This paper presents the best known bounds on the number of randomly sampled entries required to reconstruct an unknown low-rank matrix. The results improve upon prior work by Candès and Recht, Candès and Tao, and Keshavan, Montanari, and Oh. The reconstruction is achieved by minimizing the nuclear norm of the hidden matrix, which is the sum of its singular values. If the matrix satisfies a certain incoherence condition, the number of entries needed is proportional to a quadratic logarithmic factor of the matrix's parameters. The proof is concise and uses elementary analysis. The techniques are based on recent work in quantum information theory. The paper introduces the concept of coherence, which measures how aligned the matrix's row and column spaces are with the standard basis. A low coherence implies that the matrix is not mostly zero on the observed entries. The main result shows that under certain conditions, the nuclear norm minimization can uniquely recover the original matrix with high probability. The paper also discusses the implications of the results for matrix completion, showing that a small number of randomly sampled entries can be sufficient to reconstruct a low-rank matrix. The analysis uses a sampling with replacement model, which simplifies the proofs and allows for tighter bounds. The results are compared to previous work, showing improvements in the number of entries required and the assumptions made about the matrix. The paper concludes with a discussion of the implications of the results, noting that while the bounds are nearly optimal, further improvements may be possible. The simplicity of the argument arises from the use of sampling with replacement rather than Bernoulli sampling, and the paper suggests that similar techniques could be applied to other problems in matrix completion and compressed sensing.This paper presents the best known bounds on the number of randomly sampled entries required to reconstruct an unknown low-rank matrix. The results improve upon prior work by Candès and Recht, Candès and Tao, and Keshavan, Montanari, and Oh. The reconstruction is achieved by minimizing the nuclear norm of the hidden matrix, which is the sum of its singular values. If the matrix satisfies a certain incoherence condition, the number of entries needed is proportional to a quadratic logarithmic factor of the matrix's parameters. The proof is concise and uses elementary analysis. The techniques are based on recent work in quantum information theory. The paper introduces the concept of coherence, which measures how aligned the matrix's row and column spaces are with the standard basis. A low coherence implies that the matrix is not mostly zero on the observed entries. The main result shows that under certain conditions, the nuclear norm minimization can uniquely recover the original matrix with high probability. The paper also discusses the implications of the results for matrix completion, showing that a small number of randomly sampled entries can be sufficient to reconstruct a low-rank matrix. The analysis uses a sampling with replacement model, which simplifies the proofs and allows for tighter bounds. The results are compared to previous work, showing improvements in the number of entries required and the assumptions made about the matrix. The paper concludes with a discussion of the implications of the results, noting that while the bounds are nearly optimal, further improvements may be possible. The simplicity of the argument arises from the use of sampling with replacement rather than Bernoulli sampling, and the paper suggests that similar techniques could be applied to other problems in matrix completion and compressed sensing.