This paper presents the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low-rank matrix. These results improve upon previous work by Candès and Recht, Candès and Tao, and Keshavan, Montanari, and Oh. The reconstruction is achieved by minimizing the nuclear norm, which is the sum of the singular values of the matrix, subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, the number of entries required is proportional to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this result is short and self-contained, using elementary analysis and techniques from quantum information theory. The main theorem makes minimal assumptions about the low-rank matrix and has a smaller log factor compared to previous results. The paper also discusses the sampling model used and provides proofs for related theorems, demonstrating the simplicity and effectiveness of the proposed approach.This paper presents the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low-rank matrix. These results improve upon previous work by Candès and Recht, Candès and Tao, and Keshavan, Montanari, and Oh. The reconstruction is achieved by minimizing the nuclear norm, which is the sum of the singular values of the matrix, subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, the number of entries required is proportional to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this result is short and self-contained, using elementary analysis and techniques from quantum information theory. The main theorem makes minimal assumptions about the low-rank matrix and has a smaller log factor compared to previous results. The paper also discusses the sampling model used and provides proofs for related theorems, demonstrating the simplicity and effectiveness of the proposed approach.