Devices outputting a video signal through a DisplayPort connector may optionally support DisplayPort Dual-Mode (or DisplayPort++). This allows the DisplayPort cable to be used to carry an HDMI (TMDS) signal. From what I gather, support for this is widespread, even in cases where device manufacturers opt not to use the "DP++" logo.
FWIW, by "active", I mean a cable containing circuitry to convert an LVDS signal (DisplayPort) to a TMDS signal (HDMI, DVI). By contrast, a passive cable would simple rearrange the DisplayPort signal wires to fit an HDMI connector, requiring the source device to detect the HDMI device and output an HDMI signal accordingly.
There are inexpensive, passive DisplayPort-to-HDMI cables that rely on support for DisplayPort Dual-Mode in the video-outputting device. By contrast, the display just sees a regular HDMI signal on an HDMI port.
USB Type-C includes DisplayPort Alternate Mode. This allows the USB cable to be used to carry a DisplayPort signal. In a USB-IF presentation, it is stated that DisplayPort Alternate Mode does not support DisplayPort Dual-Mode:
"USB Type-C will NOT support DisplayPort Dual Mode (DP++)"
Page 23, VESA DisplayPort Alternate Mode on USB Type-C Technical Overview, Jim Choate, USB-IF
USB Type-C includes HDMI Alternate Mode. This allows the USB cable to be used to carry an HDMI signal. From what I gather, this is unsupported by almost (if not all) existing devices on the market.
Edit: everybodywiki.com lists the 2018 HP Envy 13 x360 as supporting HDMI Alternate Mode.
Edit: cablechick.com.au says:
HDMI Alternate Mode is a specification without an implementation (as far as we can tell). Our research turned up no consumer hardware that supports HDMI Alt-Mode by name. Instead, HDMI compatibility over USB-C is generally offered via DisplayPort or Thunderbolt 3 Alt-Modes using adapters.
Combined, this seems to suggest that both scenarios in which a passive USB Type-C-to-HDMI cable could be used – HDMI Alternate Mode and DisplayPort Alternate Mode carrying DisplayPort Dual-Mode HDMI – are, in fact, impossible.
Does this mean that every USB Type-C-to-HDMI cable on the market is active?