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While not incorrect, per se, the use of "entities" often leads to confusion as it usually implies, to the modern ear, mutually independent "beings", as in the Godhead comprises three separate and mutually independent beings. This is untrue. The three personal subsistences are each comprised of the one being and one essence of the one God.
The word subsistence literally means “to stand under.” Thus, this word gets at the idea that while God is one in essence, there are three subsistences—also called three persons—that stand under the essence. They are part of the essence. All three have the essence of deity.
Formally speaking, subsistence is the means of individuation of essence with respect to existence.
When defining person any such definition of "person" as relates to the Trinity must preserve the unity of God's being, include an element common to both the human and divine personality, and allows the impersonal human nature of Jesus.
We may say,
Person is an independent entity, indivisible, rational, incommunicable, not sustained by another nature but possessing in itself the principle of its operation.
Or, we may say,
The word, person, with reference to the Trinity, means the divine essence in a specific mode of existence and distinguished by this specific mode of existence from that essence and the other persons.
When we start to confuse the early church's use of the word "person" with our own notions of "person", we ultimately end up in some form of modalistic heresy. Given the baggage the word "person" carries in today's culture, I usually refer to the personal subsistences of the Godhead until everyone is on the same page related to how the word "person" is used in Trinitarian theological discourse. If we must insist on "person", then I tend to say that "A person is a distinct bearer of an essence, hence the Father, the Son, and the Spirit are distinct persons, each with His own personal attributes, while each also shares equally the attributes of deity (i.e., the divine essence)".
Thus, the divine essence does not exist independently along with the three Persons. The divine essence has no existence outside of and apart from the three Persons. For if the divine essence did, there would be no true unity, but a division that would lead into tri-theism.
The three persons (hypostaseis / prosopoi) co-inhering in the one divine nature (ousia) exist simultaneously with one another as distinct subsistences or persons. This means that the divine essence is not at one time entirely manifest as the Father (but not in or as the Son or Spirit), and then at another moment manifest exclusively as the Son, and yet again at another time solely as the Spirit. Rather, all three persons exist simultaneously, contrary to the preceding modalistic errors.
God’s essence is common to the three Persons of the Godhead, and God’s essence is not communicated from one Person to another Person. Each Person wholly partakes of the essence of God, possessing it as one undivided essence. Portions of the essence of God are not divided up to be enjoyed by each Person—as in one-third for the Father, one-third for the son, or one-third for the Spirit—instead the whole essence is enjoyed by each, as “in him the whole fullness of deity dwells” (Col. 2:9, also John 15:16).
For each subsistence in the divine essence, each of the Persons has a distinct subsistence, such that we can say that the Son is not the Father, the Father is not the Son, and the Holy Spirit is not the Father nor the Son.
Nevertheless, we can make a distinction between the three persons of the Trinity, because each member of the Godhead has unique attributes. We say the Father is God, the Son is God, and the Holy Spirit is God, but we don’t say that the Father is the Son, the Son is the Holy Spirit, or the Holy Spirit is the Father. There are distinctions between them, but the distinctions are not of the one essence of God. The distinctions are real, but they do not disturb the essence of deity.
AMR