AN LTT ACR ACFT HAD AN NMAC WITH AN SMT IN CTLED AIRSPACE.

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WE WERE INBOUND TO SBA ARPT AND WERE HANDED OFF FROM POINT MUGU TO SBA APCH. NEW CTLR CLRED US FOR THE VISUAL APCH TO THE ARPT; THEN IMMEDIATELY ISSUED A TA AND ADVISED A R TURN. IT TOOK A FEW SECONDS TO VISUALLY PICK UP THE OTHER ACFT (A LIGHT TWIN) WHICH WAS CONVERGING FROM OUR L AT APPROX OUR SAME ALT. DESPITE THE TURN; WE WERE STILL CONVERGING; AND I INCREASED THE BANK AND BEGAN A RAPID CLB OF 400-500 FT WHEN THE OTHER ACFT DISAPPEARED FROM VIEW BEHIND THE L WING. THE ACFT PASSED BELOW AND TO OUR L; ABOUT 300 FT AWAY. THE OTHER ACFT WAS NOT IN CONTACT WITH APCH CTL AT THE TIME OF THE CONFLICT; AND WAS XING A COMMON ROUTING FROM LAX TO SBA. THE ACFT DID HAVE AN ENCODING ALTIMETER AS THE CTLR HAD GIVEN US HIS ALT JUST PRIOR TO THE CONFLICT. THE TRANSFER FROM 1 APCH CTL TO ANOTHER HAD SOMEHOW DELAYED THE ISSUANCE OF AN EARLIER TFC WARNING UNTIL THE MOMENT OF IMMEDIATE CONFLICT. ON OUR PART; THE FO HAD BEEN RUNNING THROUGH THE DSCNT AND APCH CHKLISTS AT THE MOMENT OF HDOF; SO OUR CONCENTRATION HAD BEEN MORE INSIDE THE COCKPIT THAN OUTSIDE. THE OTHER ACFT STILL WOULD HAVE BEEN VERY DIFFICULT TO SEE EVEN IF WE HAD BEEN SCANNING IN HIS DIRECTION; SINCE WE WERE ABOVE A SOLID OVCST IN SOMEWHAT HAZY CONDITIONS AND THE OTHER ACFT; A WHITE COLORED LIGHT TWIN; WAS SLIGHTLY BELOW US AND BLENDED INTO THE OVCST LAYER.