How many dimensions has a space of 3 qubits?
In the realm of quantum information, the concept of qubits plays a pivotal role in quantum computing and quantum information processing. Qubits are the fundamental units of quantum information, analogous to classical bits in classical computing. A qubit can exist in a superposition of states, allowing for the representation of complex information and enabling quantum operations that surpass classical capabilities.
The question on how many dimensions has the system of 3 qubits refers to the quantum state space associated with a system composed of three qubits (the Hadamard space). To understand better, we need to consider the mathematical framework that describes the quantum states of multiple qubits. In quantum mechanics, the state of a single qubit can be represented as a linear combination of basis states, typically denoted as |0⟩ and |1⟩. These basis states form a two-dimensional complex vector space known as the Bloch sphere. This is a two dimensional, linear Hadamard space. However Hadamard space (states space of quantum systems) is defined over the complex body, i.e. linear combinations have complex coefficients. Each complex coefficient can be decomposed into a real and imaginary part, i.e. two real coefficients, with one multiplied by the imaginary number i. This doubles the number of dimensions of a Hadamard space (for example, for qubits we have 2 complex dimensions, but 4 real dimensions). Additionally one needs to account for the normalization condition of the Hadamard space. This condition asserts that the squares of coefficients modulus sum to 1. This is a single equation on real values that elimates one real degree of freedom, leaving the qubit space state with 3 real dimensions, justifying the Bloch sphere representation (which corresponds to a spherical reference frame) in a real 3-dimensional space.
When we consider a system of multiple qubits, the state space grows exponentially with the number of qubits. For a system of n qubits, the state space is 2^n-dimensional (but remains a complex space, in. terms of real dimensions the number must be doubled). In the case of three qubits, the state space is 2^3 = 8-dimensional (in complex dimensions, or 16-dimensional in real dimensions). However, it is again important to remind that the state space of a quantum system is subject to constraints due to the normalization condition, which requires the sum of the squared magnitudes of probability amplitudes to equal one (which reduces one real dimension, meaning that the real space state of three qubits system has actually 15 real dimensions).
In the context of a three-qubit system, the state space is spanned by a basis set consisting of 8 basis states (or in other words a state of three-qubit system is a linear combination of these 8 basis states with 8 complex coefficients). Each basis state corresponds to a unique combination of binary values for the three qubits. For instance, the basis states for a three-qubit system can be denoted as |000⟩, |001⟩, |010⟩, |011⟩, |100⟩, |101⟩, |110⟩, and |111⟩. These basis states form a complete orthonormal basis for the 8-dimensional state space of the three-qubit system.
The dimensionality of the quantum state space is important in quantum information processing as it determines the complexity and richness of quantum operations that can be performed on the system. Higher-dimensional state spaces enable the representation of more intricate quantum states and facilitate the implementation of advanced quantum algorithms and protocols.
The system of 3 qubits corresponds to a 8-dimensional state space (complex Hadamard space), where each dimension is associated with a distinct quantum state defined by the binary values of the individual qubits. Understanding the dimensionality of quantum state spaces is essential for harnessing the full potential of quantum computing and quantum information processing.
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