Abstract
We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.
Highlights
Unitary transformations of the states of two quantum bits play a prominent role in quantum information processing and computation [1]
After a discussion of the decomposition and parametrisation of SU (4) in Section 2, we focus on its geometric properties in Section 3, where we derive the invariant length element and Haar measure for the group, presenting the results in both the original parametrisation and in the context of the representation of two-qubit gates offered by the local invariants due to Makhlin [3]
In order to study the geometric properties of SU (4) in a way that is suitable to a quantum information context—where the emphasis is on the entangling capabilities of two-qubit operations—we have utilised a parametrisation of SU (4) that reflects the natural decomposition of twoqubit gates into local SU (2) ⊗ SU (2) and purely nonlocal SU (4)/SU (2) ⊗
Summary
Unitary transformations of the states of two quantum bits (qubits) play a prominent role in quantum information processing and computation [1]. To a three-dimensional space in which all locally-equivalent gates live, and we discuss the form of the length element and measure for two particular choices of coordinates for this space We use these derived geometric quantities to proceed towards our main objective: the calculation of the invariant volumes of the regions containing particular gates of interest in quantum information processing. After a discussion of the decomposition and parametrisation of SU (4) in Section 2, we focus on its geometric properties, where we derive the invariant length element and Haar measure for the group, presenting the results in both the original parametrisation and in the context of the representation of two-qubit gates offered by the local invariants due to Makhlin [3] The novelty of our approach is that these quantities will be in forms that are suited for the description of two-qubit gates, namely, in the coordinate system defined in the previous section, which separates the purely local gates in SU (2) ⊗ SU (2) from the entangling gates in A
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