In "normal" bias conditions, this open-base transistor will not act like a transistor and not even as a diode.

Let's start with developing a very simplistic view: Two diodes as you mentioned, connected back to back, with the inner electrode only connected to the two n-type areas. assume these diodes are close to ideal, they conduct current with nearly zero voltage drop in forward direction, and block current in reverse direction. No matter how you apply bias to the two outer p-type terminals, there is always one diode which blocks the current and one that would be able to conduct, so in effect the whole system blocks current in both directions. It could be substituted with an open connection and nobody would notice the difference.

Now, of course in the real world diodes are not ideal. They will conduct a small current in reverse direction, and they will exhibit a non-zero voltage drop in forward direction. So, assuming real-world diodes, there will be a current if a non-zero bias is applied. The current will be determined nearly entirely by the reverse leakage current of the diode, an externally applied voltage will drop almost entirely over the reverse-biased diode. The current will depend exponentially on the applied voltage. But still, in most use cases this current will be pretty small. How small depends on the exact devices used.

If however as you mention a bipolar transistor is used (it doesn't matter here if it is PNP or NPN), under certain circumstances things will change: The leakage current on the reverse-biased pn junction will act as base current to the transistor. At a certain voltage this current will be sufficient to turn the transistor on. For real devices, this is usually specified in the datasheet: It is the open-base Collector-Emittter breakdown voltage BVCEO. IF this breakdown voltage is different when terminals are interchanged, then one could argue that this could be used for rectification. Sustaining this breakdown condition however can damage the device and therefore is usually not allowed. Additionally, the exact breakdown voltages tend to vary from device to device and should not be relied on for circuit operation.

Thanks for your kind explanation, but I still have one more question that for  the device mentioned above(a bipolar PNP diode without base electrode) do you think it is equivalent if  the bias voltage is negative or positive ?

Well, regarding differences in polarity, it depends.

Assuming silicon devices, as that's what I'm familiar with, IF the device is fully symmetric in terms of both geometry and doping, then AND ONLY THEN I'd expect exactly the same behaviour regardless of the polarity. This however would be a pretty rare case as for a bipolar transistor usually the doping density in emitter and collector differs by orders of magnitude. In commercial bipolar transistors, the doping is such that the base-emitter diode breaks down in reverse direction already at low voltages, sometimes even less than 5V, while the base-collector diode often withstands voltages on the order of a few hundred volts. If the bias applied is between base-emitter and base-collector breakdown voltages, one could possibly use this device for rectification. But as said before, I would not use this behaviour if I could avoid it. I wouldn't trust any device relying on that.

Assuming that both PN junction diodes are identical, the current flow is mainly generated from thermionic emmission and the measurement is conducted in dark. Then, the only current would pass in both negative and positive voltage sweeping is the dark saturation current