Tensor Products

Abstract

The tensor product is a useful tool in mathematics and quantum mechanics. Not only does it help to reduce nonlinear maps into linear ones but it also helps in ways such as but not limited to explaining Einstein’s field equations. This brief paper will explore first the basics of tensor products and then some applications of it.

1. The Basics

1.1 The Bra-Ket Notation:

Bra–ket notation is a notation for linear algebra specifically designed to reduce calculations that often appear in quantum mechanics. Kets |r> are column vectors that represent quantum states in any number of dimensions, with values (a,b,c) being complex numbers (left picture). Bras <a| are the conjugate transposes of kets, with the rows and columns being swapped (right picture).

1.2 Inner/Outer/Tensor Products:

Inner Product <ΨΦ>: Also called an overlap between quantum states, the inner product is a product of two quantum states bra Psi <Ψ| and ket Phi |Φ>. It’s analogous to the scalar product in vector spaces.

Outer Product |ΨΦ|: Also called a projection, the outer product is the product of two quantum states ket Psi|Ψ> and bra Phi <Φ. The result is a matrix.

Tensor Product |ΨΦ>: Producing a column vector with cardinality 2ⁿ (n is the number of qubits), the tensor product is a product of two quantum states, ket Psi|Ψ> and ket Phi |Φ>. Normalization applies, with each superposition’s probability being its coefficient times its conjugate or magnitude squared.

1.3 Properties:

(i) A ⊗( B + C ) = A ⊗ B + A ⊗ C

(ii) ( A + B ) ⊗ C = A ⊗ C + B ⊗ C

(iii) ( k A ) ⊗ B = A ⊗ ( k B ) = k ( A ⊗ B )

(iv) ( A ⊗ B ) ⊗ C = A ⊗ ( B ⊗ C )

Where k is a scalar and A, B, and C are matrices.

1.4 Common Misconception: Tensor Product vs Kronecker Product

Seemingly the same thing, the two ideas represent operations on different objects: tensor products work on linear maps between vector spaces while Kronecker products work on matrices. The difference can be seen like this: the Kronecker product of two matrices represents the tensor product of two linear maps.

2. Applications

2.1 Quantum Teleportation:

Let’s assume Alice wants to send the qubit state | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ . Hence, she needs to pass information regarding α and β to Bob. However, the no-cloning theorem in quantum mechanics states that you cannot make a copy of an unknown quantum state (can copy classical bits though). As a result, Alice cannot directly make a copy of her qubit and send it to Bob. Through using two classical bits and an entangled qubit pair, however, Alice could transfer her qubit state to Bob (the qubit on her end A1 will disappear so it will not violate the no-cloning theorem). The following is a simple presentation including tensor products.

Alice has an arbitrary qubit A1 in a superposition state.

She also shares a maximally entangled state with Bob (A2 & B), which can be represented in any of the four bell states, or specific quantum states of two qubits that represent the simplest quantum entanglement, shown below. Let’s assume Alice and Bob decide on the first one. Tensor products are used to represent the bell states.

Hence, the total initial state (state of the three participles) becomes the following (Tensor product is used in the equations):

Using a trick, we rewrite each outcome of the qubits as superpositions of the bell states and then plug the values into the total initial states.

Rewriting and simplifying the equation, we get the following:

Alice then measures her two qubits (A1 & A2). This would result in two classical bits, identifying one of the four bell states. Each of the four outcomes has a probability of

(½ )^2=¼. Depending on Alice’s measurement (left picture), Bob’s qubit’s state will be determined (right picture).

If Alice measures the first state (the uppermost one), the quantum teleportation is done. Bob can use a unitary to obtain Alice’s message, however. Through Bob telling Alice about which qubit state he observed, Alice could inform Bob which quantum mechanics gate to perform. As a result, Bob will receive a copy of Alice’s initial message. In this example, the tensor product is used in showing the total states of the three qubits.