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To transcode a 4K MP4 file to 1080p without significant quality loss, you can use software that employs advanced algorithms to maintain the integrity of the video. Here are some methods and tools you can consider:

1. VideoProc Converter AI: This tool uses a motion-adaptive pixel scaling algorithm to convert 4K videos to 1080P without perceptible quality loss.
2. VLC Media Player: Not only for playback, VLC can also convert 4K videos to 1080p. You can adjust the settings to ensure minimal quality loss².
3. Adobe Premiere Pro: If you have access to professional editing software like Adobe Premiere, it offers robust features for video conversion with high-quality output.
4. HDconvert: An online service that can convert 4K videos to 1080p, offering customization options to balance quality and file size³.

Remember to choose the output format and resolution carefully, and consider adjusting the bitrate and codec settings to optimize the balance between quality and file size. It's also a good idea to test a small portion of the video first to ensure the results meet your expectations.