}}} be a column vector. Say if (if initial entry of is non-zero) and say if but (if is the first non-zero entry of ). Let then define . Thus has first columns and first rows consisting entirely of zeros. The st measures the number of initial zeros of and the number of initial rows and columns of .
Theorem 3.1
Let be an idempotent. Then is uniquely a sum where each is a rank idempotent and .
Proof: Algorithm 3.1 shows that has such an expression for exists. The process involves constructing first an idempotent such that is an idempotent but also that . Now work with and proceed until the zero matrix is obtained.
Suppose now where and where the and are rank idempotents. Then as this is the rank of . Now look at the first non-zero column of which is the first non-zero column of . From this get and then by induction that for .
These results are applied below in Section 3.1 to derive a uniqiue expression for a mixed matrix as the sum of rank matrices of the form given in Theorem 3.1.
3.1 Mixed matrix
Suppose with where are mutually orthogonal unit (column) vectors in . Complete to an orthogonal unit basis for . Let and for . Now where is a complete orthogonal set of idempotents. Basic matrices with the same may be collected together. Note that as is a complete orthogonal set of idempotents.
It is easy to check that is the pseudo inverse of which is useful for certain calculations.
A particular case of this is where is a mixed density matrix where are orthogonal unit vectors and . It is known that this expression for is not unique.
But where the are rank idempotents. Then where . Now by Lemma 2.4 the basic products with the same can be gathered together to get where is a complete orthogonal set of idempotents with , the are unique and . This expression for is unique by Proposition 2.12. This means that where the are orthogonal idempotents and the is now a unique expression for . Each is a sum of orthogonal rank idempotents and is of rank .
Note that and occurs with multiplicity equal to rank .
Using Theorem 3.1 the following unique expression is obtained.
Theorem 3.2
A density matrix is uniquely of the form with distinct and such for each .
3.2 Roots
A nice formula for roots of normal matrices may now be obtained directly from their expression as a product of basic matrices. Roots of unitary matrices/quantum gates will then follow.
Powers of normal matrices are easy to obtain:
Proposition 3.2
Let the normal matrix be expressed as a product of basic matrices by . Then .
Proof: The proof follows directly by induction and noting that the commute.
Let be the expression of an root of as a product of basic matrices. The are mutually orthogonal idempotents and the .
Then . Hence and so is an root of unity. Hence where for . Thus and so
[TABLE]
This gives normal matrices which the roots of . The can be any set of orthogonal idempotents and the can be any integers between and .
Proposition 3.3
The roots of which are normal consist of matrices of the following type:
[TABLE]
where is an orthogonal set of idempotents in and are integers satisfying
Suppose is a set of orthogonal symmetric idempotent matrices in . If is not complete then there exists such that is a complete orthogonal set of idempotents. The set is not unique but we shall refer to an as a complementary set of and say that completes . In particular is the minimal set which completes .
Suppose is the root of a basic matrix and is the expression for as a product of basic matrices where the are all different. Let be the expression for as a product of basic matrices. Some of the may be roots of and we separate these out so that where is an root of and commutes with each ; reordering the may be necessary. Now is a product of basic matrices involving only idempotents in a complementary set of .
Then . Now compare this with . By uniqueness and possible reordering get that and . Thus if and then and . Now for and so .
However a normal matrix has many roots and we now proceed to directly find all these.
Suppose . Now may not be and in this case define . Then is a complete orthogonal set of idempotents. Now and we need to attach this to : . Suppose roots of are required. Apply the usual way of obtaining an root of unity in and write: for positive integers from [math] to . Then and this is for any between [math] and . Here refers to real of the the modulus of . This gives all the roots of and they are expressed in a unique form as a product of basic matrices. If the expression for has basic matrices, including , then get different roots of .
3.2.1 Examples:
Consider the examples 2.1.
Example 1 uses the matrix with and . The square roots of are then plus a product of these using any square roots of obtained from a complementary set of . The only complementary set of is . A basic matrix involving , besides the identity, which is a square root of is . Thus the square roots of are where can be [math] or . 2. 2.
Example 2 uses the orthogonal matrix . Then with . Here is a complete orthogonal set of idempotents so there is no complementary set to consider.
The third roots of are: where can have values from .
4 Logic gates
Expressions as products of basic matrices are found for common quantum logic gates.
- •
Hadamard Gate The unitary matrix of the the Hadamard Gate is . Then and has eigenvalues . Thus by Proposition 2.8 can be expressed as . Solving for get . Thus expresses as a product of (minimal) basic unitary matrices where is as above. is itself a minimal basic matrix. Then with is a complete orthogonal set of idempotents. Now and so ‘reverses’ and leaves alone.
Roots of are immediate: For example a square root of is . All the square roots of may be obtained by methods of Section 3.2 and these are where . The action of on the idempotents is given by ; is left alone by and is changed to .
- •
Pauli gates: The Pauli gates are symmetric unitary matrices and hence satisfy and have eigenvalues . Thus by Proposition 2.8 they have the form for an idempotent . It is now a matter of establishing in these cases.
- –
Pauli X-gate is and has eigenvalues . Thus for a symmetric idempotent . Solving for gives giving that . Hence . In this form a square root of is easy to find and one such is . By section 3.2 all square roots of are where and .
- –
Pauli Y-gate is . Then where .
Roots of Y-gate are easy to obtain: for example is a fourth root and all fourth roots may be directly found by structures of Section 3.2.
- –
Pauli Z-gate is . Then where .
The Pauli gates are themselves (minimal) basic matrices.
- •
Phase shift gate: The Phase shift unitary matrix is and has eigenvalues . Applying the general result gives where . This expresses as a minimal basic unitary matrix. Roots of are easily obtained by the method of section 3.2. For example a third root of is but there are others, see Section 3.2.
- •
NOT and Square root of NOT: The NOT gate is . This has eigenvalues . Then .where .
Then (one value) of . For example it is possible to define .
- •
Swap, Square root of Swop: The Swap gate has matrix . This has eigenvalues , (three times) and . Then with it is seen that .
Then a square root is , where is the idempotent obtained for and this is as usually given.
could be considered as ‘a square root of a square root of Swap’.
- •
CNOT matrix: CNOT matrix is . This has eigenvalues , three times, and eigenvalue . Let . Then . has rank and , corresponding to eigenvalue , has rank . All the square roots of CNOT are .
- •
Bell non-symmetric: The Bell non-symmetric matrix is . This has eigenvalues . Then it is easy to show that where . Thus . Note that is complete.
5 Build matrices
Normal matrices are built from basic matrices of the form ; unitary matrices or quantum logic gates are built from basic matrices of the form . A particular property may be required; for example a quantum logic gate which is an root and/or a quantum logic gate with particular eigenvalues and multiplicities may be required. Requirements may be met in a constructive manner.
It is shown in Proposition 2.8 that a symmetric (unitary matrix)/(quantum logic gate) is of the form for a symmetric idempotent . This makes building symmetric unitary matrices/logic gates particularly easy.
For example the matrix is the unitary matrix related to the Hadamard matrix . is symmetric and so where is easily shown to be . It is clear that and thus also. A square root of is but there are three other square roots which are easily worked out using the methods developed in Section 3.2. is not separable and roots of are not separable.
A unitary matrix which is also a root of may be built by the methods of Section 3.2; in this case and . Suppose for example quantum logic gates (unitary matrices) in are required which are roots of . Clearly where is any idempotent and , is a basic unitary quantum gate which is a root of . Now v_{1}=\frac{1}{2}(1,1,1,-1)\text{{}^{\text{T}}},v_{2}=\frac{1}{2}(1,1,-1,1)\text{{}^{\text{T}}},v_{3}=\frac{1}{2}(1,-1,1,1)\text{{}^{\text{T}}},v_{1}=\frac{1}{2}(-1,1,1,1)\text{{}^{\text{T}}} is an orthonormal basis for and these give an orthogonal complete set of idempotents . From these, gates which are roots of may be formed: where . (If all then the identity matrix is obtained.)
Normal including unitary and Hermitian matrices may naturally be built from a set of unit orthogonal vectors. Let be a set of orthogonal unit column vectors in and set . Build which is then normal. The basic matrices with the same scalars may be amalgamated by Lemma 2.4 to get a unique expression where the are distinct. Each is a sum of a subset of . Note that need not be a full basis for ; in this case is not a complete set of orthogonal idempotents. By letting a complete set of orthogonal idempotents is obtained. To build a unitary matrix take and to build a Hermitian matrix take the to be real. If no then no and is invertible with inverse . If some , say , then . In this case the pseudo inverse of is .
Sets of orthogonal unit column vectors may be obtained from roots of unity from which unitary matrices and logic gates for various requirements may be built; details are omitted.
Comparisons may be made with the famous 1D unique building blocks for paraunitary matrices due to Belvitch and Vaidyanathan, see [5] pages 302-322.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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