TFC CONFLICT BENEATH THE LAX CLASS B AIRSPACE.

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FLT X TOOK OFF FROM RWY 30 AT LGB AT XA31Z. INITIAL CLB CLRNC WAS RWY HDG TO 1500 FT; THEN L TURN TO 180 DEG HDG; CLB TO 3000 FT. HDOF TO SOCAL DEP OCCURRED AT ABOUT 1700 FT AND THE CTLR CLRED FLT X TO LEVEL AT 4000 FT. AFTER LEVELING AT 4000 FT; THE CTLR CLRED FLT X TO CLB TO 7000 FT. IMMEDIATELY UPON INITIATING THE CLB; A TCASII CONTACT POPPED UP AT 12 O'CLOCK POS AND ABOUT 5 MI. THE CTLR THEN ADVISED FLT THAT WE HAD A VFR ACFT AT 12 O'CLOCK AND 4500 FT; AND TO INCREASE OUR RATE OF CLB. AT THIS POINT; FLT X WAS AT 4200 FT AND CLBING; AND THE TCASII CONTACT TURNED INTO A TFC ALERT AT 12 O'CLOCK; 3 MI; AND 300 FT ABOVE. CAPT DIRECTED THE FO TO START AN IMMEDIATE L TURN TO AVOID THE VFR ACFT; WHICH WE NEVER SAW BECAUSE IT WENT UNDER THE ACFT NOSE WHEN WE TURNED L. ESTIMATED LATERAL SEPARATION WAS LESS THAN 2 MI AND CO-ALT. FLT X ROLLED OUT ON 140 DEG HDG; AND CTLR ADVISED THAT WE WERE STILL ASSIGNED A 180 DEG HDG UNLESS RESPONDING TO AN RA. WE ADVISED THAT WE HAD; AND RETURNED TO A 180 DEG HDG WHEN CLR OF TFC. THIS SIT HIGHLIGHTS A SIGNIFICANT SAFETY PROB IN THAT CTLRS ARE NOT LEGALLY RESPONSIBLE FOR SEPARATING IFR AND VFR ACFT. THE CTLR IN THIS CASE HAD A NONCHALANT ATTITUDE TOWARD THE SAFETY RISK POSED BY THE VFR ACFT AND GAVE A CLB CLRNC THAT NEVER WOULD HAVE BEEN ISSUED IF THE OTHER ACFT WERE ON AN IFR CLRNC. FLT X REACTED TO A TA; NOT RA; BASED ON TCASII DISPLAY AND CTLR'S CORROBORATION OF VFR ACFT LOCATION.