Quantum machine learning aspires to overcome intractability that currently limits its applicability to practical applications. However, quantum machine learning itself is limited by low effective dimensions achievable in state-of-the-art experiments. Here, we demonstrate highly successful classifications of real-life images using photonic qubits, combining a quantum tensor-network representation of hand-written digits and entanglement-based optimization. Specifically, we focus on binary classification for hand-written zeroes and ones, whose features are cast into the tensor-network representation, further reduced by optimization based on entanglement entropy and encoded into two-qubit photonic states. We then demonstrate image classification with a high success rate exceeding 98%, through successive gate operations and projective measurements. Although we work with photons, our approach is amenable to other physical realizations such as nitrogen-vacancy centers, nuclear spins, and trapped ions, and our scheme can be scaled to efficient multi-qubit encodings of features in the tensor-product representation, thereby setting the stage for quantum-enhanced multi-class classification.