Tutorial - Unitary matrices and orthonormal basis

sharat Batra

CS4268 - Quantum Computing Tutorial - 1

General Information

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Block equations to be wrapped in double dollar signs ($$).

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General Information (Useful for All Questions)

Stochastic Matrix: A square matrix is stochastic if and only if:
All its entries are non-negative reals.
The entries of each of its columns sum to 1.
Unitary Matrix: A square matrix $ U $ is unitary if and only if it satisfies: $$ U U^\dagger = U^\dagger U = I $$ where $ U^\dagger $ is the conjugate transpose of $ U $, and $ I $ is the identity matrix.
Inner Product:
For two column vectors $ v $ and $ u $, their inner product is defined as: $$ \langle v, u \rangle = v^\dagger u, \quad \langle u, v \rangle = u^\dagger v $$
$ p $-Norm of a Vector:
The $ p $-norm of a vector $ v = (v_1, v_2, \dots, v_n) $ in a complex $ n $-dimensional vector space is given by: $$ | v |_p = \left( |v_1|^p + |v_2|^p + \dots + |v_n|^p \right)^{1/p} $$
Orthonormal Set:
A set of vectors $ \{ v_1, v_2, \dots, v_n \} $ is orthonormal if:
Each vector has an $ \ell_2 $-norm of 1.
The inner products of distinct vectors are 0.
Singular Value Decomposition (SVD):
For every $ n \times n $ complex matrix $ A $ of rank $ r \leq n $, there exist unique positive real values $ s_1, s_2, \dots, s_r $ and orthonormal sets $ \{ v_1, v_2, \dots, v_r \} $, $ \{ w_1, w_2, \dots, w_r \} $ such that: $$ A = \sum_{i=1}^{r} s_i w_i v_i^\dagger $$ where $ s_i $ are singular values, and $ \{ v_i \}, \{ w_i \} $ are right and left singular vectors, respectively.

Question 1: Unitary Matrices and Orthonormality

Question:

A matrix $ U $ is unitary if it satisfies: $$ U^\dagger U = U U^\dagger = I $$ Given the orthonormal states: $$ v_1 = \frac{3}{5} \alpha \otimes \alpha + \frac{4}{5} \beta \otimes \beta $$ $$ v_2 = \frac{4}{5} \alpha \otimes \alpha - \frac{3}{5} \beta \otimes \beta $$ $$ v_3 = \frac{2}{\sqrt{5}} \alpha \otimes \beta + \frac{i}{\sqrt{5}} \beta \otimes \alpha $$ $$ v_4 = \frac{1}{\sqrt{5}} \alpha \otimes \beta - \frac{2i}{\sqrt{5}} \beta \otimes \alpha $$ Verify that these states form an orthonormal basis, i.e., show that: $$ \langle v_i, v_j \rangle = \delta_{ij} $$ where $ \delta_{ij} $ is the Kronecker delta.

Answer:

We compute the inner product for each pair: $$ \langle v_i, v_j \rangle = \sum_{k} (v_i^k)^* v_j^k $$ After computing for all pairs, we find: $$ \langle v_i, v_j \rangle = 1 \text{ for } i = j, \quad 0 \text{ for } i \neq j $$ Thus, the given states are mutually orthonormal, forming a valid basis.

Takeaway:

The set of vectors given is orthonormal and forms a valid quantum basis.
Unitary matrices preserve the inner product and are essential in quantum mechanics.

Question 2: Orthonormal Basis and Identity Matrix

Question:

Given an orthonormal set $ \{v_1, v_2, ..., v_n\} $, show that the sum of their outer products forms the identity matrix: $$ A = \sum_{i=1}^{n} v_i v_i^\dagger = I_{n\times n} $$ Verify this property by applying $ A $ to an arbitrary vector $ w $.

Answer:

Let $ w $ be any arbitrary vector expressed in terms of the basis: $$ w = \sum_{i=1}^{n} \alpha_i v_i $$ Applying $ A $: $$ A w = \sum_{i=1}^{n} v_i v_i^\dagger \sum_{j=1}^{n} \alpha_j v_j $$ Since $ A w = w $, $ A $ acts as the identity matrix.

Takeaway:

Any set of orthonormal basis vectors satisfies $ \sum v_i v_i^\dagger = I $.
This property is crucial in quantum state representation.

Question 3: Unitary Matrix Properties

Question:

Show that a unitary matrix $ U $ satisfies the singular value decomposition (SVD) property: $$ U = \sum_{i=1}^{r} s_i w_i v_i^\dagger $$ where $ s_i $ are singular values and $ \{v_i\} $, $ \{w_i\} $ are orthonormal sets. Prove that for unitary matrices, all singular values satisfy $ s_i = 1 $.

Answer:

Using SVD: $$ U^\dagger U = \sum_{i=1}^{r} s_i^2 w_i w_i^\dagger = I $$ which is only true if $ s_i = 1 $ for all $ i $. Thus, unitary matrices preserve norms.

Takeaway:

Unitary matrices have singular values equal to 1.
They preserve the length of quantum states, making them essential for quantum operations.

Question 4: Stochastic Matrices and Norm Preservation

Question:

A column stochastic matrix $ S $ satisfies: $$ \sum_{i=1}^{n} s_{ij} = 1, \quad \forall j $$ Show that for any vector $ v $, its $ \ell_1 $-norm remains unchanged under $ S $: $$ |Sv|_1 = |v|_1 $$

Answer:

\[ Applying $ S $: \[ w = S v, \quad w_i = \sum_{j=1}^{n} s_{ij} v_j \] Then: \[ \|w\|_1 = \sum_{i=1}^{n} |w_i| = \sum_{i=1}^{n} \sum_{j=1}^{n} s_{ij} |v_j| \] Since $ S $ is stochastic: \[ \sum_{i=1}^{n} s_{ij} = 1 \] Thus, $ \|w\|_1 = \|v\|_1 $, proving norm preservation. \]

Takeaway:

Stochastic matrices preserve the $ \ell_1 $-norm.
They model probabilistic transformations in quantum mechanics.